id	sid	tid	token	lemma	pos
ejde-203	1	1	special	special	ADJ
ejde-203	1	2	issue	issue	NOUN
ejde-203	1	3	in	in	ADP
ejde-203	1	4	honor	honor	NOUN
ejde-203	1	5	of	of	ADP
ejde-203	1	6	alan	alan	PROPN
ejde-203	1	7	c.	c.	PROPN
ejde-203	1	8	lazer	lazer	PROPN
ejde-203	1	9	electronic	electronic	ADJ
ejde-203	1	10	journal	journal	NOUN
ejde-203	1	11	of	of	ADP
ejde-203	1	12	differential	differential	ADJ
ejde-203	1	13	equations	equation	NOUN
ejde-203	1	14	,	,	PUNCT
ejde-203	1	15	special	special	ADJ
ejde-203	1	16	issue	issue	NOUN
ejde-203	1	17	01	01	NUM
ejde-203	1	18	(	(	PUNCT
ejde-203	1	19	2021	2021	NUM
ejde-203	1	20	)	)	PUNCT
ejde-203	1	21	,	,	PUNCT
ejde-203	1	22	pp	pp	ADP
ejde-203	1	23	.	.	PUNCT
ejde-203	2	1	13–21	13–21	NUM
ejde-203	2	2	.	.	PUNCT
ejde-203	3	1	issn	issn	PROPN
ejde-203	3	2	:	:	PUNCT
ejde-203	3	3	1072	1072	NUM
ejde-203	3	4	-	-	SYM
ejde-203	3	5	6691	6691	NUM
ejde-203	3	6	.	.	PUNCT
ejde-203	4	1	url	url	PROPN
ejde-203	4	2	:	:	PUNCT
ejde-203	4	3	https://ejde.math.txstate.edu	https://ejde.math.txstate.edu	PROPN
ejde-203	4	4	or	or	CCONJ
ejde-203	4	5	https://ejde.math.unt.edu	https://ejde.math.unt.edu	PROPN
ejde-203	4	6	oscillation	oscillation	NOUN
ejde-203	4	7	time	time	NOUN
ejde-203	4	8	and	and	CCONJ
ejde-203	4	9	damping	damp	VERB
ejde-203	4	10	coefficients	coefficient	NOUN
ejde-203	4	11	in	in	ADP
ejde-203	4	12	a	a	DET
ejde-203	4	13	nonlinear	nonlinear	ADJ
ejde-203	4	14	pendulum	pendulum	NOUN
ejde-203	4	15	jaime	jaime	NOUN
ejde-203	4	16	arango	arango	PROPN
ejde-203	4	17	in	in	ADP
ejde-203	4	18	memory	memory	NOUN
ejde-203	4	19	of	of	ADP
ejde-203	4	20	prof	prof	PROPN
ejde-203	4	21	.	.	PUNCT
ejde-203	5	1	alan	alan	PROPN
ejde-203	5	2	lazer	lazer	PROPN
ejde-203	5	3	(	(	PUNCT
ejde-203	5	4	1938–2020	1938–2020	NUM
ejde-203	5	5	)	)	PUNCT
ejde-203	5	6	abstract	abstract	NOUN
ejde-203	5	7	.	.	PUNCT
ejde-203	6	1	we	we	PRON
ejde-203	6	2	establish	establish	VERB
ejde-203	6	3	a	a	DET
ejde-203	6	4	relationship	relationship	NOUN
ejde-203	6	5	between	between	ADP
ejde-203	6	6	the	the	DET
ejde-203	6	7	normalized	normalized	ADJ
ejde-203	6	8	damping	damp	VERB
ejde-203	6	9	coefficients	coefficient	NOUN
ejde-203	6	10	and	and	CCONJ
ejde-203	6	11	the	the	DET
ejde-203	6	12	time	time	NOUN
ejde-203	6	13	that	that	PRON
ejde-203	6	14	takes	take	VERB
ejde-203	6	15	a	a	DET
ejde-203	6	16	nonlinear	nonlinear	ADJ
ejde-203	6	17	pendulum	pendulum	NOUN
ejde-203	6	18	to	to	PART
ejde-203	6	19	complete	complete	VERB
ejde-203	6	20	one	one	NUM
ejde-203	6	21	oscillation	oscillation	NOUN
ejde-203	6	22	starting	start	VERB
ejde-203	6	23	from	from	ADP
ejde-203	6	24	an	an	DET
ejde-203	6	25	initial	initial	ADJ
ejde-203	6	26	position	position	NOUN
ejde-203	6	27	with	with	ADP
ejde-203	6	28	no	no	DET
ejde-203	6	29	velocity	velocity	NOUN
ejde-203	6	30	.	.	PUNCT
ejde-203	7	1	we	we	PRON
ejde-203	7	2	provide	provide	VERB
ejde-203	7	3	sufficient	sufficient	ADJ
ejde-203	7	4	conditions	condition	NOUN
ejde-203	7	5	on	on	ADP
ejde-203	7	6	the	the	DET
ejde-203	7	7	nonlinear	nonlinear	ADJ
ejde-203	7	8	restitution	restitution	NOUN
ejde-203	7	9	force	force	NOUN
ejde-203	7	10	so	so	SCONJ
ejde-203	7	11	that	that	SCONJ
ejde-203	7	12	this	this	DET
ejde-203	7	13	oscillation	oscillation	NOUN
ejde-203	7	14	time	time	NOUN
ejde-203	7	15	does	do	AUX
ejde-203	7	16	not	not	PART
ejde-203	7	17	depend	depend	VERB
ejde-203	7	18	monotonically	monotonically	ADV
ejde-203	7	19	on	on	ADP
ejde-203	7	20	the	the	DET
ejde-203	7	21	viscosity	viscosity	NOUN
ejde-203	7	22	damping	damp	VERB
ejde-203	7	23	coefficient	coefficient	NOUN
ejde-203	7	24	.	.	PUNCT
ejde-203	8	1	1	1	X
ejde-203	8	2	.	.	X
ejde-203	8	3	introduction	introduction	NOUN
ejde-203	8	4	the	the	DET
ejde-203	8	5	pendulum	pendulum	NOUN
ejde-203	8	6	is	be	AUX
ejde-203	8	7	perhaps	perhaps	ADV
ejde-203	8	8	the	the	DET
ejde-203	8	9	oldest	old	ADJ
ejde-203	8	10	and	and	CCONJ
ejde-203	8	11	fruitful	fruitful	ADJ
ejde-203	8	12	paradigm	paradigm	NOUN
ejde-203	8	13	for	for	ADP
ejde-203	8	14	the	the	DET
ejde-203	8	15	study	study	NOUN
ejde-203	8	16	of	of	ADP
ejde-203	8	17	an	an	DET
ejde-203	8	18	oscillating	oscillate	VERB
ejde-203	8	19	system	system	NOUN
ejde-203	8	20	.	.	PUNCT
ejde-203	9	1	the	the	DET
ejde-203	9	2	apparent	apparent	ADJ
ejde-203	9	3	regularity	regularity	NOUN
ejde-203	9	4	of	of	ADP
ejde-203	9	5	an	an	DET
ejde-203	9	6	oscillating	oscillate	VERB
ejde-203	9	7	mass	mass	NOUN
ejde-203	9	8	going	go	VERB
ejde-203	9	9	back	back	ADV
ejde-203	9	10	and	and	CCONJ
ejde-203	9	11	forth	forth	ADP
ejde-203	9	12	the	the	DET
ejde-203	9	13	equilibrium	equilibrium	NOUN
ejde-203	9	14	position	position	NOUN
ejde-203	9	15	has	have	VERB
ejde-203	9	16	fascinated	fascinate	VERB
ejde-203	9	17	scientists	scientist	NOUN
ejde-203	9	18	since	since	SCONJ
ejde-203	9	19	well	well	ADV
ejde-203	9	20	before	before	ADP
ejde-203	9	21	galileo	galileo	PROPN
ejde-203	9	22	.	.	PUNCT
ejde-203	10	1	there	there	PRON
ejde-203	10	2	are	be	VERB
ejde-203	10	3	plenty	plenty	NOUN
ejde-203	10	4	of	of	ADP
ejde-203	10	5	mathematical	mathematical	ADJ
ejde-203	10	6	models	model	NOUN
ejde-203	10	7	accounting	account	VERB
ejde-203	10	8	for	for	ADP
ejde-203	10	9	almost	almost	ADV
ejde-203	10	10	any	any	PRON
ejde-203	10	11	observed	observed	ADJ
ejde-203	10	12	behavior	behavior	NOUN
ejde-203	10	13	of	of	ADP
ejde-203	10	14	the	the	DET
ejde-203	10	15	pendulum	pendulum	NOUN
ejde-203	10	16	’s	’s	PART
ejde-203	10	17	oscillation	oscillation	NOUN
ejde-203	10	18	.	.	PUNCT
ejde-203	11	1	from	from	ADP
ejde-203	11	2	the	the	DET
ejde-203	11	3	sheer	sheer	ADJ
ejde-203	11	4	amount	amount	NOUN
ejde-203	11	5	of	of	ADP
ejde-203	11	6	the	the	DET
ejde-203	11	7	literature	literature	NOUN
ejde-203	11	8	on	on	ADP
ejde-203	11	9	the	the	DET
ejde-203	11	10	subject	subject	NOUN
ejde-203	11	11	,	,	PUNCT
ejde-203	11	12	one	one	PRON
ejde-203	11	13	would	would	AUX
ejde-203	11	14	expect	expect	VERB
ejde-203	11	15	that	that	SCONJ
ejde-203	11	16	there	there	PRON
ejde-203	11	17	is	be	VERB
ejde-203	11	18	no	no	DET
ejde-203	11	19	reasonable	reasonable	ADJ
ejde-203	11	20	question	question	NOUN
ejde-203	11	21	already	already	ADV
ejde-203	11	22	answered	answer	VERB
ejde-203	11	23	.	.	PUNCT
ejde-203	12	1	and	and	CCONJ
ejde-203	12	2	that	that	PRON
ejde-203	12	3	might	might	AUX
ejde-203	12	4	be	be	AUX
ejde-203	12	5	true	true	ADJ
ejde-203	12	6	.	.	PUNCT
ejde-203	13	1	yet	yet	ADV
ejde-203	13	2	,	,	PUNCT
ejde-203	13	3	for	for	ADP
ejde-203	13	4	whatever	whatever	DET
ejde-203	13	5	reason	reason	NOUN
ejde-203	13	6	,	,	PUNCT
ejde-203	13	7	it	it	PRON
ejde-203	13	8	is	be	AUX
ejde-203	13	9	not	not	PART
ejde-203	13	10	impossible	impossible	ADJ
ejde-203	13	11	to	to	PART
ejde-203	13	12	take	take	VERB
ejde-203	13	13	on	on	ADP
ejde-203	13	14	a	a	DET
ejde-203	13	15	question	question	NOUN
ejde-203	13	16	whose	whose	DET
ejde-203	13	17	answer	answer	NOUN
ejde-203	13	18	does	do	AUX
ejde-203	13	19	not	not	PART
ejde-203	13	20	seem	seem	VERB
ejde-203	13	21	to	to	PART
ejde-203	13	22	follow	follow	VERB
ejde-203	13	23	immediately	immediately	ADV
ejde-203	13	24	from	from	ADP
ejde-203	13	25	the	the	DET
ejde-203	13	26	classical	classical	ADJ
ejde-203	13	27	sources	source	NOUN
ejde-203	13	28	.	.	PUNCT
ejde-203	14	1	in	in	ADP
ejde-203	14	2	a	a	DET
ejde-203	14	3	typical	typical	ADJ
ejde-203	14	4	experimental	experimental	ADJ
ejde-203	14	5	setup	setup	NOUN
ejde-203	14	6	with	with	ADP
ejde-203	14	7	no	no	DET
ejde-203	14	8	noticeable	noticeable	ADJ
ejde-203	14	9	damping	damping	NOUN
ejde-203	14	10	,	,	PUNCT
ejde-203	14	11	the	the	DET
ejde-203	14	12	oscillations	oscillation	NOUN
ejde-203	14	13	of	of	ADP
ejde-203	14	14	a	a	DET
ejde-203	14	15	pendulum	pendulum	NOUN
ejde-203	14	16	are	be	AUX
ejde-203	14	17	periodic	periodic	ADJ
ejde-203	14	18	.	.	PUNCT
ejde-203	15	1	now	now	ADV
ejde-203	15	2	,	,	PUNCT
ejde-203	15	3	when	when	SCONJ
ejde-203	15	4	the	the	DET
ejde-203	15	5	damping	damping	NOUN
ejde-203	15	6	can	can	AUX
ejde-203	15	7	not	not	PART
ejde-203	15	8	be	be	AUX
ejde-203	15	9	neglected	neglect	VERB
ejde-203	15	10	,	,	PUNCT
ejde-203	15	11	we	we	PRON
ejde-203	15	12	still	still	ADV
ejde-203	15	13	observe	observe	VERB
ejde-203	15	14	oscillations	oscillation	NOUN
ejde-203	15	15	,	,	PUNCT
ejde-203	15	16	even	even	ADV
ejde-203	15	17	though	though	SCONJ
ejde-203	15	18	they	they	PRON
ejde-203	15	19	are	be	AUX
ejde-203	15	20	not	not	PART
ejde-203	15	21	periodic	periodic	ADJ
ejde-203	15	22	.	.	PUNCT
ejde-203	16	1	however	however	ADV
ejde-203	16	2	,	,	PUNCT
ejde-203	16	3	we	we	PRON
ejde-203	16	4	can	can	AUX
ejde-203	16	5	measure	measure	VERB
ejde-203	16	6	the	the	DET
ejde-203	16	7	time	time	NOUN
ejde-203	16	8	spent	spend	VERB
ejde-203	16	9	by	by	ADP
ejde-203	16	10	a	a	DET
ejde-203	16	11	complete	complete	ADJ
ejde-203	16	12	oscillation	oscillation	NOUN
ejde-203	16	13	,	,	PUNCT
ejde-203	16	14	and	and	CCONJ
ejde-203	16	15	this	this	DET
ejde-203	16	16	time	time	NOUN
ejde-203	16	17	is	be	AUX
ejde-203	16	18	a	a	DET
ejde-203	16	19	natural	natural	ADJ
ejde-203	16	20	generalization	generalization	NOUN
ejde-203	16	21	of	of	ADP
ejde-203	16	22	the	the	DET
ejde-203	16	23	period	period	NOUN
ejde-203	16	24	.	.	PUNCT
ejde-203	17	1	but	but	CCONJ
ejde-203	17	2	,	,	PUNCT
ejde-203	17	3	how	how	SCONJ
ejde-203	17	4	does	do	AUX
ejde-203	17	5	this	this	DET
ejde-203	17	6	oscillation	oscillation	NOUN
ejde-203	17	7	time	time	NOUN
ejde-203	17	8	depend	depend	VERB
ejde-203	17	9	on	on	ADP
ejde-203	17	10	the	the	DET
ejde-203	17	11	characteristic	characteristic	NOUN
ejde-203	17	12	of	of	ADP
ejde-203	17	13	the	the	DET
ejde-203	17	14	medium	medium	NOUN
ejde-203	17	15	,	,	PUNCT
ejde-203	17	16	say	say	VERB
ejde-203	17	17	on	on	ADP
ejde-203	17	18	the	the	DET
ejde-203	17	19	viscosity	viscosity	NOUN
ejde-203	17	20	of	of	ADP
ejde-203	17	21	the	the	DET
ejde-203	17	22	surrounding	surround	VERB
ejde-203	17	23	atmosphere	atmosphere	NOUN
ejde-203	17	24	?	?	PUNCT
ejde-203	18	1	it	it	PRON
ejde-203	18	2	seems	seem	VERB
ejde-203	18	3	that	that	SCONJ
ejde-203	18	4	there	there	PRON
ejde-203	18	5	is	be	VERB
ejde-203	18	6	no	no	DET
ejde-203	18	7	much	much	ADJ
ejde-203	18	8	information	information	NOUN
ejde-203	18	9	on	on	ADP
ejde-203	18	10	how	how	SCONJ
ejde-203	18	11	the	the	DET
ejde-203	18	12	damping	damping	NOUN
ejde-203	18	13	affects	affect	VERB
ejde-203	18	14	the	the	DET
ejde-203	18	15	oscillation	oscillation	NOUN
ejde-203	18	16	time	time	NOUN
ejde-203	18	17	.	.	PUNCT
ejde-203	19	1	there	there	PRON
ejde-203	19	2	are	be	VERB
ejde-203	19	3	plenty	plenty	NOUN
ejde-203	19	4	of	of	ADP
ejde-203	19	5	new	new	ADJ
ejde-203	19	6	publications	publication	NOUN
ejde-203	19	7	regarding	regard	VERB
ejde-203	19	8	damping	damp	VERB
ejde-203	19	9	and	and	CCONJ
ejde-203	19	10	oscillations	oscillation	NOUN
ejde-203	19	11	,	,	PUNCT
ejde-203	19	12	ranging	range	VERB
ejde-203	19	13	from	from	ADP
ejde-203	19	14	analytical	analytical	ADJ
ejde-203	19	15	solutions	solution	NOUN
ejde-203	19	16	[	[	X
ejde-203	19	17	3	3	NUM
ejde-203	19	18	,	,	PUNCT
ejde-203	19	19	5	5	NUM
ejde-203	19	20	,	,	PUNCT
ejde-203	19	21	6	6	NUM
ejde-203	19	22	]	]	PUNCT
ejde-203	19	23	,	,	PUNCT
ejde-203	19	24	to	to	ADP
ejde-203	19	25	very	very	ADV
ejde-203	19	26	clever	clever	ADJ
ejde-203	19	27	experimental	experimental	ADJ
ejde-203	19	28	setups	setup	NOUN
ejde-203	19	29	[	[	X
ejde-203	19	30	4	4	NUM
ejde-203	19	31	]	]	PUNCT
ejde-203	19	32	.	.	PUNCT
ejde-203	20	1	the	the	DET
ejde-203	20	2	nature	nature	NOUN
ejde-203	20	3	of	of	ADP
ejde-203	20	4	the	the	DET
ejde-203	20	5	damping	damping	NOUN
ejde-203	20	6	has	have	AUX
ejde-203	20	7	been	be	AUX
ejde-203	20	8	also	also	ADV
ejde-203	20	9	extensively	extensively	ADV
ejde-203	20	10	considered	consider	VERB
ejde-203	20	11	[	[	PUNCT
ejde-203	20	12	2	2	NUM
ejde-203	20	13	,	,	PUNCT
ejde-203	20	14	8	8	NUM
ejde-203	20	15	]	]	PUNCT
ejde-203	20	16	,	,	PUNCT
ejde-203	20	17	but	but	CCONJ
ejde-203	20	18	the	the	DET
ejde-203	20	19	dependence	dependence	NOUN
ejde-203	20	20	of	of	ADP
ejde-203	20	21	the	the	DET
ejde-203	20	22	oscillation	oscillation	NOUN
ejde-203	20	23	time	time	NOUN
ejde-203	20	24	on	on	ADP
ejde-203	20	25	the	the	DET
ejde-203	20	26	damping	damping	NOUN
ejde-203	20	27	,	,	PUNCT
ejde-203	20	28	or	or	CCONJ
ejde-203	20	29	on	on	ADP
ejde-203	20	30	the	the	DET
ejde-203	20	31	non	non	NOUN
ejde-203	20	32	-	-	NOUN
ejde-203	20	33	linearity	linearity	ADJ
ejde-203	20	34	,	,	PUNCT
ejde-203	20	35	seems	seem	VERB
ejde-203	20	36	to	to	PART
ejde-203	20	37	be	be	AUX
ejde-203	20	38	less	less	ADV
ejde-203	20	39	investigated	investigate	VERB
ejde-203	20	40	.	.	PUNCT
ejde-203	21	1	2010	2010	NUM
ejde-203	21	2	mathematics	mathematic	NOUN
ejde-203	21	3	subject	subject	NOUN
ejde-203	21	4	classification	classification	NOUN
ejde-203	21	5	.	.	PUNCT
ejde-203	22	1	34c15	34c15	NUM
ejde-203	22	2	,	,	PUNCT
ejde-203	22	3	34c25	34c25	NUM
ejde-203	22	4	.	.	PUNCT
ejde-203	23	1	key	key	ADJ
ejde-203	23	2	words	word	NOUN
ejde-203	23	3	and	and	CCONJ
ejde-203	23	4	phrases	phrase	NOUN
ejde-203	23	5	.	.	PUNCT
ejde-203	24	1	oscillation	oscillation	NOUN
ejde-203	24	2	time	time	NOUN
ejde-203	24	3	;	;	PUNCT
ejde-203	24	4	damped	damped	VERB
ejde-203	24	5	oscillation	oscillation	NOUN
ejde-203	24	6	.	.	PUNCT
ejde-203	25	1	c	c	X
ejde-203	25	2	©	©	NOUN
ejde-203	25	3	2021	2021	NUM
ejde-203	25	4	this	this	DET
ejde-203	25	5	work	work	NOUN
ejde-203	25	6	is	be	AUX
ejde-203	25	7	licensed	license	VERB
ejde-203	25	8	under	under	ADP
ejde-203	25	9	a	a	DET
ejde-203	25	10	cc	cc	NOUN
ejde-203	25	11	by	by	ADP
ejde-203	25	12	4.0	4.0	NUM
ejde-203	25	13	license	license	NOUN
ejde-203	25	14	.	.	PUNCT
ejde-203	26	1	published	publish	VERB
ejde-203	26	2	october	october	PROPN
ejde-203	26	3	6	6	NUM
ejde-203	26	4	,	,	PUNCT
ejde-203	26	5	2021	2021	NUM
ejde-203	26	6	.	.	PUNCT
ejde-203	27	1	13	13	NUM
ejde-203	27	2	14	14	NUM
ejde-203	27	3	j.	j.	PROPN
ejde-203	27	4	arango	arango	PROPN
ejde-203	27	5	ejde	ejde	PROPN
ejde-203	27	6	/	/	SYM
ejde-203	27	7	si/01	si/01	PROPN
ejde-203	27	8	to	to	PART
ejde-203	27	9	analyze	analyze	VERB
ejde-203	27	10	the	the	DET
ejde-203	27	11	oscillation	oscillation	NOUN
ejde-203	27	12	time	time	NOUN
ejde-203	27	13	,	,	PUNCT
ejde-203	27	14	we	we	PRON
ejde-203	27	15	choose	choose	VERB
ejde-203	27	16	a	a	DET
ejde-203	27	17	standard	standard	ADJ
ejde-203	27	18	mathematical	mathematical	ADJ
ejde-203	27	19	model	model	NOUN
ejde-203	27	20	appearing	appear	VERB
ejde-203	27	21	in	in	ADP
ejde-203	27	22	almost	almost	ADV
ejde-203	27	23	any	any	PRON
ejde-203	27	24	textbook	textbook	NOUN
ejde-203	27	25	of	of	ADP
ejde-203	27	26	ode	ode	PROPN
ejde-203	27	27	(	(	PUNCT
ejde-203	27	28	see	see	VERB
ejde-203	27	29	for	for	ADP
ejde-203	27	30	example	example	NOUN
ejde-203	27	31	[	[	X
ejde-203	27	32	1	1	NUM
ejde-203	27	33	]	]	NUM
ejde-203	27	34	)	)	PUNCT
ejde-203	27	35	,	,	PUNCT
ejde-203	27	36	ẍ+	ẍ+	PROPN
ejde-203	27	37	2αẋ+	2αẋ+	NUM
ejde-203	27	38	x	x	SYM
ejde-203	27	39	(	(	PUNCT
ejde-203	27	40	1	1	NUM
ejde-203	27	41	+	+	NUM
ejde-203	27	42	f(x	f(x	PROPN
ejde-203	27	43	)	)	PUNCT
ejde-203	27	44	)	)	PUNCT
ejde-203	28	1	=	=	PUNCT
ejde-203	28	2	0	0	NUM
ejde-203	28	3	,	,	PUNCT
ejde-203	28	4	(	(	PUNCT
ejde-203	28	5	1.1	1.1	NUM
ejde-203	28	6	)	)	PUNCT
ejde-203	28	7	where	where	SCONJ
ejde-203	28	8	x	x	NOUN
ejde-203	28	9	=	=	SYM
ejde-203	28	10	x(t	x(t	PROPN
ejde-203	28	11	)	)	PUNCT
ejde-203	28	12	measures	measure	VERB
ejde-203	28	13	the	the	DET
ejde-203	28	14	pendulum	pendulum	NOUN
ejde-203	28	15	’s	’s	PART
ejde-203	28	16	deviation	deviation	NOUN
ejde-203	28	17	from	from	ADP
ejde-203	28	18	the	the	DET
ejde-203	28	19	vertical	vertical	ADJ
ejde-203	28	20	axis	axis	NOUN
ejde-203	28	21	of	of	ADP
ejde-203	28	22	equilibrium	equilibrium	NOUN
ejde-203	28	23	and	and	CCONJ
ejde-203	28	24	α	α	PRON
ejde-203	28	25	≥	≥	NOUN
ejde-203	28	26	0	0	NUM
ejde-203	28	27	denote	denote	VERB
ejde-203	28	28	the	the	DET
ejde-203	28	29	viscous	viscous	ADJ
ejde-203	28	30	damping	damp	VERB
ejde-203	28	31	coefficient	coefficient	NOUN
ejde-203	28	32	.	.	PUNCT
ejde-203	29	1	the	the	DET
ejde-203	29	2	term	term	NOUN
ejde-203	29	3	xf(x	xf(x	PRON
ejde-203	29	4	)	)	PUNCT
ejde-203	29	5	models	model	VERB
ejde-203	29	6	the	the	DET
ejde-203	29	7	nonlinear	nonlinear	ADJ
ejde-203	29	8	part	part	NOUN
ejde-203	29	9	of	of	ADP
ejde-203	29	10	the	the	DET
ejde-203	29	11	restoring	restore	VERB
ejde-203	29	12	force	force	NOUN
ejde-203	29	13	.	.	PUNCT
ejde-203	30	1	we	we	PRON
ejde-203	30	2	’ve	’ve	AUX
ejde-203	30	3	rescaled	rescale	VERB
ejde-203	30	4	the	the	DET
ejde-203	30	5	time	time	NOUN
ejde-203	30	6	so	so	SCONJ
ejde-203	30	7	that	that	SCONJ
ejde-203	30	8	the	the	DET
ejde-203	30	9	period	period	NOUN
ejde-203	30	10	of	of	ADP
ejde-203	30	11	the	the	DET
ejde-203	30	12	linear	linear	PROPN
ejde-203	30	13	undamped	undampe	VERB
ejde-203	30	14	oscillation	oscillation	NOUN
ejde-203	30	15	is	be	AUX
ejde-203	30	16	exactly	exactly	ADV
ejde-203	30	17	2π	2π	NOUN
ejde-203	30	18	.	.	PUNCT
ejde-203	31	1	the	the	DET
ejde-203	31	2	mathematics	mathematic	NOUN
ejde-203	31	3	of	of	ADP
ejde-203	31	4	the	the	DET
ejde-203	31	5	solution	solution	NOUN
ejde-203	31	6	x	x	PUNCT
ejde-203	31	7	=	=	SYM
ejde-203	31	8	x(t	x(t	PROPN
ejde-203	31	9	)	)	PUNCT
ejde-203	31	10	is	be	AUX
ejde-203	31	11	classical	classical	ADJ
ejde-203	31	12	.	.	PUNCT
ejde-203	32	1	if	if	SCONJ
ejde-203	32	2	f	f	PROPN
ejde-203	32	3	is	be	AUX
ejde-203	32	4	smooth	smooth	ADJ
ejde-203	32	5	and	and	CCONJ
ejde-203	32	6	x0	x0	PROPN
ejde-203	32	7	and	and	CCONJ
ejde-203	32	8	v0	v0	NOUN
ejde-203	32	9	are	be	AUX
ejde-203	32	10	given	give	VERB
ejde-203	32	11	real	real	ADJ
ejde-203	32	12	values	value	NOUN
ejde-203	32	13	,	,	PUNCT
ejde-203	32	14	then	then	ADV
ejde-203	32	15	there	there	PRON
ejde-203	32	16	exists	exist	VERB
ejde-203	32	17	a	a	DET
ejde-203	32	18	unique	unique	ADJ
ejde-203	32	19	solution	solution	NOUN
ejde-203	32	20	satisfying	satisfy	VERB
ejde-203	32	21	the	the	DET
ejde-203	32	22	given	give	VERB
ejde-203	32	23	conditions	condition	NOUN
ejde-203	32	24	x(0	x(0	PRON
ejde-203	32	25	)	)	PUNCT
ejde-203	33	1	=	=	PUNCT
ejde-203	33	2	x0	x0	PROPN
ejde-203	33	3	and	and	CCONJ
ejde-203	33	4	ẋ(0	ẋ(0	NOUN
ejde-203	33	5	)	)	PUNCT
ejde-203	33	6	=	=	SYM
ejde-203	33	7	v0	v0	NOUN
ejde-203	33	8	.	.	PUNCT
ejde-203	34	1	moreover	moreover	ADV
ejde-203	34	2	,	,	PUNCT
ejde-203	34	3	if	if	SCONJ
ejde-203	34	4	f(0	f(0	NOUN
ejde-203	34	5	)	)	PUNCT
ejde-203	34	6	=	=	SYM
ejde-203	34	7	0	0	NUM
ejde-203	34	8	,	,	PUNCT
ejde-203	34	9	then	then	ADV
ejde-203	34	10	x	x	PUNCT
ejde-203	34	11	=	=	SYM
ejde-203	34	12	0	0	NUM
ejde-203	34	13	is	be	AUX
ejde-203	34	14	a	a	DET
ejde-203	34	15	stable	stable	ADJ
ejde-203	34	16	equilibrium	equilibrium	NOUN
ejde-203	34	17	solution	solution	NOUN
ejde-203	34	18	of	of	ADP
ejde-203	34	19	(	(	PUNCT
ejde-203	34	20	1.1	1.1	NUM
ejde-203	34	21	)	)	PUNCT
ejde-203	34	22	.	.	PUNCT
ejde-203	35	1	as	as	ADP
ejde-203	35	2	a	a	DET
ejde-203	35	3	consequence	consequence	NOUN
ejde-203	35	4	,	,	PUNCT
ejde-203	35	5	x(t	x(t	PROPN
ejde-203	35	6	)	)	PUNCT
ejde-203	35	7	is	be	AUX
ejde-203	35	8	defined	define	VERB
ejde-203	35	9	for	for	ADP
ejde-203	35	10	all	all	DET
ejde-203	35	11	t	t	PROPN
ejde-203	35	12	≥	≥	NOUN
ejde-203	35	13	0	0	NUM
ejde-203	35	14	provided	provide	VERB
ejde-203	35	15	|x0|	|x0|	PROPN
ejde-203	35	16	�	�	PROPN
ejde-203	35	17	1	1	NUM
ejde-203	35	18	and	and	CCONJ
ejde-203	35	19	|v0|	|v0|	NOUN
ejde-203	35	20	�	�	PROPN
ejde-203	35	21	1	1	NUM
ejde-203	35	22	.	.	PUNCT
ejde-203	35	23	notice	notice	VERB
ejde-203	35	24	that	that	SCONJ
ejde-203	35	25	the	the	DET
ejde-203	35	26	points	point	NOUN
ejde-203	35	27	of	of	ADP
ejde-203	35	28	vanishing	vanish	VERB
ejde-203	35	29	derivative	derivative	NOUN
ejde-203	35	30	of	of	ADP
ejde-203	35	31	a	a	DET
ejde-203	35	32	solution	solution	NOUN
ejde-203	35	33	x	x	PUNCT
ejde-203	35	34	=	=	SYM
ejde-203	35	35	x(t	x(t	PROPN
ejde-203	35	36	)	)	PUNCT
ejde-203	35	37	to	to	ADP
ejde-203	35	38	(	(	PUNCT
ejde-203	35	39	1.1	1.1	NUM
ejde-203	35	40	)	)	PUNCT
ejde-203	35	41	are	be	AUX
ejde-203	35	42	isolated	isolate	VERB
ejde-203	35	43	and	and	CCONJ
ejde-203	35	44	those	those	DET
ejde-203	35	45	points	point	NOUN
ejde-203	35	46	correspond	correspond	VERB
ejde-203	35	47	either	either	ADV
ejde-203	35	48	to	to	ADP
ejde-203	35	49	local	local	ADJ
ejde-203	35	50	maxima	maxima	NOUN
ejde-203	35	51	or	or	CCONJ
ejde-203	35	52	to	to	ADP
ejde-203	35	53	local	local	ADJ
ejde-203	35	54	minima	minima	PROPN
ejde-203	35	55	.	.	PUNCT
ejde-203	36	1	denote	denote	VERB
ejde-203	36	2	by	by	ADP
ejde-203	36	3	τ(x0	τ(x0	PROPN
ejde-203	36	4	,	,	PUNCT
ejde-203	36	5	α	α	NOUN
ejde-203	36	6	)	)	PUNCT
ejde-203	36	7	the	the	DET
ejde-203	36	8	amount	amount	NOUN
ejde-203	36	9	of	of	ADP
ejde-203	36	10	time	time	NOUN
ejde-203	36	11	spent	spend	VERB
ejde-203	36	12	(	(	PUNCT
ejde-203	36	13	by	by	ADP
ejde-203	36	14	the	the	DET
ejde-203	36	15	mass	mass	NOUN
ejde-203	36	16	)	)	PUNCT
ejde-203	36	17	completing	complete	VERB
ejde-203	36	18	one	one	NUM
ejde-203	36	19	oscillation	oscillation	NOUN
ejde-203	36	20	starting	start	VERB
ejde-203	36	21	from	from	ADP
ejde-203	36	22	x0	x0	PROPN
ejde-203	36	23	with	with	ADP
ejde-203	36	24	vanishing	vanish	VERB
ejde-203	36	25	velocity	velocity	NOUN
ejde-203	36	26	(	(	PUNCT
ejde-203	36	27	v0	v0	NOUN
ejde-203	36	28	=	=	SYM
ejde-203	36	29	0	0	NUM
ejde-203	36	30	)	)	PUNCT
ejde-203	36	31	.	.	PUNCT
ejde-203	37	1	to	to	PART
ejde-203	37	2	be	be	AUX
ejde-203	37	3	precise	precise	ADJ
ejde-203	37	4	,	,	PUNCT
ejde-203	37	5	if	if	SCONJ
ejde-203	37	6	x	x	ADP
ejde-203	37	7	=	=	SYM
ejde-203	37	8	x(t	x(t	PROPN
ejde-203	37	9	)	)	PUNCT
ejde-203	37	10	starts	start	VERB
ejde-203	37	11	from	from	ADP
ejde-203	37	12	x0	x0	PROPN
ejde-203	37	13	with	with	ADP
ejde-203	37	14	vanishing	vanish	VERB
ejde-203	37	15	velocity	velocity	NOUN
ejde-203	37	16	,	,	PUNCT
ejde-203	37	17	then	then	ADV
ejde-203	37	18	x	x	PRON
ejde-203	37	19	reaches	reach	VERB
ejde-203	37	20	a	a	DET
ejde-203	37	21	local	local	ADJ
ejde-203	37	22	maximum	maximum	NOUN
ejde-203	37	23	at	at	ADP
ejde-203	37	24	t	t	NOUN
ejde-203	37	25	=	=	SYM
ejde-203	37	26	0	0	NUM
ejde-203	37	27	,	,	PUNCT
ejde-203	37	28	and	and	CCONJ
ejde-203	37	29	the	the	DET
ejde-203	37	30	oscillation	oscillation	NOUN
ejde-203	37	31	is	be	AUX
ejde-203	37	32	completed	complete	VERB
ejde-203	37	33	when	when	SCONJ
ejde-203	37	34	x	x	PRON
ejde-203	37	35	reaches	reach	VERB
ejde-203	37	36	the	the	DET
ejde-203	37	37	next	next	ADJ
ejde-203	37	38	local	local	ADJ
ejde-203	37	39	maximum	maximum	NOUN
ejde-203	37	40	.	.	PUNCT
ejde-203	38	1	certainly	certainly	ADV
ejde-203	38	2	,	,	PUNCT
ejde-203	38	3	the	the	DET
ejde-203	38	4	oscillation	oscillation	NOUN
ejde-203	38	5	time	time	NOUN
ejde-203	38	6	generalizes	generalize	VERB
ejde-203	38	7	the	the	DET
ejde-203	38	8	period	period	NOUN
ejde-203	38	9	of	of	ADP
ejde-203	38	10	solutions	solution	NOUN
ejde-203	38	11	for	for	ADP
ejde-203	38	12	the	the	DET
ejde-203	38	13	undamped	undampe	VERB
ejde-203	38	14	model	model	NOUN
ejde-203	38	15	(	(	PUNCT
ejde-203	38	16	α	α	NOUN
ejde-203	38	17	=	=	NOUN
ejde-203	38	18	0	0	NUM
ejde-203	38	19	)	)	PUNCT
ejde-203	38	20	.	.	PUNCT
ejde-203	39	1	in	in	ADP
ejde-203	39	2	this	this	DET
ejde-203	39	3	investigation	investigation	NOUN
ejde-203	39	4	we	we	PRON
ejde-203	39	5	analyze	analyze	VERB
ejde-203	39	6	the	the	DET
ejde-203	39	7	dependence	dependence	NOUN
ejde-203	39	8	of	of	ADP
ejde-203	39	9	τ	τ	PROPN
ejde-203	39	10	on	on	ADP
ejde-203	39	11	x0	x0	PROPN
ejde-203	39	12	and	and	CCONJ
ejde-203	39	13	on	on	ADP
ejde-203	39	14	α	α	NOUN
ejde-203	39	15	under	under	ADP
ejde-203	39	16	the	the	DET
ejde-203	39	17	following	follow	VERB
ejde-203	39	18	working	work	VERB
ejde-203	39	19	assumption	assumption	NOUN
ejde-203	39	20	.	.	PUNCT
ejde-203	40	1	(	(	PUNCT
ejde-203	40	2	a1	a1	NOUN
ejde-203	40	3	)	)	PUNCT
ejde-203	40	4	on	on	ADP
ejde-203	40	5	a	a	DET
ejde-203	40	6	small	small	ADJ
ejde-203	40	7	ε	ε	NOUN
ejde-203	40	8	-	-	PUNCT
ejde-203	40	9	neighborhood	neighborhood	NOUN
ejde-203	40	10	of	of	ADP
ejde-203	40	11	0	0	NUM
ejde-203	40	12	the	the	DET
ejde-203	40	13	function	function	NOUN
ejde-203	40	14	f	f	PROPN
ejde-203	40	15	is	be	AUX
ejde-203	40	16	even	even	ADV
ejde-203	40	17	and	and	CCONJ
ejde-203	40	18	for	for	ADP
ejde-203	40	19	some	some	DET
ejde-203	40	20	constant	constant	ADJ
ejde-203	40	21	a	a	DET
ejde-203	40	22	>	>	X
ejde-203	40	23	0	0	NUM
ejde-203	41	1	we	we	PRON
ejde-203	41	2	have	have	VERB
ejde-203	41	3	f(x	f(x	NOUN
ejde-203	41	4	)	)	PUNCT
ejde-203	42	1	=	=	PUNCT
ejde-203	42	2	−a	−a	NOUN
ejde-203	42	3	x2	x2	NOUN
ejde-203	43	1	+	+	NOUN
ejde-203	43	2	o(|x|4	o(|x|4	NUM
ejde-203	43	3	)	)	PUNCT
ejde-203	43	4	,	,	PUNCT
ejde-203	43	5	we	we	PRON
ejde-203	43	6	shall	shall	AUX
ejde-203	43	7	show	show	VERB
ejde-203	43	8	that	that	SCONJ
ejde-203	43	9	for	for	SCONJ
ejde-203	43	10	x0	x0	PROPN
ejde-203	43	11	fixed	fix	VERB
ejde-203	43	12	,	,	PUNCT
ejde-203	43	13	τ	τ	PROPN
ejde-203	43	14	reaches	reach	VERB
ejde-203	43	15	a	a	DET
ejde-203	43	16	positive	positive	ADJ
ejde-203	43	17	minimum	minimum	NOUN
ejde-203	43	18	at	at	ADP
ejde-203	43	19	some	some	DET
ejde-203	43	20	0	0	NUM
ejde-203	43	21	<	<	X
ejde-203	43	22	α0	α0	ADJ
ejde-203	43	23	<	<	X
ejde-203	43	24	1	1	NUM
ejde-203	43	25	.	.	PUNCT
ejde-203	44	1	it	it	PRON
ejde-203	44	2	does	do	AUX
ejde-203	44	3	not	not	PART
ejde-203	44	4	seem	seem	VERB
ejde-203	44	5	obvious	obvious	ADJ
ejde-203	44	6	that	that	SCONJ
ejde-203	44	7	an	an	DET
ejde-203	44	8	increase	increase	NOUN
ejde-203	44	9	in	in	ADP
ejde-203	44	10	the	the	DET
ejde-203	44	11	damping	damp	VERB
ejde-203	44	12	coefficient	coefficient	NOUN
ejde-203	44	13	α	α	PRON
ejde-203	44	14	might	might	AUX
ejde-203	44	15	cause	cause	VERB
ejde-203	44	16	a	a	DET
ejde-203	44	17	decrease	decrease	NOUN
ejde-203	44	18	in	in	ADP
ejde-203	44	19	τ	τ	PROPN
ejde-203	44	20	.	.	PUNCT
ejde-203	45	1	it	it	PRON
ejde-203	45	2	is	be	AUX
ejde-203	45	3	also	also	ADV
ejde-203	45	4	worth	worth	ADJ
ejde-203	45	5	noticing	notice	VERB
ejde-203	45	6	that	that	SCONJ
ejde-203	45	7	the	the	DET
ejde-203	45	8	existence	existence	NOUN
ejde-203	45	9	of	of	ADP
ejde-203	45	10	a	a	DET
ejde-203	45	11	minimum	minimum	NOUN
ejde-203	45	12	of	of	ADP
ejde-203	45	13	τ	τ	PROPN
ejde-203	45	14	is	be	AUX
ejde-203	45	15	a	a	DET
ejde-203	45	16	consequence	consequence	NOUN
ejde-203	45	17	the	the	DET
ejde-203	45	18	sign	sign	NOUN
ejde-203	45	19	of	of	ADP
ejde-203	45	20	the	the	DET
ejde-203	45	21	constant	constant	ADJ
ejde-203	45	22	a	a	PRON
ejde-203	45	23	in	in	ADP
ejde-203	45	24	the	the	DET
ejde-203	45	25	above	above	ADJ
ejde-203	45	26	assumption	assumption	NOUN
ejde-203	45	27	.	.	PUNCT
ejde-203	46	1	indeed	indeed	ADV
ejde-203	46	2	,	,	PUNCT
ejde-203	46	3	according	accord	VERB
ejde-203	46	4	to	to	ADP
ejde-203	46	5	numerical	numerical	ADJ
ejde-203	46	6	experiments	experiment	NOUN
ejde-203	46	7	carried	carry	VERB
ejde-203	46	8	out	out	ADP
ejde-203	46	9	by	by	ADP
ejde-203	46	10	the	the	DET
ejde-203	46	11	author	author	NOUN
ejde-203	46	12	,	,	PUNCT
ejde-203	46	13	τ	τ	PROPN
ejde-203	46	14	does	do	AUX
ejde-203	46	15	not	not	PART
ejde-203	46	16	reach	reach	VERB
ejde-203	46	17	a	a	DET
ejde-203	46	18	positive	positive	ADJ
ejde-203	46	19	minimum	minimum	NOUN
ejde-203	46	20	if	if	SCONJ
ejde-203	46	21	a	a	DET
ejde-203	46	22	<	<	X
ejde-203	46	23	0	0	NUM
ejde-203	46	24	.	.	PUNCT
ejde-203	47	1	the	the	DET
ejde-203	47	2	author	author	NOUN
ejde-203	47	3	is	be	AUX
ejde-203	47	4	not	not	PART
ejde-203	47	5	aware	aware	ADJ
ejde-203	47	6	of	of	ADP
ejde-203	47	7	a	a	DET
ejde-203	47	8	similar	similar	ADJ
ejde-203	47	9	result	result	NOUN
ejde-203	47	10	in	in	ADP
ejde-203	47	11	the	the	DET
ejde-203	47	12	current	current	ADJ
ejde-203	47	13	literature	literature	NOUN
ejde-203	47	14	nor	nor	CCONJ
ejde-203	47	15	whether	whether	SCONJ
ejde-203	47	16	this	this	DET
ejde-203	47	17	phenomena	phenomena	NOUN
ejde-203	47	18	has	have	AUX
ejde-203	47	19	been	be	AUX
ejde-203	47	20	experimentally	experimentally	ADV
ejde-203	47	21	addressed	address	VERB
ejde-203	47	22	.	.	PUNCT
ejde-203	48	1	this	this	DET
ejde-203	48	2	article	article	NOUN
ejde-203	48	3	was	be	AUX
ejde-203	48	4	written	write	VERB
ejde-203	48	5	with	with	ADP
ejde-203	48	6	the	the	DET
ejde-203	48	7	aim	aim	NOUN
ejde-203	48	8	at	at	ADP
ejde-203	48	9	the	the	DET
ejde-203	48	10	mathematical	mathematical	ADJ
ejde-203	48	11	pendulum	pendulum	NOUN
ejde-203	48	12	x(1	x(1	PROPN
ejde-203	49	1	+	+	PROPN
ejde-203	49	2	f(x	f(x	PROPN
ejde-203	49	3	)	)	PUNCT
ejde-203	49	4	)	)	PUNCT
ejde-203	50	1	=	=	SYM
ejde-203	50	2	sinx	sinx	X
ejde-203	50	3	.	.	PUNCT
ejde-203	51	1	in	in	ADP
ejde-203	51	2	that	that	DET
ejde-203	51	3	case	case	NOUN
ejde-203	51	4	,	,	PUNCT
ejde-203	51	5	figure	figure	NOUN
ejde-203	51	6	1	1	NUM
ejde-203	51	7	summarizes	summarize	NOUN
ejde-203	51	8	our	our	PRON
ejde-203	51	9	findings	finding	NOUN
ejde-203	51	10	by	by	ADP
ejde-203	51	11	picturing	picture	VERB
ejde-203	51	12	the	the	DET
ejde-203	51	13	numerically	numerically	ADV
ejde-203	51	14	simulated	simulate	VERB
ejde-203	51	15	value	value	NOUN
ejde-203	51	16	for	for	ADP
ejde-203	51	17	τ(x0	τ(x0	NOUN
ejde-203	51	18	,	,	PUNCT
ejde-203	51	19	α	α	NOUN
ejde-203	51	20	)	)	PUNCT
ejde-203	51	21	.	.	PUNCT
ejde-203	52	1	interestingly	interestingly	ADV
ejde-203	52	2	,	,	PUNCT
ejde-203	52	3	our	our	PRON
ejde-203	52	4	qualitative	qualitative	ADJ
ejde-203	52	5	analysis	analysis	NOUN
ejde-203	52	6	accurately	accurately	ADV
ejde-203	52	7	reflects	reflect	VERB
ejde-203	52	8	variations	variation	NOUN
ejde-203	52	9	of	of	ADP
ejde-203	52	10	τ	τ	PROPN
ejde-203	52	11	that	that	PRON
ejde-203	52	12	are	be	AUX
ejde-203	52	13	not	not	PART
ejde-203	52	14	easy	easy	ADJ
ejde-203	52	15	to	to	PART
ejde-203	52	16	spot	spot	VERB
ejde-203	52	17	numerically	numerically	ADV
ejde-203	52	18	.	.	PUNCT
ejde-203	53	1	for	for	ADP
ejde-203	53	2	instance	instance	NOUN
ejde-203	53	3	,	,	PUNCT
ejde-203	53	4	the	the	DET
ejde-203	53	5	minimum	minimum	NOUN
ejde-203	53	6	of	of	ADP
ejde-203	53	7	τ(x0	τ(x0	NOUN
ejde-203	53	8	,	,	PUNCT
ejde-203	53	9	α	α	NOUN
ejde-203	53	10	)	)	PUNCT
ejde-203	53	11	for	for	ADP
ejde-203	53	12	x0	x0	PROPN
ejde-203	53	13	=	=	SYM
ejde-203	53	14	0.1	0.1	NUM
ejde-203	53	15	is	be	AUX
ejde-203	53	16	not	not	PART
ejde-203	53	17	evident	evident	ADJ
ejde-203	53	18	in	in	ADP
ejde-203	53	19	figure	figure	NOUN
ejde-203	53	20	1	1	NUM
ejde-203	53	21	.	.	PUNCT
ejde-203	54	1	the	the	DET
ejde-203	54	2	arguments	argument	NOUN
ejde-203	54	3	and	and	CCONJ
ejde-203	54	4	proofs	proof	NOUN
ejde-203	54	5	in	in	ADP
ejde-203	54	6	this	this	DET
ejde-203	54	7	article	article	NOUN
ejde-203	54	8	are	be	AUX
ejde-203	54	9	entirely	entirely	ADV
ejde-203	54	10	based	base	VERB
ejde-203	54	11	on	on	ADP
ejde-203	54	12	well	well	ADV
ejde-203	54	13	established	establish	VERB
ejde-203	54	14	techniques	technique	NOUN
ejde-203	54	15	of	of	ADP
ejde-203	54	16	ode	ode	PROPN
ejde-203	54	17	theory	theory	NOUN
ejde-203	54	18	.	.	PUNCT
ejde-203	55	1	however	however	ADV
ejde-203	55	2	,	,	PUNCT
ejde-203	55	3	the	the	DET
ejde-203	55	4	main	main	ADJ
ejde-203	55	5	result	result	NOUN
ejde-203	55	6	(	(	PUNCT
ejde-203	55	7	theorem	theorem	ADJ
ejde-203	55	8	3.2	3.2	NUM
ejde-203	55	9	)	)	PUNCT
ejde-203	55	10	rests	rest	VERB
ejde-203	55	11	on	on	ADP
ejde-203	55	12	delicate	delicate	ADJ
ejde-203	55	13	estimates	estimate	NOUN
ejde-203	55	14	involving	involve	VERB
ejde-203	55	15	a	a	DET
ejde-203	55	16	differential	differential	ADJ
ejde-203	55	17	equation	equation	NOUN
ejde-203	55	18	describing	describe	VERB
ejde-203	55	19	the	the	DET
ejde-203	55	20	dependence	dependence	NOUN
ejde-203	55	21	of	of	ADP
ejde-203	55	22	the	the	DET
ejde-203	55	23	solution	solution	NOUN
ejde-203	55	24	x	x	PUNCT
ejde-203	55	25	=	=	SYM
ejde-203	55	26	x(t	x(t	PROPN
ejde-203	55	27	)	)	PUNCT
ejde-203	55	28	with	with	ADP
ejde-203	55	29	respect	respect	NOUN
ejde-203	55	30	to	to	ADP
ejde-203	55	31	α	α	NOUN
ejde-203	55	32	.	.	PROPN
ejde-203	55	33	2	2	NUM
ejde-203	55	34	.	.	NOUN
ejde-203	55	35	underdamped	underdampe	VERB
ejde-203	55	36	oscillations	oscillation	NOUN
ejde-203	55	37	definitions	definition	NOUN
ejde-203	55	38	of	of	ADP
ejde-203	55	39	underdamped	underdamped	ADJ
ejde-203	55	40	oscillations	oscillation	NOUN
ejde-203	55	41	in	in	ADP
ejde-203	55	42	linear	linear	PROPN
ejde-203	55	43	systems	system	NOUN
ejde-203	55	44	naturally	naturally	ADV
ejde-203	55	45	carry	carry	VERB
ejde-203	55	46	over	over	ADP
ejde-203	55	47	to	to	ADP
ejde-203	55	48	solutions	solution	NOUN
ejde-203	55	49	of	of	ADP
ejde-203	55	50	(	(	PUNCT
ejde-203	55	51	1.1	1.1	NUM
ejde-203	55	52	)	)	PUNCT
ejde-203	55	53	.	.	PUNCT
ejde-203	56	1	from	from	ADP
ejde-203	56	2	now	now	ADV
ejde-203	56	3	on	on	ADV
ejde-203	56	4	,	,	PUNCT
ejde-203	56	5	x	x	X
ejde-203	56	6	(	(	PUNCT
ejde-203	56	7	·	·	PUNCT
ejde-203	56	8	,	,	PUNCT
ejde-203	56	9	x0	x0	PROPN
ejde-203	56	10	,	,	PUNCT
ejde-203	56	11	α	α	X
ejde-203	56	12	)	)	PUNCT
ejde-203	56	13	stands	stand	VERB
ejde-203	56	14	for	for	ADP
ejde-203	56	15	the	the	DET
ejde-203	56	16	unique	unique	ADJ
ejde-203	56	17	solution	solution	NOUN
ejde-203	56	18	to	to	ADP
ejde-203	56	19	(	(	PUNCT
ejde-203	56	20	1.1	1.1	NUM
ejde-203	56	21	)	)	PUNCT
ejde-203	56	22	satisfying	satisfy	VERB
ejde-203	56	23	the	the	DET
ejde-203	56	24	initial	initial	ADJ
ejde-203	56	25	condition	condition	NOUN
ejde-203	56	26	x(0	x(0	PROPN
ejde-203	56	27	)	)	PUNCT
ejde-203	57	1	=	=	PUNCT
ejde-203	57	2	x0	x0	PROPN
ejde-203	57	3	and	and	CCONJ
ejde-203	57	4	ẋ(0	ẋ(0	NOUN
ejde-203	57	5	)	)	PUNCT
ejde-203	58	1	=	=	SYM
ejde-203	58	2	0	0	X
ejde-203	58	3	.	.	PUNCT
ejde-203	59	1	we	we	PRON
ejde-203	59	2	also	also	ADV
ejde-203	59	3	write	write	VERB
ejde-203	59	4	τ(x0	τ(x0	NOUN
ejde-203	59	5	,	,	PUNCT
ejde-203	59	6	α	α	NOUN
ejde-203	59	7	)	)	PUNCT
ejde-203	59	8	to	to	PART
ejde-203	59	9	highlight	highlight	VERB
ejde-203	59	10	the	the	DET
ejde-203	59	11	dependence	dependence	NOUN
ejde-203	59	12	of	of	ADP
ejde-203	59	13	the	the	DET
ejde-203	59	14	oscillation	oscillation	NOUN
ejde-203	59	15	time	time	NOUN
ejde-203	59	16	on	on	ADP
ejde-203	59	17	x0	x0	PROPN
ejde-203	59	18	and	and	CCONJ
ejde-203	59	19	α	α	X
ejde-203	59	20	.	.	PUNCT
ejde-203	60	1	we	we	PRON
ejde-203	60	2	will	will	AUX
ejde-203	60	3	write	write	VERB
ejde-203	60	4	simply	simply	ADV
ejde-203	60	5	τ	τ	PROPN
ejde-203	60	6	or	or	CCONJ
ejde-203	60	7	x	x	SYM
ejde-203	60	8	when	when	SCONJ
ejde-203	60	9	no	no	DET
ejde-203	60	10	confusion	confusion	NOUN
ejde-203	60	11	can	can	AUX
ejde-203	60	12	arise	arise	VERB
ejde-203	60	13	.	.	PUNCT
ejde-203	61	1	it	it	PRON
ejde-203	61	2	is	be	AUX
ejde-203	61	3	convenient	convenient	ADJ
ejde-203	61	4	to	to	PART
ejde-203	61	5	represent	represent	VERB
ejde-203	61	6	(	(	PUNCT
ejde-203	61	7	1.1	1.1	NUM
ejde-203	61	8	)	)	PUNCT
ejde-203	61	9	in	in	ADP
ejde-203	61	10	the	the	DET
ejde-203	61	11	phase	phase	NOUN
ejde-203	61	12	ejde-2021	ejde-2021	ADJ
ejde-203	61	13	/	/	SYM
ejde-203	61	14	si/01	si/01	PROPN
ejde-203	61	15	oscillation	oscillation	NOUN
ejde-203	61	16	time	time	NOUN
ejde-203	61	17	and	and	CCONJ
ejde-203	61	18	damping	damp	VERB
ejde-203	61	19	15	15	NUM
ejde-203	61	20	0.0	0.0	NUM
ejde-203	61	21	0.1	0.1	NUM
ejde-203	61	22	0.2	0.2	NUM
ejde-203	61	23	0.3	0.3	NUM
ejde-203	61	24	0.4	0.4	NUM
ejde-203	61	25	0.5	0.5	NUM
ejde-203	61	26	damping	damp	VERB
ejde-203	61	27	coefficient	coefficient	NOUN
ejde-203	61	28	α	α	NUM
ejde-203	61	29	2π	2π	PROPN
ejde-203	61	30	6.4	6.4	NUM
ejde-203	61	31	6.9	6.9	NUM
ejde-203	61	32	7.5	7.5	NUM
ejde-203	61	33	τ	τ	X
ejde-203	61	34	(	(	PUNCT
ejde-203	61	35	x	x	SYM
ejde-203	61	36	0	0	NUM
ejde-203	61	37	,	,	PUNCT
ejde-203	61	38	α	α	NOUN
ejde-203	61	39	,	,	PUNCT
ejde-203	61	40	0	0	NUM
ejde-203	61	41	)	)	PUNCT
ejde-203	61	42	x0	x0	PROPN
ejde-203	62	1	=	=	PUNCT
ejde-203	62	2	0.1	0.1	NUM
ejde-203	62	3	x0	x0	NOUN
ejde-203	62	4	=	=	PUNCT
ejde-203	63	1	0.6	0.6	NUM
ejde-203	63	2	x0	x0	NOUN
ejde-203	63	3	=	=	PUNCT
ejde-203	63	4	1.2	1.2	NUM
ejde-203	63	5	x0	x0	NOUN
ejde-203	63	6	=	=	NOUN
ejde-203	63	7	1.6	1.6	NUM
ejde-203	63	8	figure	figure	NOUN
ejde-203	63	9	1	1	NUM
ejde-203	63	10	.	.	PUNCT
ejde-203	63	11	numerical	numerical	PROPN
ejde-203	63	12	simulation	simulation	PROPN
ejde-203	63	13	of	of	ADP
ejde-203	63	14	τ(x0	τ(x0	PROPN
ejde-203	63	15	,	,	PUNCT
ejde-203	63	16	α	α	NOUN
ejde-203	63	17	)	)	PUNCT
ejde-203	63	18	depending	depend	VERB
ejde-203	63	19	on	on	ADP
ejde-203	63	20	α	α	NOUN
ejde-203	63	21	for	for	ADP
ejde-203	63	22	several	several	ADJ
ejde-203	63	23	values	value	NOUN
ejde-203	63	24	of	of	ADP
ejde-203	63	25	x0	x0	PROPN
ejde-203	63	26	with	with	ADP
ejde-203	63	27	x(1	x(1	PROPN
ejde-203	63	28	+	+	PROPN
ejde-203	63	29	f(x	f(x	PROPN
ejde-203	63	30	)	)	PUNCT
ejde-203	63	31	)	)	PUNCT
ejde-203	64	1	=	=	SYM
ejde-203	64	2	sinx	sinx	X
ejde-203	64	3	.	.	PUNCT
ejde-203	65	1	space	space	NOUN
ejde-203	65	2	(	(	PUNCT
ejde-203	65	3	x	x	NOUN
ejde-203	65	4	,	,	PUNCT
ejde-203	65	5	v	v	NOUN
ejde-203	65	6	)	)	PUNCT
ejde-203	65	7	with	with	ADP
ejde-203	65	8	ẋ	ẋ	PROPN
ejde-203	65	9	=	=	SYM
ejde-203	65	10	v	v	PROPN
ejde-203	65	11	:	:	PUNCT
ejde-203	65	12	ẋ	ẋ	PROPN
ejde-203	65	13	=	=	PUNCT
ejde-203	66	1	v	v	ADP
ejde-203	66	2	v̇	v̇	NOUN
ejde-203	66	3	=	=	PUNCT
ejde-203	67	1	−2αv	−2αv	NOUN
ejde-203	67	2	−	−	PROPN
ejde-203	67	3	x−	x−	PROPN
ejde-203	67	4	x	x	SYM
ejde-203	67	5	f(x	f(x	PROPN
ejde-203	67	6	)	)	PUNCT
ejde-203	67	7	.	.	PUNCT
ejde-203	68	1	(	(	PUNCT
ejde-203	68	2	2.1	2.1	NUM
ejde-203	68	3	)	)	PUNCT
ejde-203	68	4	this	this	DET
ejde-203	68	5	equation	equation	NOUN
ejde-203	68	6	is	be	AUX
ejde-203	68	7	explicitly	explicitly	ADV
ejde-203	68	8	solvable	solvable	ADJ
ejde-203	68	9	whenever	whenever	SCONJ
ejde-203	68	10	f	f	PROPN
ejde-203	68	11	≡	≡	PROPN
ejde-203	68	12	0	0	NUM
ejde-203	68	13	,	,	PUNCT
ejde-203	68	14	and	and	CCONJ
ejde-203	68	15	in	in	ADP
ejde-203	68	16	that	that	DET
ejde-203	68	17	case	case	NOUN
ejde-203	68	18	,	,	PUNCT
ejde-203	68	19	its	its	PRON
ejde-203	68	20	solution	solution	NOUN
ejde-203	68	21	is	be	AUX
ejde-203	68	22	xl(t	xl(t	PRON
ejde-203	68	23	)	)	PUNCT
ejde-203	69	1	=	=	SYM
ejde-203	69	2	e−αt	e−αt	PROPN
ejde-203	69	3	ω	ω	X
ejde-203	69	4	(	(	PUNCT
ejde-203	69	5	ω	ω	PROPN
ejde-203	69	6	cosωt+	cosωt+	X
ejde-203	69	7	α	α	NOUN
ejde-203	69	8	sinω	sinω	VERB
ejde-203	69	9	t)x0	t)x0	PROPN
ejde-203	69	10	vl(t	vl(t	PUNCT
ejde-203	69	11	)	)	PUNCT
ejde-203	69	12	=	=	PRON
ejde-203	69	13	−	−	PROPN
ejde-203	69	14	e−αt	e−αt	PROPN
ejde-203	69	15	ω	ω	X
ejde-203	69	16	sinωtx0	sinωtx0	NOUN
ejde-203	69	17	(	(	PUNCT
ejde-203	69	18	2.2	2.2	NUM
ejde-203	69	19	)	)	PUNCT
ejde-203	69	20	where	where	SCONJ
ejde-203	69	21	ω	ω	NOUN
ejde-203	69	22	=	=	NOUN
ejde-203	69	23	√	√	PROPN
ejde-203	69	24	1−	1−	NUM
ejde-203	69	25	α2	α2	ADJ
ejde-203	69	26	.	.	PUNCT
ejde-203	70	1	moreover	moreover	ADV
ejde-203	70	2	,	,	PUNCT
ejde-203	70	3	the	the	DET
ejde-203	70	4	oscillation	oscillation	NOUN
ejde-203	70	5	time	time	NOUN
ejde-203	70	6	τl	τl	ADJ
ejde-203	70	7	is	be	AUX
ejde-203	70	8	τl	τl	ADJ
ejde-203	70	9	=	=	PUNCT
ejde-203	70	10	2π	2π	NUM
ejde-203	70	11	ω	ω	X
ejde-203	70	12	=	=	SYM
ejde-203	71	1	2π√	2π√	PROPN
ejde-203	71	2	1−	1−	NUM
ejde-203	71	3	α2	α2	ADJ
ejde-203	71	4	.	.	PUNCT
ejde-203	72	1	notice	notice	VERB
ejde-203	72	2	that	that	SCONJ
ejde-203	72	3	τl	τl	ADJ
ejde-203	72	4	is	be	AUX
ejde-203	72	5	an	an	DET
ejde-203	72	6	increasing	increase	VERB
ejde-203	72	7	function	function	NOUN
ejde-203	72	8	that	that	PRON
ejde-203	72	9	solely	solely	ADV
ejde-203	72	10	depends	depend	VERB
ejde-203	72	11	on	on	ADP
ejde-203	72	12	α	α	X
ejde-203	72	13	.	.	PUNCT
ejde-203	73	1	though	though	SCONJ
ejde-203	73	2	a	a	DET
ejde-203	73	3	closed	close	VERB
ejde-203	73	4	-	-	PUNCT
ejde-203	73	5	form	form	NOUN
ejde-203	73	6	solution	solution	NOUN
ejde-203	73	7	of	of	ADP
ejde-203	73	8	(	(	PUNCT
ejde-203	73	9	1.1	1.1	NUM
ejde-203	73	10	)	)	PUNCT
ejde-203	73	11	is	be	AUX
ejde-203	73	12	either	either	CCONJ
ejde-203	73	13	not	not	PART
ejde-203	73	14	known	know	VERB
ejde-203	73	15	or	or	CCONJ
ejde-203	73	16	impractical	impractical	ADJ
ejde-203	73	17	,	,	PUNCT
ejde-203	73	18	we	we	PRON
ejde-203	73	19	could	could	AUX
ejde-203	73	20	express	express	VERB
ejde-203	73	21	the	the	DET
ejde-203	73	22	relevant	relevant	ADJ
ejde-203	73	23	solutions	solution	NOUN
ejde-203	73	24	implicitly	implicitly	ADV
ejde-203	73	25	.	.	PUNCT
ejde-203	74	1	to	to	ADP
ejde-203	74	2	that	that	DET
ejde-203	74	3	end	end	NOUN
ejde-203	74	4	,	,	PUNCT
ejde-203	74	5	we	we	PRON
ejde-203	74	6	rewrite	rewrite	VERB
ejde-203	74	7	(	(	PUNCT
ejde-203	74	8	2.1	2.1	NUM
ejde-203	74	9	)	)	PUNCT
ejde-203	74	10	so	so	SCONJ
ejde-203	74	11	that	that	SCONJ
ejde-203	74	12	the	the	DET
ejde-203	74	13	nonlinear	nonlinear	ADJ
ejde-203	74	14	term	term	NOUN
ejde-203	74	15	−x	−x	PROPN
ejde-203	74	16	f(x	f(x	PROPN
ejde-203	74	17	)	)	PUNCT
ejde-203	74	18	assumes	assume	VERB
ejde-203	74	19	the	the	DET
ejde-203	74	20	role	role	NOUN
ejde-203	74	21	of	of	ADP
ejde-203	74	22	a	a	DET
ejde-203	74	23	non	non	ADJ
ejde-203	74	24	homogeneous	homogeneous	ADJ
ejde-203	74	25	forcing	forcing	NOUN
ejde-203	74	26	term	term	NOUN
ejde-203	74	27	.	.	PUNCT
ejde-203	75	1	the	the	DET
ejde-203	75	2	expression	expression	NOUN
ejde-203	75	3	for	for	ADP
ejde-203	75	4	the	the	DET
ejde-203	75	5	solution	solution	NOUN
ejde-203	75	6	(	(	PUNCT
ejde-203	75	7	x	x	X
ejde-203	75	8	,	,	PUNCT
ejde-203	75	9	v	v	NOUN
ejde-203	75	10	)	)	PUNCT
ejde-203	75	11	is	be	AUX
ejde-203	75	12	implicitly	implicitly	ADV
ejde-203	75	13	given	give	VERB
ejde-203	75	14	by	by	ADP
ejde-203	75	15	x(t	x(t	NOUN
ejde-203	75	16	)	)	PUNCT
ejde-203	76	1	=	=	SYM
ejde-203	76	2	xl(t)−	xl(t)−	PROPN
ejde-203	76	3	1	1	NUM
ejde-203	76	4	ω	ω	NUM
ejde-203	76	5	∫	∫	PROPN
ejde-203	76	6	t	t	PROPN
ejde-203	76	7	0	0	NUM
ejde-203	76	8	e−α(t−s	e−α(t−s	PROPN
ejde-203	76	9	)	)	PUNCT
ejde-203	76	10	sinω(t−	sinω(t−	NOUN
ejde-203	76	11	s)x(s)f(x(s	s)x(s)f(x(s	NOUN
ejde-203	76	12	)	)	PUNCT
ejde-203	76	13	)	)	PUNCT
ejde-203	76	14	ds	ds	PROPN
ejde-203	76	15	v(t	v(t	NOUN
ejde-203	76	16	)	)	PUNCT
ejde-203	77	1	=	=	PRON
ejde-203	77	2	vl(t)−	vl(t)−	PROPN
ejde-203	77	3	1	1	NUM
ejde-203	77	4	ω	ω	NUM
ejde-203	77	5	∫	∫	PROPN
ejde-203	77	6	t	t	PROPN
ejde-203	77	7	0	0	PUNCT
ejde-203	78	1	e−α(t−s)(ω	e−α(t−s)(ω	NOUN
ejde-203	78	2	cosω(t−	cosω(t−	NOUN
ejde-203	79	1	s)−	s)−	NOUN
ejde-203	80	1	α	α	PROPN
ejde-203	80	2	sinω(t−	sinω(t−	NOUN
ejde-203	80	3	s))x(s)f(x(s	s))x(s)f(x(s	NOUN
ejde-203	80	4	)	)	PUNCT
ejde-203	80	5	)	)	PUNCT
ejde-203	81	1	ds	ds	INTJ
ejde-203	81	2	(	(	PUNCT
ejde-203	81	3	2.3	2.3	NUM
ejde-203	81	4	)	)	PUNCT
ejde-203	81	5	next	next	ADV
ejde-203	81	6	,	,	PUNCT
ejde-203	81	7	we	we	PRON
ejde-203	81	8	estimate	estimate	VERB
ejde-203	81	9	the	the	DET
ejde-203	81	10	solutions	solution	NOUN
ejde-203	81	11	of	of	ADP
ejde-203	81	12	(	(	PUNCT
ejde-203	81	13	2.1	2.1	NUM
ejde-203	81	14	)	)	PUNCT
ejde-203	81	15	in	in	ADP
ejde-203	81	16	the	the	DET
ejde-203	81	17	conservative	conservative	ADJ
ejde-203	81	18	case	case	NOUN
ejde-203	81	19	(	(	PUNCT
ejde-203	81	20	α	α	NOUN
ejde-203	81	21	=	=	NOUN
ejde-203	81	22	0	0	NUM
ejde-203	81	23	)	)	PUNCT
ejde-203	81	24	in	in	ADP
ejde-203	81	25	which	which	PRON
ejde-203	81	26	all	all	DET
ejde-203	81	27	solutions	solution	NOUN
ejde-203	81	28	are	be	AUX
ejde-203	81	29	periodic	periodic	ADJ
ejde-203	81	30	and	and	CCONJ
ejde-203	81	31	the	the	DET
ejde-203	81	32	period	period	NOUN
ejde-203	81	33	is	be	AUX
ejde-203	81	34	given	give	VERB
ejde-203	81	35	by	by	ADP
ejde-203	81	36	τ	τ	PROPN
ejde-203	81	37	≡	≡	PROPN
ejde-203	81	38	τ(x0	τ(x0	NOUN
ejde-203	81	39	,	,	PUNCT
ejde-203	81	40	0	0	NUM
ejde-203	81	41	)	)	PUNCT
ejde-203	81	42	.	.	PUNCT
ejde-203	82	1	lemma	lemma	PROPN
ejde-203	82	2	2.1	2.1	NUM
ejde-203	82	3	.	.	PUNCT
ejde-203	83	1	if	if	SCONJ
ejde-203	83	2	(	(	PUNCT
ejde-203	83	3	x	x	NOUN
ejde-203	83	4	,	,	PUNCT
ejde-203	83	5	v	v	NOUN
ejde-203	83	6	)	)	PUNCT
ejde-203	83	7	stands	stand	VERB
ejde-203	83	8	for	for	ADP
ejde-203	83	9	the	the	DET
ejde-203	83	10	solution	solution	NOUN
ejde-203	83	11	to	to	ADP
ejde-203	83	12	(	(	PUNCT
ejde-203	83	13	2.1	2.1	NUM
ejde-203	83	14	)	)	PUNCT
ejde-203	83	15	with	with	ADP
ejde-203	83	16	α	α	NOUN
ejde-203	83	17	=	=	SYM
ejde-203	83	18	0	0	NUM
ejde-203	83	19	that	that	SCONJ
ejde-203	83	20	satisfies	satisfy	VERB
ejde-203	83	21	(	(	PUNCT
ejde-203	83	22	x(0	x(0	PROPN
ejde-203	83	23	)	)	PUNCT
ejde-203	83	24	,	,	PUNCT
ejde-203	83	25	v(0	v(0	NOUN
ejde-203	83	26	)	)	PUNCT
ejde-203	83	27	)	)	PUNCT
ejde-203	84	1	=	=	PRON
ejde-203	84	2	(	(	PUNCT
ejde-203	84	3	x0	x0	PROPN
ejde-203	84	4	,	,	PUNCT
ejde-203	84	5	0	0	NUM
ejde-203	84	6	)	)	PUNCT
ejde-203	84	7	,	,	PUNCT
ejde-203	84	8	then	then	ADV
ejde-203	84	9	there	there	PRON
ejde-203	84	10	exists	exist	VERB
ejde-203	84	11	δ	δ	PROPN
ejde-203	84	12	>	>	X
ejde-203	84	13	0	0	PUNCT
ejde-203	85	1	so	so	SCONJ
ejde-203	85	2	that	that	SCONJ
ejde-203	85	3	for	for	SCONJ
ejde-203	85	4	all	all	DET
ejde-203	85	5	|x0|	|x0|	NOUN
ejde-203	85	6	≤	≤	ADJ
ejde-203	85	7	δ	δ	PROPN
ejde-203	85	8	and	and	CCONJ
ejde-203	85	9	all	all	DET
ejde-203	85	10	16	16	NUM
ejde-203	85	11	j.	j.	PROPN
ejde-203	85	12	arango	arango	PROPN
ejde-203	85	13	ejde	ejde	PROPN
ejde-203	85	14	/	/	SYM
ejde-203	85	15	si/01	si/01	PROPN
ejde-203	85	16	0	0	NUM
ejde-203	85	17	≤	≤	NUM
ejde-203	85	18	t	t	PROPN
ejde-203	85	19	≤	≤	NUM
ejde-203	85	20	τ	τ	PUNCT
ejde-203	85	21	we	we	PRON
ejde-203	85	22	have	have	VERB
ejde-203	85	23	x(t	x(t	PROPN
ejde-203	85	24	)	)	PUNCT
ejde-203	86	1	=	=	PUNCT
ejde-203	86	2	x0	x0	PROPN
ejde-203	86	3	cos	cos	PROPN
ejde-203	86	4	t+r1(t	t+r1(t	PROPN
ejde-203	86	5	,	,	PUNCT
ejde-203	86	6	x0	x0	PROPN
ejde-203	86	7	)	)	PUNCT
ejde-203	86	8	,	,	PUNCT
ejde-203	86	9	v(t	v(t	NOUN
ejde-203	86	10	)	)	PUNCT
ejde-203	86	11	=	=	SYM
ejde-203	86	12	−x0	−x0	NOUN
ejde-203	86	13	sin	sin	NOUN
ejde-203	86	14	t+r2(t	t+r2(t	NOUN
ejde-203	86	15	,	,	PUNCT
ejde-203	86	16	x0	x0	PROPN
ejde-203	86	17	)	)	PUNCT
ejde-203	86	18	,	,	PUNCT
ejde-203	86	19	(	(	PUNCT
ejde-203	86	20	2.4	2.4	NUM
ejde-203	86	21	)	)	PUNCT
ejde-203	86	22	where	where	SCONJ
ejde-203	86	23	|ri(t	|ri(t	X
ejde-203	86	24	,	,	PUNCT
ejde-203	86	25	x0)|	x0)|	PROPN
ejde-203	86	26	≤	≤	NUM
ejde-203	86	27	const	const	NOUN
ejde-203	86	28	.	.	PUNCT
ejde-203	87	1	|x30|	|x30|	NOUN
ejde-203	87	2	,	,	PUNCT
ejde-203	87	3	i	i	PRON
ejde-203	87	4	=	=	NOUN
ejde-203	87	5	1	1	NUM
ejde-203	87	6	,	,	PUNCT
ejde-203	87	7	2	2	NUM
ejde-203	87	8	.	.	PUNCT
ejde-203	87	9	proof	proof	NOUN
ejde-203	87	10	.	.	PUNCT
ejde-203	88	1	letting	let	VERB
ejde-203	88	2	α	α	PRON
ejde-203	88	3	=	=	SYM
ejde-203	88	4	0	0	NUM
ejde-203	88	5	in	in	ADP
ejde-203	88	6	(	(	PUNCT
ejde-203	88	7	2.3	2.3	NUM
ejde-203	88	8	)	)	PUNCT
ejde-203	88	9	we	we	PRON
ejde-203	88	10	obtain	obtain	VERB
ejde-203	88	11	r1(t	r1(t	NOUN
ejde-203	88	12	,	,	PUNCT
ejde-203	88	13	x0	x0	NUM
ejde-203	88	14	)	)	PUNCT
ejde-203	89	1	=	=	PUNCT
ejde-203	90	1	−	−	PROPN
ejde-203	90	2	∫	∫	PROPN
ejde-203	90	3	t	t	PROPN
ejde-203	90	4	0	0	NUM
ejde-203	91	1	cos(t−	cos(t−	PRON
ejde-203	91	2	s)x(s)f(x(s	s)x(s)f(x(s	NOUN
ejde-203	91	3	)	)	PUNCT
ejde-203	91	4	)	)	PUNCT
ejde-203	92	1	ds	ds	PROPN
ejde-203	92	2	.	.	PUNCT
ejde-203	92	3	(	(	PUNCT
ejde-203	92	4	2.5	2.5	NUM
ejde-203	92	5	)	)	PUNCT
ejde-203	92	6	since	since	SCONJ
ejde-203	92	7	(	(	PUNCT
ejde-203	92	8	0	0	NUM
ejde-203	92	9	,	,	PUNCT
ejde-203	92	10	0	0	NUM
ejde-203	92	11	)	)	PUNCT
ejde-203	92	12	is	be	AUX
ejde-203	92	13	a	a	DET
ejde-203	92	14	stable	stable	ADJ
ejde-203	92	15	equilibrium	equilibrium	NOUN
ejde-203	92	16	solution	solution	NOUN
ejde-203	92	17	to	to	ADP
ejde-203	92	18	(	(	PUNCT
ejde-203	92	19	2.1	2.1	NUM
ejde-203	92	20	)	)	PUNCT
ejde-203	92	21	,	,	PUNCT
ejde-203	92	22	there	there	PRON
ejde-203	92	23	exists	exist	VERB
ejde-203	92	24	δ	δ	PROPN
ejde-203	92	25	>	>	X
ejde-203	92	26	0	0	PUNCT
ejde-203	92	27	and	and	CCONJ
ejde-203	92	28	ε	ε	PROPN
ejde-203	92	29	>	>	X
ejde-203	92	30	0	0	PUNCT
ejde-203	93	1	so	so	SCONJ
ejde-203	93	2	that	that	SCONJ
ejde-203	93	3	any	any	DET
ejde-203	93	4	solution	solution	NOUN
ejde-203	93	5	(	(	PUNCT
ejde-203	93	6	x	x	X
ejde-203	93	7	,	,	PUNCT
ejde-203	93	8	v	v	NOUN
ejde-203	93	9	)	)	PUNCT
ejde-203	93	10	to	to	ADP
ejde-203	93	11	(	(	PUNCT
ejde-203	93	12	2.1	2.1	NUM
ejde-203	93	13	)	)	PUNCT
ejde-203	93	14	starting	start	VERB
ejde-203	93	15	at	at	ADP
ejde-203	93	16	(	(	PUNCT
ejde-203	93	17	x0	x0	PROPN
ejde-203	93	18	,	,	PUNCT
ejde-203	93	19	0	0	NUM
ejde-203	93	20	)	)	PUNCT
ejde-203	93	21	,	,	PUNCT
ejde-203	93	22	with	with	ADP
ejde-203	93	23	|x0|	|x0|	PROPN
ejde-203	93	24	≤	≤	NUM
ejde-203	93	25	δ	δ	PROPN
ejde-203	93	26	satisfies	satisfy	VERB
ejde-203	93	27	|x(t)|	|x(t)|	PROPN
ejde-203	93	28	≤	≤	PROPN
ejde-203	93	29	ε	ε	PROPN
ejde-203	93	30	.	.	PUNCT
ejde-203	94	1	now	now	ADV
ejde-203	94	2	write	write	VERB
ejde-203	94	3	f	f	PROPN
ejde-203	94	4	(	(	PUNCT
ejde-203	94	5	z	z	NOUN
ejde-203	94	6	)	)	PUNCT
ejde-203	94	7	=	=	PUNCT
ejde-203	94	8	−zf(z	−zf(z	X
ejde-203	94	9	)	)	PUNCT
ejde-203	94	10	and	and	CCONJ
ejde-203	94	11	notice	notice	VERB
ejde-203	94	12	that	that	SCONJ
ejde-203	94	13	for	for	ADP
ejde-203	94	14	some	some	DET
ejde-203	94	15	ξ	ξ	PROPN
ejde-203	94	16	∈	∈	PROPN
ejde-203	94	17	(	(	PUNCT
ejde-203	94	18	−ε	−ε	NOUN
ejde-203	94	19	,	,	PUNCT
ejde-203	94	20	ε	ε	PROPN
ejde-203	94	21	)	)	PUNCT
ejde-203	94	22	we	we	PRON
ejde-203	94	23	have	have	VERB
ejde-203	94	24	f	f	PROPN
ejde-203	94	25	(	(	PUNCT
ejde-203	94	26	x(s	x(s	PROPN
ejde-203	94	27	)	)	PUNCT
ejde-203	94	28	)	)	PUNCT
ejde-203	95	1	=	=	SYM
ejde-203	95	2	f	f	PROPN
ejde-203	95	3	(	(	PUNCT
ejde-203	95	4	x0	x0	PROPN
ejde-203	95	5	cos	cos	PROPN
ejde-203	95	6	s+r1(s	s+r1(s	PROPN
ejde-203	95	7	,	,	PUNCT
ejde-203	95	8	x0	x0	PROPN
ejde-203	95	9	)	)	PUNCT
ejde-203	95	10	)	)	PUNCT
ejde-203	96	1	=	=	SYM
ejde-203	96	2	f	f	PROPN
ejde-203	96	3	(	(	PUNCT
ejde-203	96	4	x0	x0	PROPN
ejde-203	96	5	cos	cos	PROPN
ejde-203	96	6	s	s	PROPN
ejde-203	96	7	)	)	PUNCT
ejde-203	96	8	+	+	ADJ
ejde-203	96	9	r1(s	r1(s	ADJ
ejde-203	96	10	,	,	PUNCT
ejde-203	96	11	x0)f	x0)f	ADJ
ejde-203	96	12	′(ξ	′(ξ	PROPN
ejde-203	96	13	)	)	PUNCT
ejde-203	96	14	.	.	PUNCT
ejde-203	97	1	next	next	ADJ
ejde-203	97	2	,	,	PUNCT
ejde-203	97	3	identity	identity	NOUN
ejde-203	97	4	(	(	PUNCT
ejde-203	97	5	2.5	2.5	NUM
ejde-203	97	6	)	)	PUNCT
ejde-203	97	7	,	,	PUNCT
ejde-203	97	8	assumption	assumption	NOUN
ejde-203	97	9	(	(	PUNCT
ejde-203	97	10	?	?	PUNCT
ejde-203	97	11	?	?	PUNCT
ejde-203	97	12	)	)	PUNCT
ejde-203	97	13	and	and	CCONJ
ejde-203	97	14	some	some	DET
ejde-203	97	15	standard	standard	ADJ
ejde-203	97	16	estimations	estimation	NOUN
ejde-203	97	17	yield	yield	VERB
ejde-203	97	18	|r1(t	|r1(t	PROPN
ejde-203	97	19	,	,	PUNCT
ejde-203	97	20	x0)|	x0)|	PROPN
ejde-203	97	21	≤	≤	NOUN
ejde-203	98	1	2a|x30|+	2a|x30|+	NUM
ejde-203	98	2	c2	c2	PROPN
ejde-203	98	3	∫	∫	PROPN
ejde-203	98	4	t	t	PROPN
ejde-203	98	5	0	0	NUM
ejde-203	98	6	|r1(s	|r1(s	PROPN
ejde-203	98	7	,	,	PUNCT
ejde-203	98	8	x0)|	x0)|	NOUN
ejde-203	99	1	ds	ds	PROPN
ejde-203	99	2	where	where	SCONJ
ejde-203	99	3	c2	c2	PROPN
ejde-203	99	4	=	=	PROPN
ejde-203	99	5	maxz∈[−ε	maxz∈[−ε	PROPN
ejde-203	99	6	,	,	PUNCT
ejde-203	99	7	ε	ε	PROPN
ejde-203	99	8	]	]	PUNCT
ejde-203	99	9	|f	|f	PROPN
ejde-203	99	10	′(z)|	′(z)|	PROPN
ejde-203	99	11	.	.	PUNCT
ejde-203	100	1	the	the	DET
ejde-203	100	2	first	first	ADJ
ejde-203	100	3	claim	claim	NOUN
ejde-203	100	4	follows	follow	VERB
ejde-203	100	5	now	now	ADV
ejde-203	100	6	from	from	ADP
ejde-203	100	7	gronwall	gronwall	PROPN
ejde-203	100	8	’s	’s	PART
ejde-203	100	9	inequality	inequality	NOUN
ejde-203	100	10	.	.	PUNCT
ejde-203	101	1	the	the	DET
ejde-203	101	2	proof	proof	NOUN
ejde-203	101	3	of	of	ADP
ejde-203	101	4	the	the	DET
ejde-203	101	5	estimation	estimation	NOUN
ejde-203	101	6	for	for	ADP
ejde-203	101	7	r2	r2	PROPN
ejde-203	101	8	is	be	AUX
ejde-203	101	9	analogous	analogous	ADJ
ejde-203	101	10	.	.	PUNCT
ejde-203	102	1	�	�	PROPN
ejde-203	102	2	at	at	ADP
ejde-203	102	3	this	this	DET
ejde-203	102	4	point	point	NOUN
ejde-203	102	5	it	it	PRON
ejde-203	102	6	is	be	AUX
ejde-203	102	7	appropriated	appropriate	VERB
ejde-203	102	8	to	to	PART
ejde-203	102	9	define	define	VERB
ejde-203	102	10	the	the	DET
ejde-203	102	11	half	half	ADJ
ejde-203	102	12	oscillation	oscillation	NOUN
ejde-203	102	13	time	time	NOUN
ejde-203	102	14	τ̂	τ̂	PUNCT
ejde-203	102	15	=	=	SYM
ejde-203	102	16	τ̂(x0	τ̂(x0	NOUN
ejde-203	102	17	,	,	PUNCT
ejde-203	102	18	α	α	NOUN
ejde-203	102	19	)	)	PUNCT
ejde-203	102	20	to	to	PART
ejde-203	102	21	be	be	AUX
ejde-203	102	22	the	the	DET
ejde-203	102	23	time	time	NOUN
ejde-203	102	24	spent	spend	VERB
ejde-203	102	25	by	by	ADP
ejde-203	102	26	the	the	DET
ejde-203	102	27	solution	solution	NOUN
ejde-203	102	28	x(t	x(t	PROPN
ejde-203	102	29	,	,	PUNCT
ejde-203	102	30	x0	x0	PROPN
ejde-203	102	31	,	,	PUNCT
ejde-203	102	32	α	α	X
ejde-203	102	33	)	)	PUNCT
ejde-203	102	34	,	,	PUNCT
ejde-203	102	35	t	t	PROPN
ejde-203	102	36	≥	≥	NUM
ejde-203	102	37	0	0	NUM
ejde-203	102	38	,	,	PUNCT
ejde-203	102	39	reaching	reach	VERB
ejde-203	102	40	the	the	DET
ejde-203	102	41	next	next	ADJ
ejde-203	102	42	local	local	ADJ
ejde-203	102	43	minimum	minimum	NOUN
ejde-203	102	44	.	.	PUNCT
ejde-203	103	1	if	if	SCONJ
ejde-203	103	2	α	α	PRON
ejde-203	103	3	=	=	NOUN
ejde-203	103	4	0	0	NUM
ejde-203	103	5	and	and	CCONJ
ejde-203	103	6	f	f	PROPN
ejde-203	103	7	is	be	AUX
ejde-203	103	8	even	even	ADV
ejde-203	103	9	,	,	PUNCT
ejde-203	103	10	the	the	DET
ejde-203	103	11	symmetry	symmetry	NOUN
ejde-203	103	12	of	of	ADP
ejde-203	103	13	the	the	DET
ejde-203	103	14	solution	solution	NOUN
ejde-203	103	15	(	(	PUNCT
ejde-203	103	16	1.1	1.1	NUM
ejde-203	103	17	)	)	PUNCT
ejde-203	103	18	yields	yield	NOUN
ejde-203	103	19	.	.	PUNCT
ejde-203	104	1	2τ̂	2τ̂	NUM
ejde-203	104	2	=	=	SYM
ejde-203	104	3	τ	τ	PROPN
ejde-203	104	4	.	.	PUNCT
ejde-203	105	1	lemma	lemma	PROPN
ejde-203	105	2	2.2	2.2	NUM
ejde-203	105	3	.	.	PUNCT
ejde-203	106	1	if	if	SCONJ
ejde-203	106	2	τ̂	τ̂	NUM
ejde-203	106	3	=	=	SYM
ejde-203	106	4	τ̂(x0	τ̂(x0	NOUN
ejde-203	106	5	,	,	PUNCT
ejde-203	106	6	α	α	NOUN
ejde-203	106	7	)	)	PUNCT
ejde-203	106	8	denote	denote	VERB
ejde-203	106	9	the	the	DET
ejde-203	106	10	half	half	ADJ
ejde-203	106	11	oscillation	oscillation	NOUN
ejde-203	106	12	time	time	NOUN
ejde-203	106	13	and	and	CCONJ
ejde-203	106	14	a	a	PRON
ejde-203	106	15	is	be	AUX
ejde-203	106	16	the	the	DET
ejde-203	106	17	constant	constant	ADJ
ejde-203	106	18	of	of	ADP
ejde-203	106	19	assumption	assumption	NOUN
ejde-203	106	20	(	(	PUNCT
ejde-203	106	21	a1	a1	NOUN
ejde-203	106	22	)	)	PUNCT
ejde-203	106	23	,	,	PUNCT
ejde-203	106	24	then	then	ADV
ejde-203	106	25	τ̂(x0	τ̂(x0	NUM
ejde-203	106	26	,	,	PUNCT
ejde-203	106	27	α	α	NOUN
ejde-203	106	28	)	)	PUNCT
ejde-203	106	29	>	>	X
ejde-203	107	1	π√	π√	PROPN
ejde-203	107	2	1−	1−	NUM
ejde-203	107	3	α2	α2	ADJ
ejde-203	107	4	and	and	CCONJ
ejde-203	107	5	lim	lim	PROPN
ejde-203	107	6	x0→0	x0→0	PROPN
ejde-203	107	7	+	+	PROPN
ejde-203	107	8	τ̂(x0	τ̂(x0	PROPN
ejde-203	107	9	,	,	PUNCT
ejde-203	107	10	0	0	NUM
ejde-203	107	11	)	)	PUNCT
ejde-203	107	12	=	=	PUNCT
ejde-203	108	1	π	π	X
ejde-203	108	2	+	+	CCONJ
ejde-203	108	3	aπ	aπ	NUM
ejde-203	108	4	8	8	NUM
ejde-203	108	5	x20	x20	NOUN
ejde-203	108	6	+	+	CCONJ
ejde-203	108	7	o(x30	o(x30	NOUN
ejde-203	108	8	)	)	PUNCT
ejde-203	108	9	.	.	PUNCT
ejde-203	109	1	proof	proof	NOUN
ejde-203	109	2	.	.	PUNCT
ejde-203	110	1	we	we	PRON
ejde-203	110	2	introduce	introduce	VERB
ejde-203	110	3	introduce	introduce	VERB
ejde-203	110	4	the	the	DET
ejde-203	110	5	polar	polar	ADJ
ejde-203	110	6	coordinates	coordinate	NOUN
ejde-203	111	1	r	r	NOUN
ejde-203	111	2	=	=	PUNCT
ejde-203	111	3	√	√	PROPN
ejde-203	111	4	x2	x2	PROPN
ejde-203	112	1	+	+	CCONJ
ejde-203	112	2	v2	v2	PROPN
ejde-203	112	3	,	,	PUNCT
ejde-203	112	4	tan	tan	NOUN
ejde-203	112	5	θ	θ	NOUN
ejde-203	113	1	=	=	PUNCT
ejde-203	113	2	x	x	SYM
ejde-203	113	3	v	v	NOUN
ejde-203	113	4	,	,	PUNCT
ejde-203	113	5	to	to	PART
ejde-203	113	6	obtain	obtain	VERB
ejde-203	113	7	θ̇	θ̇	PRON
ejde-203	113	8	=	=	SYM
ejde-203	113	9	−(1	−(1	NOUN
ejde-203	113	10	+	+	CCONJ
ejde-203	113	11	α	α	PRON
ejde-203	113	12	sin	sin	NOUN
ejde-203	113	13	2θ	2θ	NUM
ejde-203	113	14	+	+	CCONJ
ejde-203	113	15	sin2	sin2	NOUN
ejde-203	113	16	θf(x	θf(x	NOUN
ejde-203	113	17	)	)	PUNCT
ejde-203	113	18	)	)	PUNCT
ejde-203	114	1	ṙ	ṙ	NOUN
ejde-203	114	2	=	=	SYM
ejde-203	114	3	−v	−v	NOUN
ejde-203	114	4	r	r	NOUN
ejde-203	114	5	(	(	PUNCT
ejde-203	114	6	2αv	2αv	NOUN
ejde-203	114	7	+	+	CCONJ
ejde-203	114	8	xf(x	xf(x	NUM
ejde-203	114	9	)	)	PUNCT
ejde-203	114	10	)	)	PUNCT
ejde-203	114	11	(	(	PUNCT
ejde-203	114	12	2.6	2.6	NUM
ejde-203	114	13	)	)	PUNCT
ejde-203	114	14	from	from	ADP
ejde-203	114	15	these	these	DET
ejde-203	114	16	equations	equation	NOUN
ejde-203	114	17	,	,	PUNCT
ejde-203	114	18	we	we	PRON
ejde-203	114	19	obtain	obtain	VERB
ejde-203	114	20	the	the	DET
ejde-203	114	21	following	follow	VERB
ejde-203	114	22	expression	expression	NOUN
ejde-203	114	23	for	for	ADP
ejde-203	114	24	the	the	DET
ejde-203	114	25	half	half	ADJ
ejde-203	114	26	oscillation	oscillation	NOUN
ejde-203	114	27	time	time	NOUN
ejde-203	114	28	,	,	PUNCT
ejde-203	114	29	τ̂	τ̂	PUNCT
ejde-203	114	30	:	:	PUNCT
ejde-203	114	31	=	=	SYM
ejde-203	114	32	τ̂(x0	τ̂(x0	NOUN
ejde-203	114	33	,	,	PUNCT
ejde-203	114	34	α	α	NOUN
ejde-203	114	35	)	)	PUNCT
ejde-203	114	36	=	=	SYM
ejde-203	115	1	∫	∫	PROPN
ejde-203	115	2	π	π	NOUN
ejde-203	115	3	0	0	NUM
ejde-203	116	1	dθ	dθ	PROPN
ejde-203	116	2	1	1	NUM
ejde-203	117	1	+	+	NUM
ejde-203	117	2	α	α	NOUN
ejde-203	117	3	sin	sin	NOUN
ejde-203	117	4	2θ	2θ	NUM
ejde-203	117	5	+	+	CCONJ
ejde-203	117	6	sin2	sin2	NOUN
ejde-203	117	7	θf(x(θ	θf(x(θ	NOUN
ejde-203	117	8	)	)	PUNCT
ejde-203	117	9	)	)	PUNCT
ejde-203	117	10	.	.	PUNCT
ejde-203	118	1	(	(	PUNCT
ejde-203	118	2	2.7	2.7	NUM
ejde-203	118	3	)	)	PUNCT
ejde-203	118	4	now	now	ADV
ejde-203	118	5	,	,	PUNCT
ejde-203	118	6	the	the	DET
ejde-203	118	7	effect	effect	NOUN
ejde-203	118	8	of	of	ADP
ejde-203	118	9	the	the	DET
ejde-203	118	10	nonlinearity	nonlinearity	NOUN
ejde-203	118	11	on	on	ADP
ejde-203	118	12	the	the	DET
ejde-203	118	13	oscillation	oscillation	NOUN
ejde-203	118	14	time	time	NOUN
ejde-203	118	15	is	be	AUX
ejde-203	118	16	clear	clear	ADJ
ejde-203	118	17	.	.	PUNCT
ejde-203	119	1	by	by	ADP
ejde-203	119	2	assumption	assumption	NOUN
ejde-203	119	3	(	(	PUNCT
ejde-203	119	4	a1	a1	NOUN
ejde-203	119	5	)	)	PUNCT
ejde-203	119	6	we	we	PRON
ejde-203	119	7	obtain	obtain	VERB
ejde-203	119	8	τ̂(x0	τ̂(x0	NOUN
ejde-203	119	9	,	,	PUNCT
ejde-203	119	10	α	α	NOUN
ejde-203	119	11	)	)	PUNCT
ejde-203	119	12	>	>	X
ejde-203	120	1	∫	∫	PROPN
ejde-203	121	1	π	π	NOUN
ejde-203	121	2	0	0	NUM
ejde-203	122	1	dθ	dθ	PROPN
ejde-203	122	2	1	1	NUM
ejde-203	122	3	+	+	NUM
ejde-203	122	4	α	α	NOUN
ejde-203	122	5	sin	sin	NOUN
ejde-203	122	6	2θ	2θ	NUM
ejde-203	122	7	=	=	SYM
ejde-203	122	8	π√	π√	PROPN
ejde-203	122	9	1−	1−	NUM
ejde-203	122	10	α2	α2	ADJ
ejde-203	122	11	.	.	PUNCT
ejde-203	123	1	for	for	ADP
ejde-203	123	2	α	α	NOUN
ejde-203	123	3	=	=	SYM
ejde-203	123	4	0	0	NUM
ejde-203	123	5	we	we	PRON
ejde-203	123	6	use	use	VERB
ejde-203	123	7	estimation	estimation	NOUN
ejde-203	123	8	(	(	PUNCT
ejde-203	123	9	2.4	2.4	NUM
ejde-203	123	10	)	)	PUNCT
ejde-203	123	11	to	to	PART
ejde-203	123	12	obtain	obtain	VERB
ejde-203	123	13	τ̂(x0	τ̂(x0	PUNCT
ejde-203	123	14	,	,	PUNCT
ejde-203	123	15	0	0	NUM
ejde-203	123	16	)	)	PUNCT
ejde-203	123	17	=	=	SYM
ejde-203	124	1	∫	∫	PROPN
ejde-203	124	2	π	π	PROPN
ejde-203	124	3	0	0	PROPN
ejde-203	125	1	dθ	dθ	PROPN
ejde-203	125	2	1−	1−	NUM
ejde-203	125	3	a	a	DET
ejde-203	125	4	x20	x20	NOUN
ejde-203	125	5	sin2	sin2	NOUN
ejde-203	125	6	θ	θ	PROPN
ejde-203	125	7	cos2	cos2	PROPN
ejde-203	125	8	t(θ	t(θ	PROPN
ejde-203	125	9	)	)	PUNCT
ejde-203	126	1	+	+	NUM
ejde-203	126	2	o(x30	o(x30	NOUN
ejde-203	126	3	)	)	PUNCT
ejde-203	126	4	.	.	PUNCT
ejde-203	127	1	ejde-2021	ejde-2021	ADJ
ejde-203	127	2	/	/	SYM
ejde-203	127	3	si/01	si/01	PROPN
ejde-203	127	4	oscillation	oscillation	NOUN
ejde-203	127	5	time	time	NOUN
ejde-203	127	6	and	and	CCONJ
ejde-203	127	7	damping	damp	VERB
ejde-203	127	8	17	17	NUM
ejde-203	127	9	a	a	DET
ejde-203	127	10	straightforward	straightforward	ADJ
ejde-203	127	11	computation	computation	NOUN
ejde-203	127	12	yields	yield	NOUN
ejde-203	127	13	lim	lim	PROPN
ejde-203	128	1	x0→0	x0→0	PROPN
ejde-203	128	2	+	+	PROPN
ejde-203	128	3	τ̂(x0	τ̂(x0	PROPN
ejde-203	128	4	,	,	PUNCT
ejde-203	128	5	0	0	NUM
ejde-203	128	6	)	)	PUNCT
ejde-203	128	7	=	=	SYM
ejde-203	128	8	π	π	PROPN
ejde-203	128	9	,	,	PUNCT
ejde-203	128	10	lim	lim	PROPN
ejde-203	128	11	x0→0	x0→0	PROPN
ejde-203	128	12	+	+	PROPN
ejde-203	128	13	∂τ̂	∂τ̂	ADJ
ejde-203	128	14	∂x0	∂x0	PROPN
ejde-203	128	15	(	(	PUNCT
ejde-203	128	16	x0	x0	PROPN
ejde-203	128	17	,	,	PUNCT
ejde-203	128	18	0	0	NUM
ejde-203	128	19	)	)	PUNCT
ejde-203	129	1	=	=	SYM
ejde-203	129	2	0	0	X
ejde-203	129	3	.	.	PUNCT
ejde-203	130	1	now	now	ADV
ejde-203	130	2	,	,	PUNCT
ejde-203	130	3	the	the	DET
ejde-203	130	4	expression	expression	NOUN
ejde-203	130	5	for	for	ADP
ejde-203	130	6	∂2τ̂	∂2τ̂	PROPN
ejde-203	130	7	∂x2	∂x2	PROPN
ejde-203	130	8	0	0	NUM
ejde-203	130	9	(	(	PUNCT
ejde-203	130	10	x0	x0	PROPN
ejde-203	130	11	,	,	PUNCT
ejde-203	130	12	0	0	NUM
ejde-203	130	13	)	)	PUNCT
ejde-203	130	14	is	be	AUX
ejde-203	130	15	somewhat	somewhat	ADV
ejde-203	130	16	cumbersome	cumbersome	ADJ
ejde-203	130	17	.	.	PUNCT
ejde-203	131	1	however	however	ADV
ejde-203	131	2	,	,	PUNCT
ejde-203	131	3	taking	take	VERB
ejde-203	131	4	into	into	ADP
ejde-203	131	5	account	account	NOUN
ejde-203	131	6	that	that	SCONJ
ejde-203	131	7	limx0→0	limx0→0	PROPN
ejde-203	131	8	t(θ	t(θ	PROPN
ejde-203	131	9	)	)	PUNCT
ejde-203	131	10	=	=	SYM
ejde-203	131	11	θ	θ	X
ejde-203	131	12	,	,	PUNCT
ejde-203	131	13	we	we	PRON
ejde-203	131	14	readily	readily	ADV
ejde-203	131	15	obtain	obtain	VERB
ejde-203	131	16	lim	lim	PROPN
ejde-203	131	17	x0→0	x0→0	PROPN
ejde-203	131	18	+	+	PROPN
ejde-203	131	19	∂2τ̂	∂2τ̂	PROPN
ejde-203	131	20	∂x20	∂x20	NOUN
ejde-203	131	21	(	(	PUNCT
ejde-203	131	22	x0	x0	PROPN
ejde-203	131	23	,	,	PUNCT
ejde-203	131	24	0	0	NUM
ejde-203	131	25	)	)	PUNCT
ejde-203	131	26	=	=	SYM
ejde-203	132	1	∫	∫	PROPN
ejde-203	132	2	π	π	NOUN
ejde-203	132	3	0	0	PROPN
ejde-203	132	4	2a	2a	NUM
ejde-203	132	5	sin2	sin2	NOUN
ejde-203	132	6	θ	θ	PROPN
ejde-203	132	7	cos2	cos2	PROPN
ejde-203	132	8	θ	θ	PROPN
ejde-203	132	9	dθ	dθ	PROPN
ejde-203	132	10	=	=	PUNCT
ejde-203	132	11	2aπ	2aπ	NOUN
ejde-203	132	12	8	8	NUM
ejde-203	132	13	,	,	PUNCT
ejde-203	132	14	and	and	CCONJ
ejde-203	132	15	the	the	DET
ejde-203	132	16	second	second	ADJ
ejde-203	132	17	claim	claim	NOUN
ejde-203	132	18	of	of	ADP
ejde-203	132	19	the	the	DET
ejde-203	132	20	lemma	lemma	PROPN
ejde-203	132	21	follows	follow	VERB
ejde-203	132	22	by	by	ADP
ejde-203	132	23	the	the	DET
ejde-203	132	24	second	second	ADJ
ejde-203	132	25	order	order	NOUN
ejde-203	132	26	taylor	taylor	NOUN
ejde-203	132	27	expansion	expansion	NOUN
ejde-203	132	28	of	of	ADP
ejde-203	132	29	τ̂(x0	τ̂(x0	PROPN
ejde-203	132	30	,	,	PUNCT
ejde-203	132	31	0	0	NUM
ejde-203	132	32	)	)	PUNCT
ejde-203	132	33	around	around	ADP
ejde-203	132	34	x0	x0	PROPN
ejde-203	132	35	�	�	PROPN
ejde-203	132	36	a	a	DET
ejde-203	132	37	reasoning	reasoning	NOUN
ejde-203	132	38	analogous	analogous	ADJ
ejde-203	132	39	to	to	ADP
ejde-203	132	40	that	that	PRON
ejde-203	132	41	in	in	ADP
ejde-203	132	42	the	the	DET
ejde-203	132	43	proof	proof	NOUN
ejde-203	132	44	of	of	ADP
ejde-203	132	45	the	the	DET
ejde-203	132	46	preceding	precede	VERB
ejde-203	132	47	lemma	lemma	PROPN
ejde-203	132	48	shows	show	VERB
ejde-203	132	49	that	that	SCONJ
ejde-203	132	50	τ(x0	τ(x0	NOUN
ejde-203	132	51	,	,	PUNCT
ejde-203	132	52	α	α	NOUN
ejde-203	132	53	)	)	PUNCT
ejde-203	132	54	>	>	X
ejde-203	133	1	2π√	2π√	PROPN
ejde-203	133	2	1−	1−	NUM
ejde-203	133	3	α2	α2	PROPN
ejde-203	133	4	≡	≡	PROPN
ejde-203	133	5	τl	τl	PROPN
ejde-203	133	6	.	.	PUNCT
ejde-203	134	1	this	this	DET
ejde-203	134	2	last	last	ADJ
ejde-203	134	3	inequality	inequality	NOUN
ejde-203	134	4	is	be	AUX
ejde-203	134	5	illustrated	illustrate	VERB
ejde-203	134	6	in	in	ADP
ejde-203	134	7	figure	figure	NOUN
ejde-203	134	8	2	2	NUM
ejde-203	134	9	when	when	SCONJ
ejde-203	134	10	a	a	DET
ejde-203	134	11	=	=	NOUN
ejde-203	134	12	1	1	X
ejde-203	134	13	.	.	PUNCT
ejde-203	135	1	if	if	SCONJ
ejde-203	135	2	we	we	PRON
ejde-203	135	3	had	have	AUX
ejde-203	135	4	considered	consider	VERB
ejde-203	135	5	in	in	ADP
ejde-203	135	6	assumption	assumption	NOUN
ejde-203	135	7	(	(	PUNCT
ejde-203	135	8	a1	a1	NOUN
ejde-203	135	9	)	)	PUNCT
ejde-203	135	10	negative	negative	ADJ
ejde-203	135	11	values	value	NOUN
ejde-203	135	12	for	for	ADP
ejde-203	135	13	a	a	PRON
ejde-203	135	14	,	,	PUNCT
ejde-203	135	15	then	then	ADV
ejde-203	135	16	the	the	DET
ejde-203	135	17	inequality	inequality	NOUN
ejde-203	135	18	would	would	AUX
ejde-203	135	19	reverse	reverse	VERB
ejde-203	135	20	to	to	ADP
ejde-203	135	21	τ(x0	τ(x0	NOUN
ejde-203	135	22	,	,	PUNCT
ejde-203	135	23	α	α	NOUN
ejde-203	135	24	)	)	PUNCT
ejde-203	135	25	<	<	X
ejde-203	136	1	τl	τl	PROPN
ejde-203	136	2	as	as	SCONJ
ejde-203	136	3	it	it	PRON
ejde-203	136	4	is	be	AUX
ejde-203	136	5	depicted	depict	VERB
ejde-203	136	6	in	in	ADP
ejde-203	136	7	figure	figure	NOUN
ejde-203	136	8	2	2	NUM
ejde-203	136	9	.	.	NOUN
ejde-203	136	10	3	3	NUM
ejde-203	136	11	.	.	X
ejde-203	136	12	role	role	NOUN
ejde-203	136	13	of	of	ADP
ejde-203	136	14	the	the	DET
ejde-203	136	15	viscous	viscous	ADJ
ejde-203	136	16	damping	damp	VERB
ejde-203	136	17	it	it	PRON
ejde-203	136	18	is	be	AUX
ejde-203	136	19	not	not	PART
ejde-203	136	20	difficult	difficult	ADJ
ejde-203	136	21	at	at	ADV
ejde-203	136	22	all	all	ADV
ejde-203	136	23	to	to	PART
ejde-203	136	24	obtain	obtain	VERB
ejde-203	136	25	a	a	DET
ejde-203	136	26	differential	differential	ADJ
ejde-203	136	27	equation	equation	NOUN
ejde-203	136	28	describing	describe	VERB
ejde-203	136	29	the	the	DET
ejde-203	136	30	movement	movement	NOUN
ejde-203	136	31	of	of	ADP
ejde-203	136	32	the	the	DET
ejde-203	136	33	pendulum	pendulum	NOUN
ejde-203	136	34	depending	depend	VERB
ejde-203	136	35	on	on	ADP
ejde-203	136	36	the	the	DET
ejde-203	136	37	viscous	viscous	ADJ
ejde-203	136	38	damping	damp	VERB
ejde-203	136	39	coefficient	coefficient	NOUN
ejde-203	136	40	.	.	PUNCT
ejde-203	137	1	indeed	indeed	ADV
ejde-203	137	2	,	,	PUNCT
ejde-203	137	3	writing	write	VERB
ejde-203	137	4	x(t	x(t	PROPN
ejde-203	137	5	,	,	PUNCT
ejde-203	137	6	x0	x0	PROPN
ejde-203	137	7	,	,	PUNCT
ejde-203	137	8	α	α	X
ejde-203	137	9	)	)	PUNCT
ejde-203	137	10	=	=	SYM
ejde-203	138	1	∂x	∂x	PROPN
ejde-203	138	2	∂α	∂α	PROPN
ejde-203	138	3	(	(	PUNCT
ejde-203	138	4	t	t	PROPN
ejde-203	138	5	,	,	PUNCT
ejde-203	138	6	x0	x0	PROPN
ejde-203	138	7	,	,	PUNCT
ejde-203	138	8	α	α	X
ejde-203	138	9	)	)	PUNCT
ejde-203	138	10	,	,	PUNCT
ejde-203	138	11	v	v	PROPN
ejde-203	138	12	(	(	PUNCT
ejde-203	138	13	t	t	PROPN
ejde-203	138	14	,	,	PUNCT
ejde-203	138	15	x0	x0	PROPN
ejde-203	138	16	,	,	PUNCT
ejde-203	138	17	α	α	X
ejde-203	138	18	)	)	PUNCT
ejde-203	138	19	=	=	SYM
ejde-203	139	1	∂v	∂v	PROPN
ejde-203	139	2	∂α	∂α	PROPN
ejde-203	139	3	(	(	PUNCT
ejde-203	139	4	t	t	PROPN
ejde-203	139	5	,	,	PUNCT
ejde-203	139	6	x0	x0	PROPN
ejde-203	139	7	,	,	PUNCT
ejde-203	139	8	α	α	X
ejde-203	139	9	)	)	PUNCT
ejde-203	139	10	.	.	PUNCT
ejde-203	140	1	differentiating	differentiate	VERB
ejde-203	140	2	(	(	PUNCT
ejde-203	140	3	2.1	2.1	NUM
ejde-203	140	4	)	)	PUNCT
ejde-203	140	5	with	with	ADP
ejde-203	140	6	respect	respect	NOUN
ejde-203	140	7	to	to	ADP
ejde-203	140	8	α	α	PROPN
ejde-203	140	9	yields	yield	NOUN
ejde-203	140	10	ẋ	ẋ	PUNCT
ejde-203	141	1	=	=	PRON
ejde-203	141	2	v	v	ADP
ejde-203	141	3	v̇	v̇	NOUN
ejde-203	142	1	=	=	PUNCT
ejde-203	143	1	−2αv	−2αv	NOUN
ejde-203	143	2	−x	−x	VERB
ejde-203	143	3	−	−	PROPN
ejde-203	143	4	2v	2v	PROPN
ejde-203	143	5	−	−	PROPN
ejde-203	144	1	(	(	PUNCT
ejde-203	144	2	xf	xf	PROPN
ejde-203	144	3	′(x	′(x	PROPN
ejde-203	144	4	)	)	PUNCT
ejde-203	145	1	+	+	CCONJ
ejde-203	145	2	f(x))x	f(x))x	PROPN
ejde-203	145	3	.	.	PUNCT
ejde-203	146	1	(	(	PUNCT
ejde-203	146	2	3.1	3.1	NUM
ejde-203	146	3	)	)	PUNCT
ejde-203	146	4	as	as	ADP
ejde-203	146	5	for	for	ADP
ejde-203	146	6	the	the	DET
ejde-203	146	7	initial	initial	ADJ
ejde-203	146	8	conditions	condition	NOUN
ejde-203	146	9	we	we	PRON
ejde-203	146	10	have	have	VERB
ejde-203	146	11	x(0	x(0	PROPN
ejde-203	146	12	,	,	PUNCT
ejde-203	146	13	x0	x0	PROPN
ejde-203	146	14	,	,	PUNCT
ejde-203	146	15	α	α	X
ejde-203	146	16	)	)	PUNCT
ejde-203	146	17	=	=	SYM
ejde-203	146	18	0	0	NUM
ejde-203	146	19	,	,	PUNCT
ejde-203	146	20	v	v	NOUN
ejde-203	146	21	(	(	PUNCT
ejde-203	146	22	0	0	NUM
ejde-203	146	23	,	,	PUNCT
ejde-203	146	24	x0	x0	PROPN
ejde-203	146	25	,	,	PUNCT
ejde-203	146	26	α	α	X
ejde-203	146	27	)	)	PUNCT
ejde-203	146	28	=	=	SYM
ejde-203	147	1	0	0	X
ejde-203	147	2	.	.	PUNCT
ejde-203	148	1	let	let	VERB
ejde-203	148	2	us	we	PRON
ejde-203	148	3	write	write	VERB
ejde-203	148	4	g(x	g(x	NOUN
ejde-203	148	5	)	)	PUNCT
ejde-203	149	1	=	=	SYM
ejde-203	150	1	−	−	PROPN
ejde-203	150	2	d	d	X
ejde-203	150	3	dx	dx	PROPN
ejde-203	150	4	(	(	PUNCT
ejde-203	150	5	xf(x	xf(x	PROPN
ejde-203	150	6	)	)	PUNCT
ejde-203	150	7	)	)	PUNCT
ejde-203	150	8	.	.	PUNCT
ejde-203	151	1	again	again	ADV
ejde-203	151	2	,	,	PUNCT
ejde-203	151	3	as	as	SCONJ
ejde-203	151	4	we	we	PRON
ejde-203	151	5	did	do	VERB
ejde-203	151	6	with	with	ADP
ejde-203	151	7	equation	equation	NOUN
ejde-203	151	8	(	(	PUNCT
ejde-203	151	9	2.1	2.1	NUM
ejde-203	151	10	)	)	PUNCT
ejde-203	151	11	,	,	PUNCT
ejde-203	151	12	equation	equation	NOUN
ejde-203	151	13	(	(	PUNCT
ejde-203	151	14	3.1	3.1	NUM
ejde-203	151	15	)	)	PUNCT
ejde-203	151	16	can	can	AUX
ejde-203	151	17	be	be	AUX
ejde-203	151	18	seen	see	VERB
ejde-203	151	19	as	as	ADP
ejde-203	151	20	a	a	DET
ejde-203	151	21	linear	linear	ADJ
ejde-203	151	22	homogeneous	homogeneous	ADJ
ejde-203	151	23	part	part	NOUN
ejde-203	151	24	plus	plus	CCONJ
ejde-203	151	25	the	the	DET
ejde-203	151	26	forcing	force	VERB
ejde-203	151	27	term	term	NOUN
ejde-203	151	28	−2v+g(x)x	−2v+g(x)x	NOUN
ejde-203	151	29	.	.	PUNCT
ejde-203	152	1	the	the	DET
ejde-203	152	2	solution	solution	NOUN
ejde-203	152	3	x	x	X
ejde-203	152	4	,	,	PUNCT
ejde-203	152	5	v	v	NOUN
ejde-203	152	6	is	be	AUX
ejde-203	152	7	implicitly	implicitly	ADV
ejde-203	152	8	given	give	VERB
ejde-203	152	9	by	by	ADP
ejde-203	152	10	x(t	x(t	PROPN
ejde-203	152	11	)	)	PUNCT
ejde-203	152	12	=	=	SYM
ejde-203	152	13	1	1	NUM
ejde-203	152	14	ω	ω	NUM
ejde-203	152	15	∫	∫	PROPN
ejde-203	152	16	t	t	PROPN
ejde-203	152	17	0	0	NUM
ejde-203	152	18	e−α(t−s	e−α(t−s	PROPN
ejde-203	152	19	)	)	PUNCT
ejde-203	152	20	sinω(t−	sinω(t−	NOUN
ejde-203	152	21	s	s	PART
ejde-203	152	22	)	)	PUNCT
ejde-203	152	23	{	{	PUNCT
ejde-203	152	24	−	−	PROPN
ejde-203	152	25	2v(s	2v(s	NUM
ejde-203	152	26	)	)	PUNCT
ejde-203	153	1	+	+	NOUN
ejde-203	153	2	g(x(s))x(s	g(x(s))x(s	NOUN
ejde-203	153	3	)	)	PUNCT
ejde-203	153	4	}	}	PUNCT
ejde-203	153	5	ds	ds	ADP
ejde-203	153	6	v	v	NOUN
ejde-203	153	7	(	(	PUNCT
ejde-203	153	8	t	t	NOUN
ejde-203	153	9	)	)	PUNCT
ejde-203	153	10	=	=	SYM
ejde-203	153	11	1	1	NUM
ejde-203	153	12	ω	ω	NUM
ejde-203	153	13	∫	∫	PROPN
ejde-203	153	14	t	t	PROPN
ejde-203	153	15	0	0	PUNCT
ejde-203	154	1	e−α(t−s)(ω	e−α(t−s)(ω	NOUN
ejde-203	154	2	cosω(t−	cosω(t−	NOUN
ejde-203	154	3	s	s	X
ejde-203	154	4	)	)	PUNCT
ejde-203	155	1	−	−	PROPN
ejde-203	155	2	α	α	NUM
ejde-203	155	3	sinω(t−	sinω(t−	NOUN
ejde-203	155	4	s	s	PART
ejde-203	155	5	)	)	PUNCT
ejde-203	155	6	)	)	PUNCT
ejde-203	155	7	{	{	PUNCT
ejde-203	156	1	−	−	PROPN
ejde-203	156	2	2v(s	2v(s	NUM
ejde-203	156	3	)	)	PUNCT
ejde-203	157	1	+	+	NOUN
ejde-203	157	2	g(x(s))x(s	g(x(s))x(s	NOUN
ejde-203	157	3	)	)	PUNCT
ejde-203	157	4	}	}	PUNCT
ejde-203	157	5	ds	ds	VERB
ejde-203	157	6	in	in	ADP
ejde-203	157	7	particular	particular	ADJ
ejde-203	157	8	,	,	PUNCT
ejde-203	157	9	for	for	ADP
ejde-203	157	10	α	α	NOUN
ejde-203	157	11	=	=	SYM
ejde-203	157	12	0	0	PROPN
ejde-203	157	13	the	the	DET
ejde-203	157	14	above	above	ADJ
ejde-203	157	15	expressions	expression	NOUN
ejde-203	157	16	reduce	reduce	VERB
ejde-203	157	17	to	to	ADP
ejde-203	157	18	x(t	x(t	NOUN
ejde-203	157	19	)	)	PUNCT
ejde-203	158	1	=	=	SYM
ejde-203	159	1	∫	∫	PROPN
ejde-203	159	2	t	t	NOUN
ejde-203	159	3	0	0	NUM
ejde-203	160	1	sin(t−	sin(t−	PROPN
ejde-203	160	2	s	s	X
ejde-203	160	3	)	)	PUNCT
ejde-203	160	4	{	{	PUNCT
ejde-203	160	5	−	−	PROPN
ejde-203	160	6	2v(s	2v(s	NUM
ejde-203	160	7	)	)	PUNCT
ejde-203	161	1	+	+	NOUN
ejde-203	161	2	g(x(s))x(s	g(x(s))x(s	NOUN
ejde-203	161	3	)	)	PUNCT
ejde-203	161	4	}	}	PUNCT
ejde-203	161	5	ds	ds	ADP
ejde-203	161	6	v	v	NOUN
ejde-203	161	7	(	(	PUNCT
ejde-203	161	8	t	t	NOUN
ejde-203	161	9	)	)	PUNCT
ejde-203	161	10	=	=	SYM
ejde-203	162	1	∫	∫	PROPN
ejde-203	162	2	t	t	PROPN
ejde-203	162	3	0	0	NUM
ejde-203	163	1	cos	cos	PROPN
ejde-203	163	2	(	(	PUNCT
ejde-203	163	3	t−	t−	PROPN
ejde-203	163	4	s	s	X
ejde-203	163	5	)	)	PUNCT
ejde-203	163	6	{	{	PUNCT
ejde-203	163	7	−	−	PROPN
ejde-203	163	8	2v(s	2v(s	NUM
ejde-203	163	9	)	)	PUNCT
ejde-203	164	1	+	+	NOUN
ejde-203	164	2	g(x(s))x(s	g(x(s))x(s	NOUN
ejde-203	164	3	)	)	PUNCT
ejde-203	164	4	}	}	PUNCT
ejde-203	164	5	ds	ds	X
ejde-203	164	6	(	(	PUNCT
ejde-203	164	7	3.2	3.2	NUM
ejde-203	164	8	)	)	PUNCT
ejde-203	164	9	the	the	DET
ejde-203	164	10	following	follow	VERB
ejde-203	164	11	lemma	lemma	PROPN
ejde-203	164	12	plays	play	VERB
ejde-203	164	13	a	a	DET
ejde-203	164	14	major	major	ADJ
ejde-203	164	15	role	role	NOUN
ejde-203	164	16	in	in	ADP
ejde-203	164	17	the	the	DET
ejde-203	164	18	main	main	ADJ
ejde-203	164	19	result	result	NOUN
ejde-203	164	20	of	of	ADP
ejde-203	164	21	the	the	DET
ejde-203	164	22	paper	paper	NOUN
ejde-203	164	23	.	.	PUNCT
ejde-203	165	1	18	18	NUM
ejde-203	165	2	j.	j.	PROPN
ejde-203	165	3	arango	arango	PROPN
ejde-203	165	4	ejde	ejde	PROPN
ejde-203	165	5	/	/	SYM
ejde-203	165	6	si/01	si/01	PROPN
ejde-203	165	7	lemma	lemma	PROPN
ejde-203	165	8	3.1	3.1	NUM
ejde-203	165	9	.	.	PUNCT
ejde-203	166	1	under	under	ADP
ejde-203	166	2	assumption	assumption	NOUN
ejde-203	166	3	(	(	PUNCT
ejde-203	166	4	a1	a1	NOUN
ejde-203	166	5	)	)	PUNCT
ejde-203	166	6	,	,	PUNCT
ejde-203	166	7	if	if	SCONJ
ejde-203	166	8	τ̂	τ̂	ADP
ejde-203	166	9	=	=	SYM
ejde-203	166	10	τ̂(x0	τ̂(x0	NOUN
ejde-203	166	11	,	,	PUNCT
ejde-203	166	12	0	0	NUM
ejde-203	166	13	)	)	PUNCT
ejde-203	166	14	denotes	denote	VERB
ejde-203	166	15	the	the	DET
ejde-203	166	16	half	half	ADJ
ejde-203	166	17	oscillation	oscillation	NOUN
ejde-203	166	18	time	time	NOUN
ejde-203	166	19	when	when	SCONJ
ejde-203	166	20	α	α	PROPN
ejde-203	166	21	=	=	SYM
ejde-203	166	22	0	0	NUM
ejde-203	166	23	,	,	PUNCT
ejde-203	166	24	then	then	ADV
ejde-203	166	25	for	for	ADP
ejde-203	166	26	0	0	NUM
ejde-203	166	27	<	<	X
ejde-203	166	28	x0	x0	PROPN
ejde-203	166	29	�	�	PROPN
ejde-203	166	30	1	1	NUM
ejde-203	166	31	we	we	PRON
ejde-203	166	32	have	have	VERB
ejde-203	166	33	v	v	NOUN
ejde-203	166	34	(	(	PUNCT
ejde-203	166	35	τ̂	τ̂	NUM
ejde-203	166	36	,	,	PUNCT
ejde-203	166	37	x0	x0	PROPN
ejde-203	166	38	,	,	PUNCT
ejde-203	166	39	0	0	NUM
ejde-203	166	40	)	)	PUNCT
ejde-203	166	41	>	>	X
ejde-203	167	1	0	0	X
ejde-203	167	2	.	.	PUNCT
ejde-203	168	1	proof	proof	NOUN
ejde-203	168	2	.	.	PUNCT
ejde-203	169	1	we	we	PRON
ejde-203	169	2	start	start	VERB
ejde-203	169	3	with	with	ADP
ejde-203	169	4	an	an	DET
ejde-203	169	5	auxiliary	auxiliary	ADJ
ejde-203	169	6	estimate	estimate	NOUN
ejde-203	169	7	for	for	ADP
ejde-203	169	8	x(t	x(t	PROPN
ejde-203	169	9	)	)	PUNCT
ejde-203	169	10	in	in	ADP
ejde-203	169	11	equation	equation	NOUN
ejde-203	169	12	(	(	PUNCT
ejde-203	169	13	3.2	3.2	NUM
ejde-203	169	14	)	)	PUNCT
ejde-203	169	15	.	.	PUNCT
ejde-203	170	1	by	by	ADP
ejde-203	170	2	lemma	lemma	PROPN
ejde-203	170	3	(	(	PUNCT
ejde-203	170	4	2.1	2.1	NUM
ejde-203	170	5	)	)	PUNCT
ejde-203	170	6	and	and	CCONJ
ejde-203	170	7	by	by	ADP
ejde-203	170	8	assumption	assumption	NOUN
ejde-203	170	9	(	(	PUNCT
ejde-203	170	10	a1	a1	NOUN
ejde-203	170	11	)	)	PUNCT
ejde-203	170	12	,	,	PUNCT
ejde-203	170	13	for	for	ADP
ejde-203	170	14	0	0	NUM
ejde-203	170	15	<	<	X
ejde-203	170	16	t	t	PROPN
ejde-203	170	17	≤	≤	NUM
ejde-203	170	18	π	π	NOUN
ejde-203	170	19	we	we	PRON
ejde-203	170	20	have	have	VERB
ejde-203	170	21	x(t	x(t	PROPN
ejde-203	170	22	)	)	PUNCT
ejde-203	170	23	=	=	SYM
ejde-203	171	1	x0(−t	x0(−t	PUNCT
ejde-203	171	2	cos	cos	X
ejde-203	171	3	t+	t+	PROPN
ejde-203	171	4	sin	sin	PROPN
ejde-203	171	5	t	t	PROPN
ejde-203	171	6	)	)	PUNCT
ejde-203	171	7	+	+	CCONJ
ejde-203	171	8	3ax20	3ax20	NUM
ejde-203	171	9	∫	∫	NOUN
ejde-203	171	10	t	t	NOUN
ejde-203	171	11	0	0	NUM
ejde-203	172	1	sin(t−	sin(t−	PROPN
ejde-203	172	2	s	s	PROPN
ejde-203	172	3	)	)	PUNCT
ejde-203	172	4	cos2	cos2	NOUN
ejde-203	172	5	sx(s	sx(s	PROPN
ejde-203	172	6	)	)	PUNCT
ejde-203	172	7	ds+o(|x0|4	ds+o(|x0|4	NUM
ejde-203	172	8	)	)	PUNCT
ejde-203	172	9	(	(	PUNCT
ejde-203	172	10	3.3	3.3	NUM
ejde-203	172	11	)	)	PUNCT
ejde-203	172	12	notice	notice	VERB
ejde-203	172	13	that	that	SCONJ
ejde-203	172	14	x1(t	x1(t	X
ejde-203	172	15	)	)	PUNCT
ejde-203	172	16	≡	≡	PROPN
ejde-203	172	17	x0(−t	x0(−t	PUNCT
ejde-203	173	1	cos	cos	PROPN
ejde-203	173	2	t+sin	t+sin	PROPN
ejde-203	173	3	t	t	PROPN
ejde-203	173	4	)	)	PUNCT
ejde-203	173	5	does	do	AUX
ejde-203	173	6	not	not	PART
ejde-203	173	7	vanish	vanish	VERB
ejde-203	173	8	on	on	ADP
ejde-203	173	9	(	(	PUNCT
ejde-203	173	10	0	0	NUM
ejde-203	173	11	,	,	PUNCT
ejde-203	173	12	π	π	NOUN
ejde-203	173	13	)	)	PUNCT
ejde-203	173	14	and	and	CCONJ
ejde-203	173	15	that	that	DET
ejde-203	173	16	g(x(s	g(x(s	NOUN
ejde-203	173	17	)	)	PUNCT
ejde-203	173	18	)	)	PUNCT
ejde-203	174	1	>	>	X
ejde-203	174	2	0	0	PUNCT
ejde-203	174	3	provided	provide	VERB
ejde-203	174	4	0	0	NUM
ejde-203	174	5	<	<	X
ejde-203	174	6	x0	x0	PROPN
ejde-203	174	7	�	�	PROPN
ejde-203	174	8	1	1	NUM
ejde-203	174	9	.	.	PUNCT
ejde-203	174	10	further	far	ADV
ejde-203	174	11	,	,	PUNCT
ejde-203	174	12	the	the	DET
ejde-203	174	13	initial	initial	ADJ
ejde-203	174	14	conditions	condition	NOUN
ejde-203	174	15	forx(t	forx(t	NOUN
ejde-203	174	16	)	)	PUNCT
ejde-203	174	17	at	at	ADP
ejde-203	174	18	t	t	NOUN
ejde-203	174	19	=	=	SYM
ejde-203	174	20	0	0	NUM
ejde-203	174	21	and	and	CCONJ
ejde-203	174	22	equation	equation	NOUN
ejde-203	174	23	(	(	PUNCT
ejde-203	174	24	3.1	3.1	NUM
ejde-203	174	25	)	)	PUNCT
ejde-203	174	26	yield	yield	NOUN
ejde-203	174	27	that	that	PRON
ejde-203	174	28	x(0	x(0	PROPN
ejde-203	174	29	)	)	PUNCT
ejde-203	175	1	=	=	SYM
ejde-203	175	2	0	0	NUM
ejde-203	175	3	=	=	SYM
ejde-203	175	4	ẋ(0	ẋ(0	X
ejde-203	175	5	)	)	PUNCT
ejde-203	175	6	=	=	SYM
ejde-203	175	7	ẍ(0	ẍ(0	PROPN
ejde-203	175	8	)	)	PUNCT
ejde-203	175	9	and	and	CCONJ
ejde-203	175	10	...	...	PUNCT
ejde-203	175	11	x(0	x(0	PROPN
ejde-203	175	12	)	)	PUNCT
ejde-203	175	13	=	=	SYM
ejde-203	176	1	2x0(1	2x0(1	NUM
ejde-203	176	2	+	+	NUM
ejde-203	176	3	f(x0	f(x0	NOUN
ejde-203	176	4	)	)	PUNCT
ejde-203	176	5	)	)	PUNCT
ejde-203	177	1	>	>	X
ejde-203	177	2	0	0	NUM
ejde-203	177	3	,	,	PUNCT
ejde-203	177	4	meaning	mean	VERB
ejde-203	177	5	that	that	SCONJ
ejde-203	177	6	x(t	x(t	PROPN
ejde-203	177	7	)	)	PUNCT
ejde-203	177	8	is	be	AUX
ejde-203	177	9	positive	positive	ADJ
ejde-203	177	10	on	on	ADP
ejde-203	177	11	an	an	DET
ejde-203	177	12	interval	interval	NOUN
ejde-203	177	13	(	(	PUNCT
ejde-203	177	14	0	0	NUM
ejde-203	177	15	,	,	PUNCT
ejde-203	177	16	ε	ε	PROPN
ejde-203	177	17	)	)	PUNCT
ejde-203	177	18	with	with	ADP
ejde-203	177	19	ε	ε	PROPN
ejde-203	177	20	>	>	X
ejde-203	177	21	0	0	PROPN
ejde-203	177	22	.	.	PUNCT
ejde-203	178	1	we	we	PRON
ejde-203	178	2	claim	claim	VERB
ejde-203	178	3	that	that	SCONJ
ejde-203	178	4	x(t	x(t	PROPN
ejde-203	178	5	)	)	PUNCT
ejde-203	178	6	>	>	X
ejde-203	178	7	0	0	PUNCT
ejde-203	179	1	for	for	ADP
ejde-203	179	2	0	0	NUM
ejde-203	179	3	<	<	X
ejde-203	179	4	t	t	PROPN
ejde-203	179	5	≤	≤	NUM
ejde-203	179	6	π	π	X
ejde-203	179	7	.	.	PUNCT
ejde-203	180	1	on	on	ADP
ejde-203	180	2	the	the	DET
ejde-203	180	3	contrary	contrary	NOUN
ejde-203	180	4	,	,	PUNCT
ejde-203	180	5	there	there	PRON
ejde-203	180	6	exists	exist	VERB
ejde-203	180	7	ε	ε	PROPN
ejde-203	180	8	<	<	X
ejde-203	180	9	t0	t0	X
ejde-203	180	10	<	<	X
ejde-203	180	11	π	π	PROPN
ejde-203	180	12	such	such	ADJ
ejde-203	180	13	that	that	DET
ejde-203	180	14	x(t0	x(t0	NOUN
ejde-203	180	15	)	)	PUNCT
ejde-203	181	1	=	=	SYM
ejde-203	181	2	0	0	NUM
ejde-203	181	3	and	and	CCONJ
ejde-203	181	4	x(t	x(t	PROPN
ejde-203	181	5	)	)	PUNCT
ejde-203	181	6	>	>	X
ejde-203	181	7	0	0	PUNCT
ejde-203	182	1	for	for	ADP
ejde-203	182	2	t	t	PROPN
ejde-203	182	3	∈	∈	PROPN
ejde-203	182	4	(	(	PUNCT
ejde-203	182	5	0	0	NUM
ejde-203	182	6	,	,	PUNCT
ejde-203	182	7	t0	t0	NOUN
ejde-203	182	8	)	)	PUNCT
ejde-203	182	9	.	.	PUNCT
ejde-203	183	1	now	now	ADV
ejde-203	183	2	,	,	PUNCT
ejde-203	183	3	by	by	ADP
ejde-203	183	4	lemma	lemma	PROPN
ejde-203	183	5	2.2	2.2	NUM
ejde-203	183	6	we	we	PRON
ejde-203	183	7	know	know	VERB
ejde-203	183	8	that	that	PRON
ejde-203	183	9	τ̂	τ̂	PUNCT
ejde-203	183	10	>	>	X
ejde-203	184	1	π	π	X
ejde-203	184	2	.	.	PUNCT
ejde-203	185	1	therefore	therefore	ADV
ejde-203	185	2	,	,	PUNCT
ejde-203	185	3	the	the	DET
ejde-203	185	4	polar	polar	ADJ
ejde-203	185	5	angle	angle	NOUN
ejde-203	185	6	θ(t	θ(t	PROPN
ejde-203	185	7	)	)	PUNCT
ejde-203	185	8	in	in	ADP
ejde-203	185	9	(	(	PUNCT
ejde-203	185	10	2.6	2.6	NUM
ejde-203	185	11	)	)	PUNCT
ejde-203	185	12	satisfies	satisfie	NOUN
ejde-203	185	13	−π	−π	ADV
ejde-203	185	14	<	<	X
ejde-203	185	15	θ(t	θ(t	PROPN
ejde-203	185	16	)	)	PUNCT
ejde-203	185	17	<	<	X
ejde-203	185	18	0	0	NUM
ejde-203	185	19	for	for	ADP
ejde-203	185	20	all	all	PRON
ejde-203	185	21	0	0	NUM
ejde-203	185	22	<	<	X
ejde-203	185	23	t	t	X
ejde-203	185	24	<	<	X
ejde-203	185	25	π	π	X
ejde-203	185	26	and	and	CCONJ
ejde-203	185	27	a	a	DET
ejde-203	185	28	fortiori	fortiori	X
ejde-203	185	29	v(t	v(t	NOUN
ejde-203	185	30	)	)	PUNCT
ejde-203	185	31	<	<	X
ejde-203	185	32	0	0	PUNCT
ejde-203	185	33	on	on	ADP
ejde-203	185	34	(	(	PUNCT
ejde-203	185	35	0	0	NUM
ejde-203	185	36	,	,	PUNCT
ejde-203	185	37	π	π	NOUN
ejde-203	185	38	]	]	X
ejde-203	185	39	.	.	PUNCT
ejde-203	186	1	but	but	CCONJ
ejde-203	186	2	this	this	PRON
ejde-203	186	3	is	be	AUX
ejde-203	186	4	a	a	DET
ejde-203	186	5	contradiction	contradiction	NOUN
ejde-203	186	6	to	to	ADP
ejde-203	186	7	the	the	DET
ejde-203	186	8	first	first	ADJ
ejde-203	186	9	equation	equation	NOUN
ejde-203	186	10	of	of	ADP
ejde-203	186	11	(	(	PUNCT
ejde-203	186	12	3.2	3.2	NUM
ejde-203	186	13	)	)	PUNCT
ejde-203	186	14	evaluated	evaluate	VERB
ejde-203	186	15	at	at	ADP
ejde-203	186	16	t	t	NOUN
ejde-203	186	17	=	=	SYM
ejde-203	186	18	t0	t0	PROPN
ejde-203	186	19	since	since	SCONJ
ejde-203	186	20	for	for	ADP
ejde-203	186	21	s	s	PROPN
ejde-203	186	22	∈	∈	PROPN
ejde-203	186	23	(	(	PUNCT
ejde-203	186	24	0	0	NUM
ejde-203	186	25	,	,	PUNCT
ejde-203	186	26	t0	t0	PROPN
ejde-203	186	27	)	)	PUNCT
ejde-203	186	28	we	we	PRON
ejde-203	186	29	have	have	VERB
ejde-203	186	30	sin(t0	sin(t0	INTJ
ejde-203	186	31	−	−	NUM
ejde-203	186	32	s	s	NOUN
ejde-203	186	33	)	)	PUNCT
ejde-203	186	34	{	{	PUNCT
ejde-203	186	35	−	−	PROPN
ejde-203	186	36	2v(s	2v(s	NUM
ejde-203	186	37	)	)	PUNCT
ejde-203	187	1	+	+	NOUN
ejde-203	187	2	g(x(s))x(s	g(x(s))x(s	NOUN
ejde-203	187	3	)	)	PUNCT
ejde-203	187	4	}	}	PUNCT
ejde-203	188	1	>	>	X
ejde-203	188	2	0	0	X
ejde-203	188	3	.	.	PUNCT
ejde-203	189	1	next	next	ADV
ejde-203	189	2	,	,	PUNCT
ejde-203	189	3	by	by	ADP
ejde-203	189	4	(	(	PUNCT
ejde-203	189	5	3.3	3.3	NUM
ejde-203	189	6	)	)	PUNCT
ejde-203	189	7	it	it	PRON
ejde-203	189	8	follows	follow	VERB
ejde-203	189	9	immediately	immediately	ADV
ejde-203	189	10	that	that	SCONJ
ejde-203	189	11	x(t	x(t	PROPN
ejde-203	189	12	)	)	PUNCT
ejde-203	189	13	=	=	PUNCT
ejde-203	189	14	x1(t)+o(|x0|3	x1(t)+o(|x0|3	NUM
ejde-203	189	15	)	)	PUNCT
ejde-203	189	16	.	.	PUNCT
ejde-203	190	1	analogously	analogously	ADV
ejde-203	190	2	,	,	PUNCT
ejde-203	190	3	for	for	ADP
ejde-203	190	4	v	v	PROPN
ejde-203	190	5	(	(	PUNCT
ejde-203	190	6	t	t	NOUN
ejde-203	190	7	)	)	PUNCT
ejde-203	190	8	we	we	PRON
ejde-203	190	9	obtain	obtain	VERB
ejde-203	190	10	v	v	ADP
ejde-203	190	11	(	(	PUNCT
ejde-203	190	12	t	t	NOUN
ejde-203	190	13	)	)	PUNCT
ejde-203	191	1	=	=	NOUN
ejde-203	191	2	x0	x0	PROPN
ejde-203	191	3	t	t	PROPN
ejde-203	191	4	sin	sin	NOUN
ejde-203	191	5	t+	t+	PUNCT
ejde-203	191	6	3ax20	3ax20	NUM
ejde-203	191	7	∫	∫	NOUN
ejde-203	191	8	t	t	NOUN
ejde-203	191	9	0	0	NUM
ejde-203	192	1	cos(t−	cos(t−	PROPN
ejde-203	192	2	s	s	X
ejde-203	192	3	)	)	PUNCT
ejde-203	192	4	cos2	cos2	PROPN
ejde-203	192	5	sx1(s	sx1(s	PROPN
ejde-203	192	6	)	)	PUNCT
ejde-203	192	7	ds+o(|x0|4	ds+o(|x0|4	NOUN
ejde-203	192	8	)	)	PUNCT
ejde-203	192	9	≡v1(t	≡v1(t	PROPN
ejde-203	192	10	)	)	PUNCT
ejde-203	193	1	+	+	PUNCT
ejde-203	194	1	v2(t	v2(t	X
ejde-203	194	2	)	)	PUNCT
ejde-203	195	1	+	+	NOUN
ejde-203	195	2	o(|x0|4	o(|x0|4	X
ejde-203	195	3	)	)	PUNCT
ejde-203	195	4	where	where	SCONJ
ejde-203	195	5	v1(t	v1(t	X
ejde-203	195	6	)	)	PUNCT
ejde-203	195	7	≡	≡	PROPN
ejde-203	195	8	x0	x0	PROPN
ejde-203	195	9	t	t	PROPN
ejde-203	195	10	sin	sin	NOUN
ejde-203	195	11	t.	t.	PROPN
ejde-203	195	12	now	now	ADV
ejde-203	195	13	,	,	PUNCT
ejde-203	195	14	v2(t	v2(t	X
ejde-203	195	15	)	)	PUNCT
ejde-203	195	16	can	can	AUX
ejde-203	195	17	be	be	AUX
ejde-203	195	18	explicitly	explicitly	ADV
ejde-203	195	19	evaluated	evaluate	VERB
ejde-203	195	20	.	.	PUNCT
ejde-203	196	1	for	for	ADP
ejde-203	196	2	the	the	DET
ejde-203	196	3	reader	reader	NOUN
ejde-203	196	4	’s	’s	PART
ejde-203	196	5	convenience	convenience	NOUN
ejde-203	196	6	,	,	PUNCT
ejde-203	196	7	we	we	PRON
ejde-203	196	8	write	write	VERB
ejde-203	196	9	the	the	DET
ejde-203	196	10	complete	complete	ADJ
ejde-203	196	11	expression	expression	NOUN
ejde-203	196	12	for	for	ADP
ejde-203	196	13	v2	v2	NOUN
ejde-203	196	14	,	,	PUNCT
ejde-203	196	15	v2(t	v2(t	NOUN
ejde-203	196	16	)	)	PUNCT
ejde-203	197	1	=	=	NOUN
ejde-203	197	2	3a	3a	NUM
ejde-203	197	3	x30	x30	NUM
ejde-203	198	1	(	(	PUNCT
ejde-203	198	2	−	−	PROPN
ejde-203	198	3	1	1	NUM
ejde-203	198	4	32	32	NUM
ejde-203	198	5	(	(	PUNCT
ejde-203	198	6	6	6	NUM
ejde-203	198	7	t2	t2	NOUN
ejde-203	198	8	+	+	CCONJ
ejde-203	198	9	5	5	NUM
ejde-203	198	10	)	)	PUNCT
ejde-203	198	11	cos	cos	ADP
ejde-203	198	12	t−	t−	PROPN
ejde-203	198	13	3	3	NUM
ejde-203	198	14	32	32	NUM
ejde-203	198	15	t	t	PROPN
ejde-203	198	16	sin	sin	NOUN
ejde-203	198	17	3	3	NUM
ejde-203	198	18	t−	t−	PROPN
ejde-203	198	19	1	1	NUM
ejde-203	198	20	16	16	NUM
ejde-203	198	21	t	t	NOUN
ejde-203	198	22	sin	sin	NOUN
ejde-203	198	23	t−	t−	PROPN
ejde-203	198	24	17	17	NUM
ejde-203	198	25	128	128	NUM
ejde-203	198	26	cos	co	NOUN
ejde-203	198	27	3	3	NUM
ejde-203	198	28	t+	t+	NUM
ejde-203	198	29	37	37	NUM
ejde-203	198	30	128	128	NUM
ejde-203	198	31	cos	cos	PROPN
ejde-203	198	32	t	t	PROPN
ejde-203	198	33	)	)	PUNCT
ejde-203	198	34	.	.	PUNCT
ejde-203	199	1	moreover	moreover	ADV
ejde-203	199	2	,	,	PUNCT
ejde-203	199	3	it	it	PRON
ejde-203	199	4	is	be	AUX
ejde-203	199	5	somewhat	somewhat	ADV
ejde-203	199	6	tedious	tedious	ADJ
ejde-203	199	7	but	but	CCONJ
ejde-203	199	8	straightforward	straightforward	ADJ
ejde-203	199	9	to	to	PART
ejde-203	199	10	show	show	VERB
ejde-203	199	11	that	that	SCONJ
ejde-203	199	12	v2	v2	PROPN
ejde-203	199	13	is	be	AUX
ejde-203	199	14	positive	positive	ADJ
ejde-203	199	15	and	and	CCONJ
ejde-203	199	16	increasing	increase	VERB
ejde-203	199	17	on	on	ADP
ejde-203	199	18	a	a	DET
ejde-203	199	19	small	small	ADJ
ejde-203	199	20	neighborhood	neighborhood	NOUN
ejde-203	199	21	of	of	ADP
ejde-203	199	22	π	π	PROPN
ejde-203	199	23	.	.	PUNCT
ejde-203	200	1	by	by	ADP
ejde-203	200	2	lemma	lemma	PROPN
ejde-203	200	3	2.2	2.2	NUM
ejde-203	200	4	τ̂	τ̂	PUNCT
ejde-203	200	5	>	>	PUNCT
ejde-203	200	6	π	π	PROPN
ejde-203	200	7	,	,	PUNCT
ejde-203	200	8	therefore	therefore	ADV
ejde-203	200	9	v2(τ̂	v2(τ̂	ADJ
ejde-203	200	10	)	)	PUNCT
ejde-203	200	11	>	>	PUNCT
ejde-203	200	12	v2(π	v2(π	NOUN
ejde-203	200	13	)	)	PUNCT
ejde-203	200	14	=	=	NOUN
ejde-203	200	15	9ax30π	9ax30π	NOUN
ejde-203	200	16	2	2	NUM
ejde-203	200	17	16	16	NUM
ejde-203	200	18	.	.	PUNCT
ejde-203	201	1	again	again	ADV
ejde-203	201	2	,	,	PUNCT
ejde-203	201	3	by	by	ADP
ejde-203	201	4	lemma	lemma	PROPN
ejde-203	201	5	2.2	2.2	NUM
ejde-203	201	6	we	we	PRON
ejde-203	201	7	obtain	obtain	VERB
ejde-203	201	8	v1(τ̂	v1(τ̂	NOUN
ejde-203	201	9	)	)	PUNCT
ejde-203	201	10	=	=	SYM
ejde-203	201	11	v1(π	v1(π	PROPN
ejde-203	201	12	)	)	PUNCT
ejde-203	201	13	+	+	CCONJ
ejde-203	201	14	(	(	PUNCT
ejde-203	201	15	τ̂	τ̂	X
ejde-203	201	16	−	−	NOUN
ejde-203	201	17	π)v	π)v	X
ejde-203	201	18	′1(π	′1(π	PUNCT
ejde-203	201	19	)	)	PUNCT
ejde-203	201	20	+	+	NOUN
ejde-203	201	21	o(|x0|4	o(|x0|4	X
ejde-203	201	22	)	)	PUNCT
ejde-203	202	1	=	=	NOUN
ejde-203	202	2	−	−	PROPN
ejde-203	202	3	a	a	DET
ejde-203	202	4	x30	x30	NOUN
ejde-203	202	5	π	π	NOUN
ejde-203	202	6	2	2	NUM
ejde-203	202	7	8	8	NUM
ejde-203	202	8	+	+	SYM
ejde-203	202	9	o(|x0|4	o(|x0|4	NOUN
ejde-203	202	10	)	)	PUNCT
ejde-203	202	11	,	,	PUNCT
ejde-203	202	12	so	so	SCONJ
ejde-203	202	13	that	that	SCONJ
ejde-203	202	14	v	v	NOUN
ejde-203	202	15	(	(	PUNCT
ejde-203	202	16	τ̂	τ̂	NOUN
ejde-203	202	17	)	)	PUNCT
ejde-203	202	18	=	=	SYM
ejde-203	202	19	v1(τ̂	v1(τ̂	PROPN
ejde-203	202	20	)	)	PUNCT
ejde-203	202	21	+	+	SYM
ejde-203	202	22	v2(τ̂	v2(τ̂	ADJ
ejde-203	202	23	)	)	PUNCT
ejde-203	202	24	>	>	X
ejde-203	202	25	0	0	X
ejde-203	202	26	.	.	PUNCT
ejde-203	202	27	�	�	PROPN
ejde-203	202	28	now	now	ADV
ejde-203	202	29	we	we	PRON
ejde-203	202	30	are	be	AUX
ejde-203	202	31	in	in	ADP
ejde-203	202	32	a	a	DET
ejde-203	202	33	position	position	NOUN
ejde-203	202	34	to	to	PART
ejde-203	202	35	show	show	VERB
ejde-203	202	36	the	the	DET
ejde-203	202	37	main	main	ADJ
ejde-203	202	38	result	result	NOUN
ejde-203	202	39	of	of	ADP
ejde-203	202	40	the	the	DET
ejde-203	202	41	paper	paper	NOUN
ejde-203	202	42	.	.	PUNCT
ejde-203	203	1	ejde-2021	ejde-2021	ADJ
ejde-203	203	2	/	/	SYM
ejde-203	203	3	si/01	si/01	PROPN
ejde-203	203	4	oscillation	oscillation	NOUN
ejde-203	203	5	time	time	NOUN
ejde-203	203	6	and	and	CCONJ
ejde-203	203	7	damping	damp	VERB
ejde-203	203	8	19	19	NUM
ejde-203	203	9	theorem	theorem	NOUN
ejde-203	203	10	3.2	3.2	NUM
ejde-203	203	11	.	.	PUNCT
ejde-203	204	1	under	under	ADP
ejde-203	204	2	assumption	assumption	NOUN
ejde-203	204	3	(	(	PUNCT
ejde-203	204	4	a1	a1	NOUN
ejde-203	204	5	)	)	PUNCT
ejde-203	204	6	,	,	PUNCT
ejde-203	204	7	there	there	PRON
ejde-203	204	8	exists	exist	VERB
ejde-203	204	9	a	a	DET
ejde-203	204	10	δ	δ	PROPN
ejde-203	204	11	>	>	X
ejde-203	204	12	0	0	NUM
ejde-203	205	1	such	such	ADJ
ejde-203	205	2	that	that	PRON
ejde-203	205	3	for	for	ADP
ejde-203	205	4	0	0	NUM
ejde-203	205	5	<	<	X
ejde-203	205	6	x0	x0	PROPN
ejde-203	205	7	<	<	X
ejde-203	205	8	δ	δ	PROPN
ejde-203	205	9	fixed	fix	VERB
ejde-203	205	10	,	,	PUNCT
ejde-203	205	11	the	the	DET
ejde-203	205	12	oscillation	oscillation	NOUN
ejde-203	205	13	time	time	NOUN
ejde-203	205	14	τ(x0	τ(x0	PROPN
ejde-203	205	15	,	,	PUNCT
ejde-203	205	16	α	α	NOUN
ejde-203	205	17	)	)	PUNCT
ejde-203	205	18	,	,	PUNCT
ejde-203	205	19	for	for	ADP
ejde-203	205	20	0	0	NUM
ejde-203	205	21	<	<	X
ejde-203	205	22	α	α	X
ejde-203	205	23	<	<	X
ejde-203	205	24	1	1	NUM
ejde-203	205	25	,	,	PUNCT
ejde-203	205	26	reaches	reach	VERB
ejde-203	205	27	a	a	DET
ejde-203	205	28	positive	positive	ADJ
ejde-203	205	29	minimum	minimum	NOUN
ejde-203	205	30	at	at	ADP
ejde-203	205	31	some	some	DET
ejde-203	205	32	0	0	NUM
ejde-203	205	33	<	<	X
ejde-203	205	34	α	α	X
ejde-203	205	35	<	<	X
ejde-203	205	36	1	1	NUM
ejde-203	205	37	.	.	PUNCT
ejde-203	206	1	moreover	moreover	ADV
ejde-203	206	2	,	,	PUNCT
ejde-203	206	3	lim	lim	PROPN
ejde-203	206	4	α→1−	α→1−	PROPN
ejde-203	206	5	τ(x0	τ(x0	PROPN
ejde-203	206	6	,	,	PUNCT
ejde-203	206	7	α	α	NOUN
ejde-203	206	8	)	)	PUNCT
ejde-203	206	9	=	=	NOUN
ejde-203	206	10	∞.	∞.	PROPN
ejde-203	206	11	proof	proof	NOUN
ejde-203	206	12	.	.	PUNCT
ejde-203	207	1	we	we	PRON
ejde-203	207	2	let	let	VERB
ejde-203	207	3	0	0	PUNCT
ejde-203	207	4	<	<	X
ejde-203	207	5	x0	x0	PROPN
ejde-203	207	6	�	�	PROPN
ejde-203	207	7	1	1	NUM
ejde-203	207	8	fixed	fix	VERB
ejde-203	207	9	by	by	ADP
ejde-203	207	10	now	now	ADV
ejde-203	207	11	and	and	CCONJ
ejde-203	207	12	denote	denote	VERB
ejde-203	207	13	by	by	ADP
ejde-203	207	14	(	(	PUNCT
ejde-203	207	15	x	x	NOUN
ejde-203	207	16	,	,	PUNCT
ejde-203	207	17	v	v	NOUN
ejde-203	207	18	)	)	PUNCT
ejde-203	207	19	be	be	AUX
ejde-203	207	20	the	the	DET
ejde-203	207	21	solution	solution	NOUN
ejde-203	207	22	of	of	ADP
ejde-203	207	23	equation	equation	NOUN
ejde-203	207	24	(	(	PUNCT
ejde-203	207	25	2.1	2.1	NUM
ejde-203	207	26	)	)	PUNCT
ejde-203	207	27	.	.	PUNCT
ejde-203	208	1	by	by	ADP
ejde-203	208	2	definition	definition	NOUN
ejde-203	208	3	of	of	ADP
ejde-203	208	4	τ̂	τ̂	PUNCT
ejde-203	208	5	we	we	PRON
ejde-203	208	6	have	have	VERB
ejde-203	208	7	v(τ̂	v(τ̂	NOUN
ejde-203	208	8	,	,	PUNCT
ejde-203	208	9	α	α	NOUN
ejde-203	208	10	)	)	PUNCT
ejde-203	208	11	=	=	SYM
ejde-203	208	12	0	0	NUM
ejde-203	208	13	,	,	PUNCT
ejde-203	208	14	so	so	SCONJ
ejde-203	208	15	that	that	SCONJ
ejde-203	208	16	the	the	DET
ejde-203	208	17	implicit	implicit	ADJ
ejde-203	208	18	function	function	NOUN
ejde-203	208	19	theorem	theorem	VERB
ejde-203	208	20	yields	yield	NOUN
ejde-203	208	21	∂τ̂	∂τ̂	VERB
ejde-203	208	22	∂α	∂α	PROPN
ejde-203	209	1	v̇(τ̂	v̇(τ̂	INTJ
ejde-203	209	2	,	,	PUNCT
ejde-203	209	3	α	α	X
ejde-203	209	4	)	)	PUNCT
ejde-203	210	1	+	+	X
ejde-203	210	2	v	v	X
ejde-203	210	3	(	(	PUNCT
ejde-203	210	4	τ̂	τ̂	NUM
ejde-203	210	5	,	,	PUNCT
ejde-203	210	6	α	α	X
ejde-203	210	7	)	)	PUNCT
ejde-203	210	8	=	=	SYM
ejde-203	210	9	0	0	NUM
ejde-203	210	10	,	,	PUNCT
ejde-203	210	11	therefore	therefore	ADV
ejde-203	210	12	∂τ̂	∂τ̂	VERB
ejde-203	210	13	∂α	∂α	PROPN
ejde-203	210	14	=	=	SYM
ejde-203	210	15	v	v	PROPN
ejde-203	210	16	(	(	PUNCT
ejde-203	210	17	τ̂	τ̂	NUM
ejde-203	210	18	,	,	PUNCT
ejde-203	210	19	α	α	X
ejde-203	210	20	)	)	PUNCT
ejde-203	210	21	x(τ̂	x(τ̂	NOUN
ejde-203	210	22	,	,	PUNCT
ejde-203	210	23	α)(1	α)(1	X
ejde-203	211	1	+	+	CCONJ
ejde-203	211	2	f(x(τ̂	f(x(τ̂	ADJ
ejde-203	211	3	,	,	PUNCT
ejde-203	211	4	α	α	NOUN
ejde-203	211	5	)	)	PUNCT
ejde-203	211	6	)	)	PUNCT
ejde-203	211	7	)	)	PUNCT
ejde-203	211	8	.	.	PUNCT
ejde-203	212	1	since	since	SCONJ
ejde-203	212	2	x(τ̂	x(τ̂	PROPN
ejde-203	212	3	,	,	PUNCT
ejde-203	212	4	α	α	X
ejde-203	212	5	)	)	PUNCT
ejde-203	212	6	is	be	AUX
ejde-203	212	7	negative	negative	ADJ
ejde-203	212	8	,	,	PUNCT
ejde-203	212	9	it	it	PRON
ejde-203	212	10	follows	follow	VERB
ejde-203	212	11	from	from	ADP
ejde-203	212	12	lemma	lemma	PROPN
ejde-203	212	13	3.1	3.1	NUM
ejde-203	212	14	that	that	PRON
ejde-203	212	15	and	and	CCONJ
ejde-203	212	16	∂τ̂	∂τ̂	ADJ
ejde-203	212	17	∂α	∂α	PROPN
ejde-203	212	18	|α=0	|α=0	VERB
ejde-203	212	19	<	<	X
ejde-203	212	20	0	0	NUM
ejde-203	212	21	.	.	PUNCT
ejde-203	213	1	now	now	ADV
ejde-203	213	2	we	we	PRON
ejde-203	213	3	shall	shall	AUX
ejde-203	213	4	show	show	VERB
ejde-203	213	5	that	that	SCONJ
ejde-203	213	6	the	the	DET
ejde-203	213	7	last	last	ADJ
ejde-203	213	8	inequality	inequality	NOUN
ejde-203	213	9	holds	hold	VERB
ejde-203	213	10	for	for	ADP
ejde-203	213	11	the	the	DET
ejde-203	213	12	oscillation	oscillation	NOUN
ejde-203	213	13	time	time	NOUN
ejde-203	213	14	τ	τ	X
ejde-203	213	15	.	.	PUNCT
ejde-203	214	1	to	to	PART
ejde-203	214	2	do	do	VERB
ejde-203	214	3	that	that	PRON
ejde-203	214	4	,	,	PUNCT
ejde-203	214	5	we	we	PRON
ejde-203	214	6	write	write	VERB
ejde-203	214	7	x̂0	x̂0	PUNCT
ejde-203	215	1	=	=	PUNCT
ejde-203	215	2	−x(τ̂(α	−x(τ̂(α	PROPN
ejde-203	215	3	,	,	PUNCT
ejde-203	215	4	x0	x0	PROPN
ejde-203	215	5	)	)	PUNCT
ejde-203	215	6	,	,	PUNCT
ejde-203	215	7	x0	x0	PROPN
ejde-203	215	8	)	)	PUNCT
ejde-203	215	9	and	and	CCONJ
ejde-203	215	10	see	see	VERB
ejde-203	215	11	that	that	SCONJ
ejde-203	215	12	τ(α	τ(α	PRON
ejde-203	215	13	,	,	PUNCT
ejde-203	215	14	x0	x0	PROPN
ejde-203	215	15	)	)	PUNCT
ejde-203	215	16	=	=	SYM
ejde-203	215	17	τ̂(α	τ̂(α	PROPN
ejde-203	215	18	,	,	PUNCT
ejde-203	215	19	x0	x0	PROPN
ejde-203	215	20	)	)	PUNCT
ejde-203	215	21	+	+	CCONJ
ejde-203	215	22	τ̂(α	τ̂(α	PROPN
ejde-203	215	23	,	,	PUNCT
ejde-203	215	24	x̂0	x̂0	PROPN
ejde-203	215	25	)	)	PUNCT
ejde-203	215	26	.	.	PUNCT
ejde-203	216	1	that	that	PRON
ejde-203	216	2	is	be	AUX
ejde-203	216	3	to	to	PART
ejde-203	216	4	say	say	VERB
ejde-203	216	5	,	,	PUNCT
ejde-203	216	6	the	the	DET
ejde-203	216	7	half	half	ADJ
ejde-203	216	8	oscillation	oscillation	NOUN
ejde-203	216	9	time	time	NOUN
ejde-203	216	10	depends	depend	VERB
ejde-203	216	11	on	on	ADP
ejde-203	216	12	|x0|	|x0|	NOUN
ejde-203	216	13	only	only	ADV
ejde-203	216	14	.	.	PUNCT
ejde-203	217	1	notice	notice	VERB
ejde-203	217	2	that	that	SCONJ
ejde-203	217	3	x̂0	x̂0	PROPN
ejde-203	217	4	≤	≤	ADJ
ejde-203	217	5	x0	x0	PROPN
ejde-203	217	6	and	and	CCONJ
ejde-203	217	7	the	the	DET
ejde-203	217	8	equality	equality	NOUN
ejde-203	217	9	holds	hold	VERB
ejde-203	217	10	in	in	ADP
ejde-203	217	11	the	the	DET
ejde-203	217	12	conservative	conservative	ADJ
ejde-203	217	13	case	case	NOUN
ejde-203	217	14	α	α	X
ejde-203	217	15	=	=	SYM
ejde-203	217	16	0	0	PUNCT
ejde-203	218	1	only	only	ADV
ejde-203	218	2	.	.	PUNCT
ejde-203	219	1	therefore	therefore	ADV
ejde-203	219	2	,	,	PUNCT
ejde-203	219	3	∂τ	∂τ	PROPN
ejde-203	219	4	∂α	∂α	PROPN
ejde-203	219	5	(	(	PUNCT
ejde-203	219	6	α	α	NOUN
ejde-203	219	7	,	,	PUNCT
ejde-203	219	8	x0	x0	PROPN
ejde-203	219	9	)	)	PUNCT
ejde-203	220	1	=	=	SYM
ejde-203	220	2	∂τ̂	∂τ̂	NOUN
ejde-203	220	3	∂α	∂α	PROPN
ejde-203	220	4	(	(	PUNCT
ejde-203	220	5	α	α	NOUN
ejde-203	220	6	,	,	PUNCT
ejde-203	220	7	x0	x0	PROPN
ejde-203	220	8	)	)	PUNCT
ejde-203	221	1	+	+	NUM
ejde-203	221	2	∂τ̂	∂τ̂	ADJ
ejde-203	221	3	∂α	∂α	PROPN
ejde-203	221	4	(	(	PUNCT
ejde-203	221	5	α	α	PROPN
ejde-203	221	6	,	,	PUNCT
ejde-203	221	7	x̂0)−	x̂0)−	PROPN
ejde-203	221	8	∂x̂0	∂x̂0	PROPN
ejde-203	222	1	∂α	∂α	PROPN
ejde-203	222	2	(	(	PUNCT
ejde-203	222	3	α	α	NOUN
ejde-203	222	4	,	,	PUNCT
ejde-203	222	5	x0	x0	PROPN
ejde-203	222	6	)	)	PUNCT
ejde-203	222	7	∂τ̂	∂τ̂	NOUN
ejde-203	223	1	∂α	∂α	PROPN
ejde-203	223	2	(	(	PUNCT
ejde-203	223	3	α	α	NOUN
ejde-203	223	4	,	,	PUNCT
ejde-203	223	5	x0	x0	PROPN
ejde-203	223	6	)	)	PUNCT
ejde-203	223	7	=	=	SYM
ejde-203	224	1	0	0	X
ejde-203	224	2	.	.	PUNCT
ejde-203	225	1	moreover	moreover	ADV
ejde-203	225	2	,	,	PUNCT
ejde-203	225	3	since	since	SCONJ
ejde-203	225	4	∂x̂0	∂x̂0	PROPN
ejde-203	225	5	∂α	∂α	PROPN
ejde-203	225	6	(	(	PUNCT
ejde-203	225	7	α	α	NOUN
ejde-203	225	8	,	,	PUNCT
ejde-203	225	9	x0	x0	PROPN
ejde-203	225	10	)	)	PUNCT
ejde-203	225	11	=	=	SYM
ejde-203	225	12	v(τ̂(α	v(τ̂(α	PROPN
ejde-203	225	13	,	,	PUNCT
ejde-203	225	14	x0	x0	PROPN
ejde-203	225	15	)	)	PUNCT
ejde-203	225	16	,	,	PUNCT
ejde-203	225	17	x0	x0	PROPN
ejde-203	225	18	)	)	PUNCT
ejde-203	226	1	=	=	SYM
ejde-203	226	2	0	0	NUM
ejde-203	226	3	,	,	PUNCT
ejde-203	226	4	we	we	PRON
ejde-203	226	5	have	have	VERB
ejde-203	226	6	that	that	PRON
ejde-203	226	7	lim	lim	PROPN
ejde-203	226	8	x0→0	x0→0	PROPN
ejde-203	227	1	+	+	PROPN
ejde-203	227	2	∂τ	∂τ	PROPN
ejde-203	227	3	∂α	∂α	PROPN
ejde-203	227	4	(	(	PUNCT
ejde-203	227	5	α	α	NOUN
ejde-203	227	6	,	,	PUNCT
ejde-203	227	7	x0	x0	PROPN
ejde-203	227	8	)	)	PUNCT
ejde-203	228	1	=	=	SYM
ejde-203	228	2	2	2	NUM
ejde-203	228	3	lim	lim	NOUN
ejde-203	228	4	x0→0	x0→0	PROPN
ejde-203	228	5	+	+	PROPN
ejde-203	228	6	∂x̂0	∂x̂0	PROPN
ejde-203	228	7	∂α	∂α	PROPN
ejde-203	228	8	(	(	PUNCT
ejde-203	228	9	α	α	NOUN
ejde-203	228	10	,	,	PUNCT
ejde-203	228	11	x0	x0	PROPN
ejde-203	228	12	)	)	PUNCT
ejde-203	228	13	.	.	PUNCT
ejde-203	229	1	finally	finally	ADV
ejde-203	229	2	,	,	PUNCT
ejde-203	229	3	by	by	ADP
ejde-203	229	4	the	the	DET
ejde-203	229	5	first	first	ADJ
ejde-203	229	6	claim	claim	NOUN
ejde-203	229	7	of	of	ADP
ejde-203	229	8	lemma	lemma	PROPN
ejde-203	229	9	2.2	2.2	NUM
ejde-203	229	10	,	,	PUNCT
ejde-203	229	11	τ(α	τ(α	PROPN
ejde-203	229	12	,	,	PUNCT
ejde-203	229	13	x0	x0	PROPN
ejde-203	229	14	)	)	PUNCT
ejde-203	229	15	must	must	AUX
ejde-203	229	16	attain	attain	VERB
ejde-203	229	17	a	a	DET
ejde-203	229	18	minimum	minimum	NOUN
ejde-203	229	19	at	at	ADP
ejde-203	229	20	some	some	DET
ejde-203	229	21	0	0	NUM
ejde-203	229	22	<	<	X
ejde-203	229	23	α	α	X
ejde-203	229	24	<	<	X
ejde-203	229	25	1	1	NUM
ejde-203	229	26	.	.	PUNCT
ejde-203	229	27	�	�	PROPN
ejde-203	229	28	4	4	NUM
ejde-203	229	29	.	.	PUNCT
ejde-203	229	30	conclusions	conclusion	NOUN
ejde-203	229	31	and	and	CCONJ
ejde-203	229	32	final	final	ADJ
ejde-203	229	33	remarks	remark	NOUN
ejde-203	229	34	an	an	DET
ejde-203	229	35	oscillating	oscillate	VERB
ejde-203	229	36	mass	mass	NOUN
ejde-203	229	37	exhibits	exhibit	VERB
ejde-203	229	38	gradually	gradually	ADV
ejde-203	229	39	diminishing	diminish	VERB
ejde-203	229	40	amplitude	amplitude	NOUN
ejde-203	229	41	in	in	ADP
ejde-203	229	42	the	the	DET
ejde-203	229	43	presence	presence	NOUN
ejde-203	229	44	of	of	ADP
ejde-203	229	45	damping	damp	VERB
ejde-203	229	46	.	.	PUNCT
ejde-203	230	1	the	the	DET
ejde-203	230	2	time	time	NOUN
ejde-203	230	3	spent	spend	VERB
ejde-203	230	4	by	by	ADP
ejde-203	230	5	the	the	DET
ejde-203	230	6	mass	mass	NOUN
ejde-203	230	7	completing	complete	VERB
ejde-203	230	8	one	one	NUM
ejde-203	230	9	oscillation	oscillation	NOUN
ejde-203	230	10	depends	depend	VERB
ejde-203	230	11	on	on	ADP
ejde-203	230	12	several	several	ADJ
ejde-203	230	13	factors	factor	NOUN
ejde-203	230	14	,	,	PUNCT
ejde-203	230	15	as	as	ADP
ejde-203	230	16	the	the	DET
ejde-203	230	17	model	model	NOUN
ejde-203	230	18	for	for	ADP
ejde-203	230	19	the	the	DET
ejde-203	230	20	restoring	restore	VERB
ejde-203	230	21	force	force	NOUN
ejde-203	230	22	,	,	PUNCT
ejde-203	230	23	how	how	SCONJ
ejde-203	230	24	the	the	DET
ejde-203	230	25	oscillation	oscillation	NOUN
ejde-203	230	26	starts	start	VERB
ejde-203	230	27	,	,	PUNCT
ejde-203	230	28	and	and	CCONJ
ejde-203	230	29	the	the	DET
ejde-203	230	30	nature	nature	NOUN
ejde-203	230	31	of	of	ADP
ejde-203	230	32	the	the	DET
ejde-203	230	33	damping	damping	NOUN
ejde-203	230	34	.	.	PUNCT
ejde-203	231	1	for	for	ADP
ejde-203	231	2	the	the	DET
ejde-203	231	3	sake	sake	NOUN
ejde-203	231	4	of	of	ADP
ejde-203	231	5	our	our	PRON
ejde-203	231	6	discussion	discussion	NOUN
ejde-203	231	7	we	we	PRON
ejde-203	231	8	consider	consider	VERB
ejde-203	231	9	a	a	DET
ejde-203	231	10	vertical	vertical	ADJ
ejde-203	231	11	pendulum	pendulum	NOUN
ejde-203	231	12	with	with	ADP
ejde-203	231	13	a	a	DET
ejde-203	231	14	nonlinear	nonlinear	ADJ
ejde-203	231	15	restoring	restore	VERB
ejde-203	231	16	force	force	NOUN
ejde-203	231	17	resembling	resemble	VERB
ejde-203	231	18	the	the	DET
ejde-203	231	19	mathematical	mathematical	ADJ
ejde-203	231	20	pendulum	pendulum	NOUN
ejde-203	231	21	,	,	PUNCT
ejde-203	231	22	letting	let	VERB
ejde-203	231	23	the	the	DET
ejde-203	231	24	oscillation	oscillation	NOUN
ejde-203	231	25	start	start	VERB
ejde-203	231	26	at	at	ADP
ejde-203	231	27	a	a	DET
ejde-203	231	28	small	small	ADJ
ejde-203	231	29	amplitude	amplitude	NOUN
ejde-203	231	30	with	with	ADP
ejde-203	231	31	vanishing	vanish	VERB
ejde-203	231	32	velocity	velocity	NOUN
ejde-203	231	33	and	and	CCONJ
ejde-203	231	34	a	a	DET
ejde-203	231	35	viscous	viscous	ADJ
ejde-203	231	36	damping	damp	VERB
ejde-203	231	37	model	model	NOUN
ejde-203	231	38	with	with	ADP
ejde-203	231	39	a	a	PRON
ejde-203	231	40	(	(	PUNCT
ejde-203	231	41	normalized	normalize	VERB
ejde-203	231	42	)	)	PUNCT
ejde-203	231	43	viscosity	viscosity	NOUN
ejde-203	231	44	coefficient	coefficient	NOUN
ejde-203	231	45	α	α	NOUN
ejde-203	231	46	.	.	PUNCT
ejde-203	232	1	we	we	PRON
ejde-203	232	2	have	have	AUX
ejde-203	232	3	proved	prove	VERB
ejde-203	232	4	that	that	SCONJ
ejde-203	232	5	the	the	DET
ejde-203	232	6	oscillation	oscillation	NOUN
ejde-203	232	7	time	time	NOUN
ejde-203	232	8	τ	τ	PROPN
ejde-203	232	9	≡	≡	PROPN
ejde-203	232	10	τ(α	τ(α	NUM
ejde-203	232	11	)	)	PUNCT
ejde-203	232	12	does	do	AUX
ejde-203	232	13	not	not	PART
ejde-203	232	14	depend	depend	VERB
ejde-203	232	15	monotonically	monotonically	ADV
ejde-203	232	16	on	on	ADP
ejde-203	232	17	α	α	NUM
ejde-203	232	18	,	,	PUNCT
ejde-203	232	19	meaning	mean	VERB
ejde-203	232	20	that	that	SCONJ
ejde-203	232	21	there	there	PRON
ejde-203	232	22	exists	exist	VERB
ejde-203	232	23	a	a	DET
ejde-203	232	24	threshold	threshold	NOUN
ejde-203	232	25	α0	α0	PROPN
ejde-203	232	26	(	(	PUNCT
ejde-203	232	27	which	which	PRON
ejde-203	232	28	depends	depend	VERB
ejde-203	232	29	on	on	ADP
ejde-203	232	30	the	the	DET
ejde-203	232	31	starting	start	VERB
ejde-203	232	32	amplitude	amplitude	NOUN
ejde-203	232	33	of	of	ADP
ejde-203	232	34	the	the	DET
ejde-203	232	35	oscillation	oscillation	NOUN
ejde-203	232	36	)	)	PUNCT
ejde-203	232	37	such	such	ADJ
ejde-203	232	38	that	that	SCONJ
ejde-203	232	39	τ	τ	PROPN
ejde-203	232	40	reaches	reach	VERB
ejde-203	232	41	a	a	DET
ejde-203	232	42	local	local	ADJ
ejde-203	232	43	minimum	minimum	NOUN
ejde-203	232	44	at	at	ADP
ejde-203	232	45	α0	α0	PROPN
ejde-203	232	46	(	(	PUNCT
ejde-203	232	47	see	see	VERB
ejde-203	232	48	figure	figure	NOUN
ejde-203	232	49	1	1	NUM
ejde-203	232	50	)	)	PUNCT
ejde-203	232	51	.	.	PUNCT
ejde-203	233	1	it	it	PRON
ejde-203	233	2	is	be	AUX
ejde-203	233	3	worth	worth	ADJ
ejde-203	233	4	noticing	notice	VERB
ejde-203	233	5	that	that	SCONJ
ejde-203	233	6	this	this	DET
ejde-203	233	7	behavior	behavior	NOUN
ejde-203	233	8	can	can	AUX
ejde-203	233	9	not	not	PART
ejde-203	233	10	be	be	AUX
ejde-203	233	11	observed	observe	VERB
ejde-203	233	12	if	if	SCONJ
ejde-203	233	13	the	the	DET
ejde-203	233	14	restitution	restitution	NOUN
ejde-203	233	15	force	force	NOUN
ejde-203	233	16	is	be	AUX
ejde-203	233	17	linear	linear	ADJ
ejde-203	233	18	,	,	PUNCT
ejde-203	233	19	i.e.	i.e.	X
ejde-203	233	20	,	,	PUNCT
ejde-203	233	21	what	what	PRON
ejde-203	233	22	we	we	PRON
ejde-203	233	23	report	report	VERB
ejde-203	233	24	in	in	ADP
ejde-203	233	25	this	this	DET
ejde-203	233	26	paper	paper	NOUN
ejde-203	233	27	is	be	AUX
ejde-203	233	28	essentially	essentially	ADV
ejde-203	233	29	a	a	DET
ejde-203	233	30	nonlinear	nonlinear	ADJ
ejde-203	233	31	phenomenon	phenomenon	NOUN
ejde-203	233	32	.	.	PUNCT
ejde-203	234	1	the	the	DET
ejde-203	234	2	proof	proof	NOUN
ejde-203	234	3	of	of	ADP
ejde-203	234	4	existence	existence	NOUN
ejde-203	234	5	of	of	ADP
ejde-203	234	6	a	a	DET
ejde-203	234	7	positive	positive	ADJ
ejde-203	234	8	minimum	minimum	NOUN
ejde-203	234	9	for	for	ADP
ejde-203	234	10	the	the	DET
ejde-203	234	11	oscillation	oscillation	NOUN
ejde-203	234	12	time	time	NOUN
ejde-203	234	13	rests	rest	VERB
ejde-203	234	14	heavily	heavily	ADV
ejde-203	234	15	on	on	ADP
ejde-203	234	16	the	the	DET
ejde-203	234	17	fact	fact	NOUN
ejde-203	234	18	that	that	SCONJ
ejde-203	234	19	the	the	DET
ejde-203	234	20	constant	constant	ADJ
ejde-203	234	21	a	a	PRON
ejde-203	234	22	in	in	ADP
ejde-203	234	23	assumption	assumption	NOUN
ejde-203	234	24	(	(	PUNCT
ejde-203	234	25	a1	a1	NOUN
ejde-203	234	26	)	)	PUNCT
ejde-203	234	27	is	be	AUX
ejde-203	234	28	positive	positive	ADJ
ejde-203	234	29	.	.	PUNCT
ejde-203	235	1	just	just	ADV
ejde-203	235	2	to	to	PART
ejde-203	235	3	experiment	experiment	VERB
ejde-203	235	4	the	the	DET
ejde-203	235	5	effect	effect	NOUN
ejde-203	235	6	of	of	ADP
ejde-203	235	7	changing	change	VERB
ejde-203	235	8	the	the	DET
ejde-203	235	9	sign	sign	NOUN
ejde-203	235	10	of	of	ADP
ejde-203	235	11	the	the	DET
ejde-203	235	12	constant	constant	ADJ
ejde-203	235	13	a	a	X
ejde-203	235	14	,	,	PUNCT
ejde-203	235	15	we	we	PRON
ejde-203	235	16	carried	carry	VERB
ejde-203	235	17	out	out	ADP
ejde-203	235	18	some	some	DET
ejde-203	235	19	numerical	numerical	ADJ
ejde-203	235	20	20	20	NUM
ejde-203	235	21	j.	j.	PROPN
ejde-203	235	22	arango	arango	PROPN
ejde-203	235	23	ejde	ejde	NOUN
ejde-203	235	24	/	/	SYM
ejde-203	235	25	si/01	si/01	PROPN
ejde-203	235	26	figure	figure	NOUN
ejde-203	235	27	2	2	NUM
ejde-203	235	28	.	.	PUNCT
ejde-203	235	29	numerical	numerical	PROPN
ejde-203	235	30	simulation	simulation	PROPN
ejde-203	235	31	of	of	ADP
ejde-203	235	32	the	the	DET
ejde-203	235	33	oscillation	oscillation	NOUN
ejde-203	235	34	time	time	NOUN
ejde-203	235	35	τ	τ	X
ejde-203	235	36	depending	depend	VERB
ejde-203	235	37	on	on	ADP
ejde-203	235	38	the	the	DET
ejde-203	235	39	damping	damp	VERB
ejde-203	235	40	coefficient	coefficient	NOUN
ejde-203	235	41	α	α	X
ejde-203	235	42	with	with	ADP
ejde-203	235	43	starting	start	VERB
ejde-203	235	44	amplitude	amplitude	NOUN
ejde-203	235	45	x0	x0	PROPN
ejde-203	235	46	=	=	PUNCT
ejde-203	235	47	0.2	0.2	NUM
ejde-203	235	48	and	and	CCONJ
ejde-203	235	49	non	non	ADJ
ejde-203	235	50	linear	linear	ADJ
ejde-203	235	51	restoring	restore	VERB
ejde-203	235	52	term	term	NOUN
ejde-203	235	53	given	give	VERB
ejde-203	235	54	by	by	ADP
ejde-203	235	55	f(x	f(x	PROPN
ejde-203	235	56	)	)	PUNCT
ejde-203	236	1	=	=	PUNCT
ejde-203	236	2	−a	−a	NOUN
ejde-203	236	3	x2	x2	PROPN
ejde-203	236	4	,	,	PUNCT
ejde-203	236	5	a	a	DET
ejde-203	236	6	=	=	PUNCT
ejde-203	236	7	±1	±1	VERB
ejde-203	236	8	.	.	PUNCT
ejde-203	237	1	the	the	DET
ejde-203	237	2	curve	curve	NOUN
ejde-203	237	3	with	with	ADP
ejde-203	237	4	the	the	DET
ejde-203	237	5	round	round	NOUN
ejde-203	237	6	marker	marker	NOUN
ejde-203	237	7	(	(	PUNCT
ejde-203	237	8	blue	blue	NOUN
ejde-203	237	9	in	in	ADP
ejde-203	237	10	the	the	DET
ejde-203	237	11	online	online	ADJ
ejde-203	237	12	version	version	NOUN
ejde-203	237	13	)	)	PUNCT
ejde-203	237	14	corresponds	correspond	VERB
ejde-203	237	15	to	to	ADP
ejde-203	237	16	the	the	DET
ejde-203	237	17	oscillation	oscillation	NOUN
ejde-203	237	18	time	time	NOUN
ejde-203	237	19	τl	τl	NOUN
ejde-203	237	20	of	of	ADP
ejde-203	237	21	the	the	DET
ejde-203	237	22	linear	linear	ADJ
ejde-203	237	23	case	case	NOUN
ejde-203	237	24	f	f	PROPN
ejde-203	237	25	≡	≡	PROPN
ejde-203	237	26	0	0	NUM
ejde-203	237	27	simulations	simulation	NOUN
ejde-203	237	28	of	of	ADP
ejde-203	237	29	τ	τ	PROPN
ejde-203	237	30	with	with	ADP
ejde-203	237	31	the	the	DET
ejde-203	237	32	nonlinear	nonlinear	ADJ
ejde-203	237	33	term	term	NOUN
ejde-203	237	34	f(x	f(x	PROPN
ejde-203	237	35	)	)	PUNCT
ejde-203	237	36	=	=	PUNCT
ejde-203	238	1	−a	−a	NOUN
ejde-203	238	2	x2	x2	PROPN
ejde-203	238	3	for	for	ADP
ejde-203	238	4	a	a	DET
ejde-203	238	5	=	=	SYM
ejde-203	238	6	1,−1	1,−1	PROPN
ejde-203	238	7	.	.	PUNCT
ejde-203	239	1	the	the	DET
ejde-203	239	2	corresponding	corresponding	ADJ
ejde-203	239	3	equations	equation	NOUN
ejde-203	239	4	are	be	AUX
ejde-203	239	5	particular	particular	ADJ
ejde-203	239	6	cases	case	NOUN
ejde-203	239	7	of	of	ADP
ejde-203	239	8	an	an	DET
ejde-203	239	9	unforced	unforced	ADJ
ejde-203	239	10	duffing	duffing	NOUN
ejde-203	239	11	oscillator	oscillator	NOUN
ejde-203	239	12	[	[	X
ejde-203	239	13	7	7	NUM
ejde-203	239	14	]	]	PUNCT
ejde-203	239	15	.	.	PUNCT
ejde-203	240	1	the	the	DET
ejde-203	240	2	numerical	numerical	ADJ
ejde-203	240	3	results	result	NOUN
ejde-203	240	4	are	be	AUX
ejde-203	240	5	shown	show	VERB
ejde-203	240	6	in	in	ADP
ejde-203	240	7	figure	figure	NOUN
ejde-203	240	8	2	2	NUM
ejde-203	240	9	.	.	PUNCT
ejde-203	240	10	just	just	ADV
ejde-203	240	11	for	for	ADP
ejde-203	240	12	the	the	DET
ejde-203	240	13	sake	sake	NOUN
ejde-203	240	14	of	of	ADP
ejde-203	240	15	the	the	DET
ejde-203	240	16	numerical	numerical	ADJ
ejde-203	240	17	experimentation	experimentation	NOUN
ejde-203	240	18	we	we	PRON
ejde-203	240	19	also	also	ADV
ejde-203	240	20	considered	consider	VERB
ejde-203	240	21	negative	negative	ADJ
ejde-203	240	22	values	value	NOUN
ejde-203	240	23	for	for	ADP
ejde-203	240	24	α	α	NOUN
ejde-203	240	25	.	.	PUNCT
ejde-203	241	1	if	if	SCONJ
ejde-203	241	2	a	a	DET
ejde-203	241	3	=	=	NOUN
ejde-203	241	4	1	1	NUM
ejde-203	241	5	we	we	PRON
ejde-203	241	6	see	see	VERB
ejde-203	241	7	that	that	SCONJ
ejde-203	241	8	τ	τ	PROPN
ejde-203	241	9	reaches	reach	VERB
ejde-203	241	10	its	its	PRON
ejde-203	241	11	minimum	minimum	NOUN
ejde-203	241	12	at	at	ADP
ejde-203	241	13	a	a	DET
ejde-203	241	14	positive	positive	ADJ
ejde-203	241	15	value	value	NOUN
ejde-203	241	16	for	for	ADP
ejde-203	241	17	α	α	NOUN
ejde-203	241	18	.	.	PUNCT
ejde-203	242	1	by	by	ADP
ejde-203	242	2	contrast	contrast	NOUN
ejde-203	242	3	,	,	PUNCT
ejde-203	242	4	if	if	SCONJ
ejde-203	242	5	a	a	PRON
ejde-203	242	6	=	=	SYM
ejde-203	242	7	−1	−1	NOUN
ejde-203	242	8	no	no	DET
ejde-203	242	9	minimum	minimum	NOUN
ejde-203	242	10	seems	seem	VERB
ejde-203	242	11	to	to	PART
ejde-203	242	12	exist	exist	VERB
ejde-203	242	13	.	.	PUNCT
ejde-203	243	1	the	the	DET
ejde-203	243	2	curve	curve	NOUN
ejde-203	243	3	with	with	ADP
ejde-203	243	4	the	the	DET
ejde-203	243	5	round	round	NOUN
ejde-203	243	6	marker	marker	NOUN
ejde-203	243	7	(	(	PUNCT
ejde-203	243	8	blue	blue	NOUN
ejde-203	243	9	in	in	ADP
ejde-203	243	10	the	the	DET
ejde-203	243	11	online	online	ADJ
ejde-203	243	12	version	version	NOUN
ejde-203	243	13	)	)	PUNCT
ejde-203	243	14	corresponds	correspond	VERB
ejde-203	243	15	to	to	ADP
ejde-203	243	16	the	the	DET
ejde-203	243	17	oscillation	oscillation	NOUN
ejde-203	243	18	time	time	NOUN
ejde-203	243	19	of	of	ADP
ejde-203	243	20	the	the	DET
ejde-203	243	21	linear	linear	ADJ
ejde-203	243	22	case	case	NOUN
ejde-203	243	23	τl	τl	VERB
ejde-203	243	24	=	=	SYM
ejde-203	243	25	2π/	2π/	NUM
ejde-203	243	26	√	√	PROPN
ejde-203	243	27	1−	1−	NUM
ejde-203	243	28	α2	α2	ADJ
ejde-203	243	29	.	.	PUNCT
ejde-203	244	1	the	the	DET
ejde-203	244	2	numerical	numerical	ADJ
ejde-203	244	3	experimentation	experimentation	NOUN
ejde-203	244	4	of	of	ADP
ejde-203	244	5	the	the	DET
ejde-203	244	6	oscillation	oscillation	NOUN
ejde-203	244	7	time	time	NOUN
ejde-203	244	8	τ	τ	PROPN
ejde-203	244	9	(	(	PUNCT
ejde-203	244	10	not	not	PART
ejde-203	244	11	shown	show	VERB
ejde-203	244	12	in	in	ADP
ejde-203	244	13	this	this	DET
ejde-203	244	14	paper	paper	NOUN
ejde-203	244	15	)	)	PUNCT
ejde-203	244	16	assuming	assume	VERB
ejde-203	244	17	a	a	DET
ejde-203	244	18	quadratic	quadratic	ADJ
ejde-203	244	19	damping	damping	NOUN
ejde-203	244	20	exhibits	exhibit	VERB
ejde-203	244	21	the	the	DET
ejde-203	244	22	same	same	ADJ
ejde-203	244	23	behavior	behavior	NOUN
ejde-203	244	24	as	as	ADP
ejde-203	244	25	the	the	DET
ejde-203	244	26	graphics	graphic	NOUN
ejde-203	244	27	of	of	ADP
ejde-203	244	28	figure	figure	NOUN
ejde-203	244	29	2	2	NUM
ejde-203	244	30	.	.	PUNCT
ejde-203	245	1	if	if	SCONJ
ejde-203	245	2	the	the	DET
ejde-203	245	3	readers	reader	NOUN
ejde-203	245	4	are	be	AUX
ejde-203	245	5	curious	curious	ADJ
ejde-203	245	6	about	about	ADP
ejde-203	245	7	the	the	DET
ejde-203	245	8	numerical	numerical	ADJ
ejde-203	245	9	experiments	experiment	NOUN
ejde-203	245	10	,	,	PUNCT
ejde-203	245	11	they	they	PRON
ejde-203	245	12	can	can	AUX
ejde-203	245	13	a	a	DET
ejde-203	245	14	look	look	NOUN
ejde-203	245	15	at	at	ADP
ejde-203	245	16	the	the	DET
ejde-203	245	17	author	author	NOUN
ejde-203	245	18	’s	’s	PART
ejde-203	245	19	github	github	PROPN
ejde-203	245	20	page	page	NOUN
ejde-203	245	21	https://github.com/arangogithub/oscillation-time	https://github.com/arangogithub/oscillation-time	NOUN
ejde-203	245	22	,	,	PUNCT
ejde-203	245	23	and	and	CCONJ
ejde-203	245	24	download	download	VERB
ejde-203	245	25	a	a	DET
ejde-203	245	26	jupyter	jupyter	ADJ
ejde-203	245	27	notebook	notebook	NOUN
ejde-203	245	28	with	with	ADP
ejde-203	245	29	the	the	DET
ejde-203	245	30	python	python	PROPN
ejde-203	245	31	code	code	NOUN
ejde-203	245	32	featuring	feature	VERB
ejde-203	245	33	the	the	DET
ejde-203	245	34	results	result	NOUN
ejde-203	245	35	shown	show	VERB
ejde-203	245	36	in	in	ADP
ejde-203	245	37	figures	figure	NOUN
ejde-203	245	38	1	1	NUM
ejde-203	245	39	and	and	CCONJ
ejde-203	245	40	2	2	NUM
ejde-203	245	41	.	.	X
ejde-203	245	42	acknowledgements	acknowledgement	NOUN
ejde-203	245	43	.	.	PUNCT
ejde-203	246	1	the	the	DET
ejde-203	246	2	author	author	NOUN
ejde-203	246	3	would	would	AUX
ejde-203	246	4	like	like	VERB
ejde-203	246	5	to	to	PART
ejde-203	246	6	give	give	VERB
ejde-203	246	7	the	the	DET
ejde-203	246	8	reviewer	reviewer	NOUN
ejde-203	246	9	his	his	PRON
ejde-203	246	10	very	very	ADV
ejde-203	246	11	heartfelt	heartfelt	ADJ
ejde-203	246	12	thanks	thank	NOUN
ejde-203	246	13	for	for	ADP
ejde-203	246	14	carefully	carefully	ADV
ejde-203	246	15	reading	read	VERB
ejde-203	246	16	the	the	DET
ejde-203	246	17	manuscript	manuscript	NOUN
ejde-203	246	18	and	and	CCONJ
ejde-203	246	19	for	for	ADP
ejde-203	246	20	pointing	point	VERB
ejde-203	246	21	out	out	ADP
ejde-203	246	22	several	several	ADJ
ejde-203	246	23	inaccuracies	inaccuracy	NOUN
ejde-203	246	24	of	of	ADP
ejde-203	246	25	the	the	DET
ejde-203	246	26	document	document	NOUN
ejde-203	246	27	.	.	PUNCT
ejde-203	247	1	it	it	PRON
ejde-203	247	2	was	be	AUX
ejde-203	247	3	my	my	PRON
ejde-203	247	4	pleasure	pleasure	NOUN
ejde-203	247	5	to	to	PART
ejde-203	247	6	discuss	discuss	VERB
ejde-203	247	7	some	some	PRON
ejde-203	247	8	of	of	ADP
ejde-203	247	9	the	the	DET
ejde-203	247	10	present	present	ADJ
ejde-203	247	11	results	result	NOUN
ejde-203	247	12	with	with	ADP
ejde-203	247	13	prof	prof	PROPN
ejde-203	247	14	.	.	PUNCT
ejde-203	248	1	alan	alan	PROPN
ejde-203	248	2	lazer	lazer	PROPN
ejde-203	248	3	when	when	SCONJ
ejde-203	248	4	he	he	PRON
ejde-203	248	5	was	be	AUX
ejde-203	248	6	at	at	ADP
ejde-203	248	7	the	the	DET
ejde-203	248	8	university	university	PROPN
ejde-203	248	9	of	of	ADP
ejde-203	248	10	miami	miami	PROPN
ejde-203	248	11	.	.	PUNCT
ejde-203	249	1	references	reference	NOUN
ejde-203	249	2	[	[	X
ejde-203	249	3	1	1	X
ejde-203	249	4	]	]	PUNCT
ejde-203	249	5	v.	v.	PROPN
ejde-203	249	6	i.	i.	PROPN
ejde-203	249	7	arnold	arnold	PROPN
ejde-203	249	8	;	;	PUNCT
ejde-203	249	9	mathematical	mathematical	ADJ
ejde-203	249	10	methods	method	NOUN
ejde-203	249	11	of	of	ADP
ejde-203	249	12	classical	classical	ADJ
ejde-203	249	13	mechanics	mechanic	NOUN
ejde-203	249	14	.	.	PUNCT
ejde-203	250	1	springer	springer	NOUN
ejde-203	250	2	,	,	PUNCT
ejde-203	250	3	1989	1989	NUM
ejde-203	250	4	.	.	PUNCT
ejde-203	251	1	[	[	X
ejde-203	251	2	2	2	NUM
ejde-203	251	3	]	]	PUNCT
ejde-203	251	4	l.	l.	PROPN
ejde-203	251	5	cveticanin	cveticanin	PROPN
ejde-203	251	6	;	;	PUNCT
ejde-203	251	7	oscillator	oscillator	NOUN
ejde-203	251	8	with	with	ADP
ejde-203	251	9	strong	strong	ADJ
ejde-203	251	10	quadratic	quadratic	ADJ
ejde-203	251	11	damping	damp	VERB
ejde-203	251	12	force	force	NOUN
ejde-203	251	13	.	.	PUNCT
ejde-203	252	1	publ	publ	NOUN
ejde-203	252	2	.	.	PUNCT
ejde-203	253	1	inst	inst	PROPN
ejde-203	253	2	.	.	PUNCT
ejde-203	253	3	math	math	NOUN
ejde-203	253	4	.	.	PUNCT
ejde-203	254	1	(	(	PUNCT
ejde-203	254	2	beograd	beograd	PROPN
ejde-203	254	3	)	)	PUNCT
ejde-203	254	4	(	(	PUNCT
ejde-203	254	5	n.s	n.s	PROPN
ejde-203	254	6	.	.	PROPN
ejde-203	254	7	)	)	PUNCT
ejde-203	254	8	,	,	PUNCT
ejde-203	254	9	85(99	85(99	NUM
ejde-203	254	10	):	):	PUNCT
ejde-203	254	11	119–130	119–130	NUM
ejde-203	254	12	,	,	PUNCT
ejde-203	254	13	march	march	PROPN
ejde-203	254	14	2009	2009	NUM
ejde-203	254	15	.	.	PUNCT
ejde-203	255	1	[	[	X
ejde-203	255	2	3	3	NUM
ejde-203	255	3	]	]	PUNCT
ejde-203	255	4	a.	a.	NOUN
ejde-203	255	5	ghose	ghose	PROPN
ejde-203	255	6	-	-	PUNCT
ejde-203	255	7	choudhury	choudhury	PROPN
ejde-203	255	8	,	,	PUNCT
ejde-203	255	9	p.	p.	PROPN
ejde-203	255	10	guha	guha	PROPN
ejde-203	255	11	;	;	PUNCT
ejde-203	255	12	an	an	DET
ejde-203	255	13	analytic	analytic	ADJ
ejde-203	255	14	technique	technique	NOUN
ejde-203	255	15	for	for	ADP
ejde-203	255	16	the	the	DET
ejde-203	255	17	solutions	solution	NOUN
ejde-203	255	18	of	of	ADP
ejde-203	255	19	nonlinear	nonlinear	ADJ
ejde-203	255	20	oscillators	oscillator	NOUN
ejde-203	255	21	with	with	ADP
ejde-203	255	22	damping	damp	VERB
ejde-203	255	23	using	use	VERB
ejde-203	255	24	the	the	DET
ejde-203	255	25	abel	abel	PROPN
ejde-203	255	26	equation	equation	NOUN
ejde-203	255	27	arxiv	arxiv	PROPN
ejde-203	255	28	:	:	PUNCT
ejde-203	255	29	1608.02324	1608.02324	NUM
ejde-203	256	1	[	[	X
ejde-203	256	2	nlin.si	nlin.si	X
ejde-203	256	3	]	]	X
ejde-203	256	4	,	,	PUNCT
ejde-203	256	5	2016	2016	NUM
ejde-203	256	6	.	.	PUNCT
ejde-203	257	1	[	[	X
ejde-203	257	2	4	4	NUM
ejde-203	257	3	]	]	PUNCT
ejde-203	257	4	r.	r.	PROPN
ejde-203	257	5	cabrera	cabrera	PROPN
ejde-203	257	6	-	-	PUNCT
ejde-203	257	7	trujillo	trujillo	PROPN
ejde-203	257	8	,	,	PUNCT
ejde-203	257	9	n.	n.	PROPN
ejde-203	257	10	c.	c.	PROPN
ejde-203	257	11	giesselmann	giesselmann	PROPN
ejde-203	257	12	,	,	PUNCT
ejde-203	257	13	d.	d.	PROPN
ejde-203	257	14	hanstorp	hanstorp	PROPN
ejde-203	257	15	,	,	PUNCT
ejde-203	257	16	j.	j.	PROPN
ejde-203	257	17	tello	tello	PROPN
ejde-203	257	18	marmolejo	marmolejo	PROPN
ejde-203	257	19	,	,	PUNCT
ejde-203	257	20	o.	o.	PROPN
ejde-203	257	21	isaksson	isaksson	PROPN
ejde-203	257	22	;	;	PUNCT
ejde-203	257	23	a	a	DET
ejde-203	257	24	fully	fully	ADV
ejde-203	257	25	manipulable	manipulable	ADJ
ejde-203	257	26	damped	damped	NOUN
ejde-203	257	27	driven	drive	VERB
ejde-203	257	28	harmonic	harmonic	ADJ
ejde-203	257	29	oscillator	oscillator	NOUN
ejde-203	257	30	using	use	VERB
ejde-203	257	31	optical	optical	ADJ
ejde-203	257	32	levitation	levitation	NOUN
ejde-203	257	33	.	.	PUNCT
ejde-203	258	1	american	american	PROPN
ejde-203	258	2	journal	journal	PROPN
ejde-203	258	3	of	of	ADP
ejde-203	258	4	physics	physics	PROPN
ejde-203	258	5	,	,	PUNCT
ejde-203	258	6	88(6	88(6	NUM
ejde-203	258	7	):	):	PUNCT
ejde-203	258	8	490–498	490–498	NUM
ejde-203	258	9	,	,	PUNCT
ejde-203	258	10	sept	sept	PROPN
ejde-203	258	11	.	.	PROPN
ejde-203	258	12	2018	2018	NUM
ejde-203	258	13	.	.	PUNCT
ejde-203	259	1	[	[	X
ejde-203	259	2	5	5	X
ejde-203	259	3	]	]	PUNCT
ejde-203	259	4	k.	k.	PROPN
ejde-203	259	5	johannessen	johannessen	PROPN
ejde-203	259	6	;	;	PUNCT
ejde-203	259	7	an	an	DET
ejde-203	259	8	analytical	analytical	ADJ
ejde-203	259	9	solution	solution	NOUN
ejde-203	259	10	to	to	ADP
ejde-203	259	11	the	the	DET
ejde-203	259	12	equation	equation	NOUN
ejde-203	259	13	of	of	ADP
ejde-203	259	14	motion	motion	NOUN
ejde-203	259	15	for	for	ADP
ejde-203	259	16	the	the	DET
ejde-203	259	17	damped	damped	ADJ
ejde-203	259	18	nonlinear	nonlinear	ADJ
ejde-203	259	19	pendulum	pendulum	NOUN
ejde-203	259	20	.	.	PUNCT
ejde-203	260	1	european	european	PROPN
ejde-203	260	2	journal	journal	PROPN
ejde-203	260	3	of	of	ADP
ejde-203	260	4	physics	physics	PROPN
ejde-203	260	5	,	,	PUNCT
ejde-203	260	6	35(3	35(3	NUM
ejde-203	260	7	):	):	PUNCT
ejde-203	260	8	035014	035014	NUM
ejde-203	260	9	,	,	PUNCT
ejde-203	260	10	march	march	PROPN
ejde-203	260	11	2014	2014	NUM
ejde-203	260	12	.	.	PUNCT
ejde-203	261	1	ejde-2021	ejde-2021	ADJ
ejde-203	261	2	/	/	SYM
ejde-203	261	3	si/01	si/01	PROPN
ejde-203	261	4	oscillation	oscillation	NOUN
ejde-203	261	5	time	time	NOUN
ejde-203	261	6	and	and	CCONJ
ejde-203	261	7	damping	damp	VERB
ejde-203	261	8	21	21	NUM
ejde-203	261	9	[	[	SYM
ejde-203	261	10	6	6	NUM
ejde-203	261	11	]	]	X
ejde-203	261	12	d.	d.	PROPN
ejde-203	261	13	kharkongor	kharkongor	PROPN
ejde-203	261	14	,	,	PUNCT
ejde-203	261	15	m.	m.	PROPN
ejde-203	261	16	c.	c.	PROPN
ejde-203	261	17	mahato	mahato	PROPN
ejde-203	261	18	;	;	PUNCT
ejde-203	261	19	resonance	resonance	NOUN
ejde-203	261	20	oscillation	oscillation	NOUN
ejde-203	261	21	of	of	ADP
ejde-203	261	22	a	a	DET
ejde-203	261	23	damped	damped	NOUN
ejde-203	261	24	driven	drive	VERB
ejde-203	261	25	simple	simple	ADJ
ejde-203	261	26	pendulum	pendulum	NOUN
ejde-203	261	27	.	.	PUNCT
ejde-203	262	1	european	european	PROPN
ejde-203	262	2	journal	journal	PROPN
ejde-203	262	3	of	of	ADP
ejde-203	262	4	physics	physics	PROPN
ejde-203	262	5	,	,	PUNCT
ejde-203	262	6	39(6	39(6	NUM
ejde-203	262	7	):	):	PUNCT
ejde-203	262	8	065002	065002	NUM
ejde-203	262	9	,	,	PUNCT
ejde-203	262	10	sept	sept	PROPN
ejde-203	262	11	.	.	PROPN
ejde-203	262	12	2018	2018	NUM
ejde-203	262	13	.	.	PUNCT
ejde-203	263	1	[	[	X
ejde-203	263	2	7	7	X
ejde-203	263	3	]	]	X
ejde-203	263	4	s.	s.	PROPN
ejde-203	263	5	wiggins	wiggins	PROPN
ejde-203	263	6	.	.	PUNCT
ejde-203	264	1	introduction	introduction	NOUN
ejde-203	264	2	to	to	AUX
ejde-203	264	3	applied	apply	VERB
ejde-203	264	4	nonlinear	nonlinear	ADJ
ejde-203	264	5	dynamical	dynamical	ADJ
ejde-203	264	6	systems	system	NOUN
ejde-203	264	7	and	and	CCONJ
ejde-203	264	8	chaos	chaos	NOUN
ejde-203	264	9	.	.	PUNCT
ejde-203	265	1	springer	springer	NOUN
ejde-203	265	2	,	,	PUNCT
ejde-203	265	3	1990	1990	NUM
ejde-203	265	4	.	.	PUNCT
ejde-203	266	1	[	[	X
ejde-203	266	2	8	8	NUM
ejde-203	266	3	]	]	X
ejde-203	266	4	l.	l.	PROPN
ejde-203	266	5	zonetti	zonetti	PROPN
ejde-203	266	6	,	,	PUNCT
ejde-203	266	7	a.	a.	NOUN
ejde-203	266	8	camargo	camargo	PROPN
ejde-203	266	9	,	,	PUNCT
ejde-203	266	10	j.	j.	PROPN
ejde-203	266	11	sartori	sartori	PROPN
ejde-203	266	12	,	,	PUNCT
ejde-203	266	13	d.	d.	PROPN
ejde-203	266	14	de	de	PROPN
ejde-203	266	15	sousa	sousa	PROPN
ejde-203	266	16	,	,	PUNCT
ejde-203	266	17	l.	l.	PROPN
ejde-203	266	18	nunes	nunes	PROPN
ejde-203	266	19	;	;	PUNCT
ejde-203	266	20	a	a	DET
ejde-203	266	21	demonstration	demonstration	NOUN
ejde-203	266	22	of	of	ADP
ejde-203	266	23	dry	dry	ADJ
ejde-203	266	24	and	and	CCONJ
ejde-203	266	25	viscous	viscous	ADJ
ejde-203	266	26	damping	damping	NOUN
ejde-203	266	27	of	of	ADP
ejde-203	266	28	an	an	DET
ejde-203	266	29	oscillating	oscillate	VERB
ejde-203	266	30	pendulum	pendulum	NOUN
ejde-203	266	31	.	.	PUNCT
ejde-203	267	1	european	european	PROPN
ejde-203	267	2	journal	journal	PROPN
ejde-203	267	3	of	of	ADP
ejde-203	267	4	physics	physics	PROPN
ejde-203	267	5	,	,	PUNCT
ejde-203	267	6	20(2	20(2	NUM
ejde-203	267	7	):	):	PUNCT
ejde-203	267	8	85–88	85–88	NUM
ejde-203	267	9	,	,	PUNCT
ejde-203	267	10	jan	jan	PROPN
ejde-203	267	11	.	.	PROPN
ejde-203	267	12	1999	1999	NUM
ejde-203	267	13	.	.	PUNCT
ejde-203	268	1	jaime	jaime	PROPN
ejde-203	268	2	arango	arango	PROPN
ejde-203	268	3	departmento	departmento	PROPN
ejde-203	268	4	de	de	PROPN
ejde-203	268	5	mathemáticas	mathemáticas	PROPN
ejde-203	268	6	,	,	PUNCT
ejde-203	268	7	universidad	universidad	PROPN
ejde-203	268	8	del	del	PROPN
ejde-203	268	9	valle	valle	PROPN
ejde-203	268	10	,	,	PUNCT
ejde-203	268	11	cali	cali	PROPN
ejde-203	268	12	76001	76001	NUM
ejde-203	268	13	,	,	PUNCT
ejde-203	268	14	colombia	colombia	PROPN
ejde-203	268	15	email	email	NOUN
ejde-203	268	16	address	address	NOUN
ejde-203	268	17	:	:	PUNCT
ejde-203	268	18	jaime.arango@correounivalle.edu.co	jaime.arango@correounivalle.edu.co	ADJ
ejde-203	268	19	1	1	X
ejde-203	268	20	.	.	PUNCT
ejde-203	268	21	introduction	introduction	NOUN
ejde-203	268	22	2	2	NUM
ejde-203	268	23	.	.	PUNCT
ejde-203	268	24	underdamped	underdampe	VERB
ejde-203	268	25	oscillations	oscillation	NOUN
ejde-203	268	26	3	3	NUM
ejde-203	268	27	.	.	PUNCT
ejde-203	268	28	role	role	NOUN
ejde-203	268	29	of	of	ADP
ejde-203	268	30	the	the	DET
ejde-203	268	31	viscous	viscous	ADJ
ejde-203	268	32	damping	damp	VERB
ejde-203	268	33	4	4	NUM
ejde-203	268	34	.	.	PUNCT
ejde-203	269	1	conclusions	conclusion	NOUN
ejde-203	269	2	and	and	CCONJ
ejde-203	269	3	final	final	ADJ
ejde-203	269	4	remarks	remark	NOUN
ejde-203	269	5	acknowledgements	acknowledgement	NOUN
ejde-203	269	6	references	reference	NOUN