id	sid	tid	token	lemma	pos
ap-2090	1	1	acta	acta	PROPN
ap-2090	1	2	polytechnica	polytechnica	PROPN
ap-2090	1	3	doi:10.14311	doi:10.14311	PROPN
ap-2090	1	4	/	/	SYM
ap-2090	1	5	ap.2014.54.0130	ap.2014.54.0130	PROPN
ap-2090	1	6	acta	acta	PROPN
ap-2090	1	7	polytechnica	polytechnica	PROPN
ap-2090	1	8	54(2):130–132	54(2):130–132	PROPN
ap-2090	1	9	,	,	PUNCT
ap-2090	1	10	2014	2014	NUM
ap-2090	1	11	©	©	PROPN
ap-2090	1	12	czech	czech	PROPN
ap-2090	1	13	technical	technical	PROPN
ap-2090	1	14	university	university	PROPN
ap-2090	1	15	in	in	ADP
ap-2090	1	16	prague	prague	PROPN
ap-2090	1	17	,	,	PUNCT
ap-2090	1	18	2014	2014	NUM
ap-2090	1	19	available	available	ADJ
ap-2090	1	20	online	online	ADV
ap-2090	1	21	at	at	ADP
ap-2090	1	22	http://ojs.cvut.cz/ojs/index.php/ap	http://ojs.cvut.cz/ojs/index.php/ap	PROPN
ap-2090	1	23	can	can	AUX
ap-2090	1	24	one	one	NUM
ap-2090	1	25	really	really	ADV
ap-2090	1	26	study	study	VERB
ap-2090	1	27	chaos	chaos	NOUN
ap-2090	1	28	analytically	analytically	ADV
ap-2090	1	29	?	?	PUNCT
ap-2090	2	1	m.	m.	NOUN
ap-2090	2	2	howard	howard	PROPN
ap-2090	2	3	leea	leea	PROPN
ap-2090	2	4	,	,	PUNCT
ap-2090	2	5	b	b	PROPN
ap-2090	2	6	a	a	DET
ap-2090	2	7	department	department	NOUN
ap-2090	2	8	of	of	ADP
ap-2090	2	9	physics	physics	PROPN
ap-2090	2	10	and	and	CCONJ
ap-2090	2	11	astronomy	astronomy	NOUN
ap-2090	2	12	,	,	PUNCT
ap-2090	2	13	university	university	NOUN
ap-2090	2	14	of	of	ADP
ap-2090	2	15	georgia	georgia	PROPN
ap-2090	2	16	,	,	PUNCT
ap-2090	2	17	athens	athens	PROPN
ap-2090	2	18	,	,	PUNCT
ap-2090	2	19	ga	ga	PROPN
ap-2090	2	20	30602	30602	NUM
ap-2090	2	21	usa	usa	PROPN
ap-2090	2	22	b	b	PROPN
ap-2090	2	23	korea	korea	PROPN
ap-2090	2	24	institute	institute	PROPN
ap-2090	2	25	for	for	ADP
ap-2090	2	26	advanced	advanced	ADJ
ap-2090	2	27	study	study	NOUN
ap-2090	2	28	,	,	PUNCT
ap-2090	2	29	seoul	seoul	PROPN
ap-2090	2	30	130	130	NUM
ap-2090	2	31	-	-	PUNCT
ap-2090	2	32	012	012	NUM
ap-2090	2	33	,	,	PUNCT
ap-2090	2	34	korea	korea	PROPN
ap-2090	2	35	correspondence	correspondence	NOUN
ap-2090	2	36	:	:	PUNCT
ap-2090	2	37	mhlee@uga.edu	mhlee@uga.edu	NOUN
ap-2090	2	38	abstract	abstract	ADJ
ap-2090	2	39	.	.	PUNCT
ap-2090	3	1	one	one	NUM
ap-2090	3	2	generally	generally	ADV
ap-2090	3	3	thinks	think	VERB
ap-2090	3	4	that	that	SCONJ
ap-2090	3	5	chaos	chaos	NOUN
ap-2090	3	6	can	can	AUX
ap-2090	3	7	be	be	AUX
ap-2090	3	8	studied	study	VERB
ap-2090	3	9	only	only	ADV
ap-2090	3	10	numerically	numerically	ADV
ap-2090	3	11	by	by	ADP
ap-2090	3	12	aid	aid	NOUN
ap-2090	3	13	of	of	ADP
ap-2090	3	14	the	the	DET
ap-2090	3	15	computer	computer	NOUN
ap-2090	3	16	.	.	PUNCT
ap-2090	4	1	it	it	PRON
ap-2090	4	2	is	be	AUX
ap-2090	4	3	however	however	ADV
ap-2090	4	4	suggested	suggest	VERB
ap-2090	4	5	by	by	ADP
ap-2090	4	6	the	the	DET
ap-2090	4	7	theorem	theorem	NOUN
ap-2090	4	8	of	of	ADP
ap-2090	4	9	sharkovskii	sharkovskii	PROPN
ap-2090	4	10	and	and	CCONJ
ap-2090	4	11	li	li	PROPN
ap-2090	4	12	and	and	CCONJ
ap-2090	4	13	yorke	yorke	PROPN
ap-2090	4	14	that	that	SCONJ
ap-2090	4	15	in	in	ADP
ap-2090	4	16	i	i	PROPN
ap-2090	4	17	d	d	PROPN
ap-2090	4	18	continuous	continuous	ADJ
ap-2090	4	19	maps	map	NOUN
ap-2090	4	20	analytical	analytical	ADJ
ap-2090	4	21	studies	study	NOUN
ap-2090	4	22	are	be	AUX
ap-2090	4	23	possible	possible	ADJ
ap-2090	4	24	.	.	PUNCT
ap-2090	5	1	how	how	SCONJ
ap-2090	5	2	one	one	PRON
ap-2090	5	3	might	might	AUX
ap-2090	5	4	achieve	achieve	VERB
ap-2090	5	5	such	such	DET
ap-2090	5	6	a	a	DET
ap-2090	5	7	goal	goal	NOUN
ap-2090	5	8	in	in	ADP
ap-2090	5	9	one	one	NUM
ap-2090	5	10	special	special	ADJ
ap-2090	5	11	map	map	NOUN
ap-2090	5	12	is	be	AUX
ap-2090	5	13	described	describe	VERB
ap-2090	5	14	.	.	PUNCT
ap-2090	6	1	keywords	keyword	NOUN
ap-2090	6	2	:	:	PUNCT
ap-2090	6	3	chaos	chaos	NOUN
ap-2090	6	4	,	,	PUNCT
ap-2090	6	5	logistic	logistic	ADJ
ap-2090	6	6	map	map	NOUN
ap-2090	6	7	,	,	PUNCT
ap-2090	6	8	sharkovskii	sharkovskii	PROPN
ap-2090	6	9	theorem	theorem	PROPN
ap-2090	6	10	.	.	PROPN
ap-2090	7	1	1	1	NUM
ap-2090	7	2	.	.	X
ap-2090	7	3	introduction	introduction	NOUN
ap-2090	7	4	when	when	SCONJ
ap-2090	7	5	one	one	NUM
ap-2090	7	6	thinks	think	VERB
ap-2090	7	7	of	of	ADP
ap-2090	7	8	chaos	chaos	NOUN
ap-2090	7	9	,	,	PUNCT
ap-2090	7	10	what	what	PRON
ap-2090	7	11	most	most	ADV
ap-2090	7	12	likely	likely	ADV
ap-2090	7	13	come	come	VERB
ap-2090	7	14	to	to	ADP
ap-2090	7	15	mind	mind	NOUN
ap-2090	7	16	are	be	AUX
ap-2090	7	17	beautiful	beautiful	ADJ
ap-2090	7	18	pictures	picture	NOUN
ap-2090	7	19	drawn	draw	VERB
ap-2090	7	20	by	by	ADP
ap-2090	7	21	the	the	DET
ap-2090	7	22	computer	computer	NOUN
ap-2090	7	23	.	.	PUNCT
ap-2090	8	1	in	in	ADP
ap-2090	8	2	fact	fact	NOUN
ap-2090	8	3	virtually	virtually	ADV
ap-2090	8	4	all	all	DET
ap-2090	8	5	articles	article	NOUN
ap-2090	8	6	and	and	CCONJ
ap-2090	8	7	books	book	NOUN
ap-2090	8	8	on	on	ADP
ap-2090	8	9	chaos	chaos	NOUN
ap-2090	8	10	have	have	VERB
ap-2090	8	11	some	some	DET
ap-2090	8	12	such	such	ADJ
ap-2090	8	13	pictures	picture	NOUN
ap-2090	8	14	illustrating	illustrate	VERB
ap-2090	8	15	unimagined	unimagined	ADJ
ap-2090	8	16	properties	property	NOUN
ap-2090	8	17	of	of	ADP
ap-2090	8	18	nonlinear	nonlinear	ADJ
ap-2090	8	19	behavior	behavior	NOUN
ap-2090	8	20	[	[	X
ap-2090	8	21	1,2	1,2	NUM
ap-2090	8	22	]	]	PUNCT
ap-2090	8	23	.	.	PUNCT
ap-2090	9	1	thus	thus	ADV
ap-2090	9	2	one	one	NUM
ap-2090	9	3	tends	tend	VERB
ap-2090	9	4	to	to	PART
ap-2090	9	5	think	think	VERB
ap-2090	9	6	that	that	SCONJ
ap-2090	9	7	an	an	DET
ap-2090	9	8	analytical	analytical	ADJ
ap-2090	9	9	approach	approach	NOUN
ap-2090	9	10	to	to	ADP
ap-2090	9	11	chaos	chaos	NOUN
ap-2090	9	12	is	be	AUX
ap-2090	9	13	nearly	nearly	ADV
ap-2090	9	14	unthinkable	unthinkable	ADJ
ap-2090	9	15	.	.	PUNCT
ap-2090	10	1	one	one	PRON
ap-2090	10	2	might	might	AUX
ap-2090	10	3	even	even	ADV
ap-2090	10	4	ask	ask	VERB
ap-2090	10	5	why	why	SCONJ
ap-2090	10	6	one	one	PRON
ap-2090	10	7	should	should	AUX
ap-2090	10	8	want	want	VERB
ap-2090	10	9	to	to	PART
ap-2090	10	10	attempt	attempt	VERB
ap-2090	10	11	it	it	PRON
ap-2090	10	12	.	.	PUNCT
ap-2090	11	1	if	if	SCONJ
ap-2090	11	2	it	it	PRON
ap-2090	11	3	were	be	AUX
ap-2090	11	4	,	,	PUNCT
ap-2090	11	5	however	however	ADV
ap-2090	11	6	,	,	PUNCT
ap-2090	11	7	possible	possible	ADJ
ap-2090	11	8	even	even	ADV
ap-2090	11	9	in	in	ADP
ap-2090	11	10	a	a	DET
ap-2090	11	11	special	special	ADJ
ap-2090	11	12	case	case	NOUN
ap-2090	11	13	or	or	CCONJ
ap-2090	11	14	cases	case	NOUN
ap-2090	11	15	,	,	PUNCT
ap-2090	11	16	it	it	PRON
ap-2090	11	17	could	could	AUX
ap-2090	11	18	help	help	VERB
ap-2090	11	19	provide	provide	VERB
ap-2090	11	20	perhaps	perhaps	ADV
ap-2090	11	21	a	a	DET
ap-2090	11	22	deeper	deep	ADJ
ap-2090	11	23	understanding	understanding	NOUN
ap-2090	11	24	.	.	PUNCT
ap-2090	12	1	this	this	DET
ap-2090	12	2	attitude	attitude	NOUN
ap-2090	12	3	is	be	AUX
ap-2090	12	4	to	to	ADP
ap-2090	12	5	our	our	PRON
ap-2090	12	6	mind	mind	NOUN
ap-2090	12	7	something	something	PRON
ap-2090	12	8	that	that	PRON
ap-2090	12	9	we	we	PRON
ap-2090	12	10	do	do	VERB
ap-2090	12	11	or	or	CCONJ
ap-2090	12	12	try	try	VERB
ap-2090	12	13	to	to	PART
ap-2090	12	14	do	do	VERB
ap-2090	12	15	in	in	ADP
ap-2090	12	16	physics	physics	NOUN
ap-2090	12	17	.	.	PUNCT
ap-2090	13	1	such	such	DET
ap-2090	13	2	a	a	DET
ap-2090	13	3	possibility	possibility	NOUN
ap-2090	13	4	is	be	AUX
ap-2090	13	5	opened	open	VERB
ap-2090	13	6	by	by	ADP
ap-2090	13	7	a	a	DET
ap-2090	13	8	remarkable	remarkable	ADJ
ap-2090	13	9	theorem	theorem	NOUN
ap-2090	13	10	proved	prove	VERB
ap-2090	13	11	by	by	ADP
ap-2090	13	12	sharkovskii	sharkovskii	PROPN
ap-2090	13	13	[	[	X
ap-2090	13	14	3	3	NUM
ap-2090	13	15	]	]	PUNCT
ap-2090	13	16	,	,	PUNCT
ap-2090	13	17	and	and	CCONJ
ap-2090	13	18	later	later	ADV
ap-2090	13	19	independently	independently	ADV
ap-2090	13	20	by	by	ADP
ap-2090	13	21	li	li	PROPN
ap-2090	13	22	and	and	CCONJ
ap-2090	13	23	yorke	yorke	PROPN
ap-2090	14	1	[	[	X
ap-2090	14	2	4	4	NUM
ap-2090	14	3	]	]	PUNCT
ap-2090	14	4	.	.	PUNCT
ap-2090	15	1	this	this	DET
ap-2090	15	2	theorem	theorem	NOUN
ap-2090	15	3	applies	apply	VERB
ap-2090	15	4	to	to	ADP
ap-2090	15	5	one	one	NUM
ap-2090	15	6	-	-	PUNCT
ap-2090	15	7	dimensional	dimensional	ADJ
ap-2090	15	8	continuous	continuous	ADJ
ap-2090	15	9	maps	map	NOUN
ap-2090	15	10	of	of	ADP
ap-2090	15	11	real	real	ADJ
ap-2090	15	12	numbers	number	NOUN
ap-2090	15	13	in	in	ADP
ap-2090	15	14	a	a	DET
ap-2090	15	15	finite	finite	ADJ
ap-2090	15	16	interval	interval	NOUN
ap-2090	15	17	.	.	PUNCT
ap-2090	16	1	the	the	DET
ap-2090	16	2	essence	essence	NOUN
ap-2090	16	3	of	of	ADP
ap-2090	16	4	the	the	DET
ap-2090	16	5	theorem	theorem	NOUN
ap-2090	16	6	reads	read	NOUN
ap-2090	16	7	that	that	SCONJ
ap-2090	16	8	if	if	SCONJ
ap-2090	16	9	3	3	NUM
ap-2090	16	10	-	-	PUNCT
ap-2090	16	11	cycle	cycle	NOUN
ap-2090	16	12	exists	exist	VERB
ap-2090	16	13	in	in	ADP
ap-2090	16	14	one	one	NUM
ap-2090	16	15	domain	domain	NOUN
ap-2090	16	16	of	of	ADP
ap-2090	16	17	such	such	DET
ap-2090	16	18	a	a	DET
ap-2090	16	19	map	map	NOUN
ap-2090	16	20	,	,	PUNCT
ap-2090	16	21	it	it	PRON
ap-2090	16	22	implies	imply	VERB
ap-2090	16	23	the	the	DET
ap-2090	16	24	existence	existence	NOUN
ap-2090	16	25	of	of	ADP
ap-2090	16	26	all	all	DET
ap-2090	16	27	other	other	ADJ
ap-2090	16	28	cycles	cycle	NOUN
ap-2090	16	29	in	in	ADP
ap-2090	16	30	the	the	DET
ap-2090	16	31	same	same	ADJ
ap-2090	16	32	domain	domain	NOUN
ap-2090	16	33	and	and	CCONJ
ap-2090	16	34	hence	hence	ADV
ap-2090	16	35	it	it	PRON
ap-2090	16	36	implies	imply	VERB
ap-2090	16	37	the	the	DET
ap-2090	16	38	existence	existence	NOUN
ap-2090	16	39	of	of	ADP
ap-2090	16	40	chaos	chaos	NOUN
ap-2090	16	41	itself	itself	PRON
ap-2090	16	42	therein	therein	ADV
ap-2090	16	43	.	.	PUNCT
ap-2090	17	1	what	what	PRON
ap-2090	17	2	is	be	AUX
ap-2090	17	3	especially	especially	ADV
ap-2090	17	4	significant	significant	ADJ
ap-2090	17	5	,	,	PUNCT
ap-2090	17	6	and	and	CCONJ
ap-2090	17	7	not	not	PART
ap-2090	17	8	yet	yet	ADV
ap-2090	17	9	widely	widely	ADV
ap-2090	17	10	recognized	recognize	VERB
ap-2090	17	11	,	,	PUNCT
ap-2090	17	12	is	be	AUX
ap-2090	17	13	that	that	SCONJ
ap-2090	17	14	by	by	ADP
ap-2090	17	15	means	mean	NOUN
ap-2090	17	16	of	of	ADP
ap-2090	17	17	3	3	NUM
ap-2090	17	18	-	-	PUNCT
ap-2090	17	19	cycle	cycle	NOUN
ap-2090	17	20	a	a	DET
ap-2090	17	21	chaotic	chaotic	ADJ
ap-2090	17	22	domain	domain	NOUN
ap-2090	17	23	can	can	AUX
ap-2090	17	24	be	be	AUX
ap-2090	17	25	determined	determine	VERB
ap-2090	17	26	without	without	ADP
ap-2090	17	27	introducing	introduce	VERB
ap-2090	17	28	a	a	DET
ap-2090	17	29	working	work	VERB
ap-2090	17	30	definition	definition	NOUN
ap-2090	17	31	of	of	ADP
ap-2090	17	32	chaos	chaos	NOUN
ap-2090	17	33	.	.	PUNCT
ap-2090	18	1	this	this	PRON
ap-2090	18	2	is	be	AUX
ap-2090	18	3	contrary	contrary	ADJ
ap-2090	18	4	to	to	ADP
ap-2090	18	5	what	what	PRON
ap-2090	18	6	is	be	AUX
ap-2090	18	7	normally	normally	ADV
ap-2090	18	8	done	do	VERB
ap-2090	18	9	in	in	ADP
ap-2090	18	10	numerical	numerical	ADJ
ap-2090	18	11	studies	study	NOUN
ap-2090	18	12	.	.	PUNCT
ap-2090	19	1	in	in	ADP
ap-2090	19	2	these	these	DET
ap-2090	19	3	studies	study	NOUN
ap-2090	19	4	,	,	PUNCT
ap-2090	19	5	a	a	DET
ap-2090	19	6	domain	domain	NOUN
ap-2090	19	7	is	be	AUX
ap-2090	19	8	said	say	VERB
ap-2090	19	9	to	to	PART
ap-2090	19	10	be	be	AUX
ap-2090	19	11	chaotic	chaotic	ADJ
ap-2090	19	12	if	if	SCONJ
ap-2090	19	13	it	it	PRON
ap-2090	19	14	meets	meet	VERB
ap-2090	19	15	a	a	DET
ap-2090	19	16	certain	certain	ADJ
ap-2090	19	17	criterion	criterion	NOUN
ap-2090	19	18	or	or	CCONJ
ap-2090	19	19	a	a	DET
ap-2090	19	20	set	set	NOUN
ap-2090	19	21	of	of	ADP
ap-2090	19	22	criteria	criterion	NOUN
ap-2090	19	23	,	,	PUNCT
ap-2090	19	24	which	which	PRON
ap-2090	19	25	represent	represent	VERB
ap-2090	19	26	working	work	VERB
ap-2090	19	27	definitions	definition	NOUN
ap-2090	19	28	of	of	ADP
ap-2090	19	29	chaos	chaos	NOUN
ap-2090	19	30	.	.	PUNCT
ap-2090	20	1	these	these	DET
ap-2090	20	2	definitions	definition	NOUN
ap-2090	20	3	are	be	AUX
ap-2090	20	4	phenomenologically	phenomenologically	ADV
ap-2090	20	5	based	base	VERB
ap-2090	20	6	,	,	PUNCT
ap-2090	20	7	not	not	PART
ap-2090	20	8	derived	derive	VERB
ap-2090	20	9	from	from	ADP
ap-2090	20	10	some	some	DET
ap-2090	20	11	higher	high	ADJ
ap-2090	20	12	principles	principle	NOUN
ap-2090	20	13	of	of	ADP
ap-2090	20	14	chaos	chaos	NOUN
ap-2090	20	15	.	.	PUNCT
ap-2090	21	1	we	we	PRON
ap-2090	21	2	will	will	AUX
ap-2090	21	3	not	not	PART
ap-2090	21	4	go	go	VERB
ap-2090	21	5	into	into	ADP
ap-2090	21	6	why	why	SCONJ
ap-2090	21	7	3	3	NUM
ap-2090	21	8	-	-	PUNCT
ap-2090	21	9	cycle	cycle	NOUN
ap-2090	21	10	plays	play	VERB
ap-2090	21	11	this	this	DET
ap-2090	21	12	pivotal	pivotal	ADJ
ap-2090	21	13	role	role	NOUN
ap-2090	21	14	,	,	PUNCT
ap-2090	21	15	for	for	ADP
ap-2090	21	16	this	this	DET
ap-2090	21	17	subject	subject	NOUN
ap-2090	21	18	is	be	AUX
ap-2090	21	19	now	now	ADV
ap-2090	21	20	well	well	ADV
ap-2090	21	21	known	know	VERB
ap-2090	21	22	.	.	PUNCT
ap-2090	22	1	what	what	PRON
ap-2090	22	2	we	we	PRON
ap-2090	22	3	will	will	AUX
ap-2090	22	4	do	do	AUX
ap-2090	22	5	is	be	AUX
ap-2090	22	6	to	to	PART
ap-2090	22	7	demonstrate	demonstrate	VERB
ap-2090	22	8	the	the	DET
ap-2090	22	9	existence	existence	NOUN
ap-2090	22	10	of	of	ADP
ap-2090	22	11	3	3	NUM
ap-2090	22	12	-	-	PUNCT
ap-2090	22	13	cycle	cycle	NOUN
ap-2090	22	14	in	in	ADP
ap-2090	22	15	an	an	DET
ap-2090	22	16	applicable	applicable	ADJ
ap-2090	22	17	map	map	NOUN
ap-2090	22	18	and	and	CCONJ
ap-2090	22	19	how	how	SCONJ
ap-2090	22	20	we	we	PRON
ap-2090	22	21	can	can	AUX
ap-2090	22	22	learn	learn	VERB
ap-2090	22	23	about	about	ADP
ap-2090	22	24	chaos	chaos	NOUN
ap-2090	22	25	from	from	ADP
ap-2090	22	26	the	the	DET
ap-2090	22	27	content	content	NOUN
ap-2090	22	28	of	of	ADP
ap-2090	22	29	the	the	DET
ap-2090	22	30	theorem	theorem	NOUN
ap-2090	22	31	.	.	PUNCT
ap-2090	23	1	this	this	DET
ap-2090	23	2	article	article	NOUN
ap-2090	23	3	being	be	AUX
ap-2090	23	4	intended	intend	VERB
ap-2090	23	5	as	as	ADP
ap-2090	23	6	an	an	DET
ap-2090	23	7	overview	overview	NOUN
ap-2090	23	8	,	,	PUNCT
ap-2090	23	9	we	we	PRON
ap-2090	23	10	will	will	AUX
ap-2090	23	11	not	not	PART
ap-2090	23	12	present	present	VERB
ap-2090	23	13	any	any	DET
ap-2090	23	14	detailed	detailed	ADJ
ap-2090	23	15	derivations	derivation	NOUN
ap-2090	23	16	,	,	PUNCT
ap-2090	23	17	just	just	ADV
ap-2090	23	18	ideas	idea	NOUN
ap-2090	23	19	of	of	ADP
ap-2090	23	20	what	what	PRON
ap-2090	23	21	can	can	AUX
ap-2090	23	22	be	be	AUX
ap-2090	23	23	done	do	VERB
ap-2090	23	24	.	.	PUNCT
ap-2090	24	1	2	2	X
ap-2090	24	2	.	.	X
ap-2090	24	3	one	one	NUM
ap-2090	24	4	-	-	PUNCT
ap-2090	24	5	dimensional	dimensional	ADJ
ap-2090	24	6	continuous	continuous	ADJ
ap-2090	24	7	map	map	NOUN
ap-2090	24	8	of	of	ADP
ap-2090	24	9	real	real	ADJ
ap-2090	24	10	numbers	number	NOUN
ap-2090	24	11	in	in	ADP
ap-2090	24	12	an	an	DET
ap-2090	24	13	interval	interval	NOUN
ap-2090	24	14	let	let	VERB
ap-2090	24	15	xi	xi	PROPN
ap-2090	24	16	and	and	CCONJ
ap-2090	24	17	xj	xj	PROPN
ap-2090	24	18	be	be	AUX
ap-2090	24	19	two	two	NUM
ap-2090	24	20	real	real	ADJ
ap-2090	24	21	numbers	number	NOUN
ap-2090	24	22	in	in	ADP
ap-2090	24	23	(	(	PUNCT
ap-2090	24	24	0,1	0,1	NUM
ap-2090	24	25	)	)	PUNCT
ap-2090	24	26	.	.	PUNCT
ap-2090	25	1	if	if	SCONJ
ap-2090	25	2	xi	xi	PROPN
ap-2090	25	3	→	→	SYM
ap-2090	25	4	xj	xj	PROPN
ap-2090	25	5	by	by	ADP
ap-2090	25	6	an	an	DET
ap-2090	25	7	action	action	NOUN
ap-2090	25	8	f	f	NOUN
ap-2090	25	9	,	,	PUNCT
ap-2090	25	10	we	we	PRON
ap-2090	25	11	write	write	VERB
ap-2090	25	12	the	the	DET
ap-2090	25	13	action	action	NOUN
ap-2090	25	14	as	as	ADP
ap-2090	25	15	xj	xj	PROPN
ap-2090	25	16	=	=	SYM
ap-2090	25	17	f(xi	f(xi	PROPN
ap-2090	25	18	)	)	PUNCT
ap-2090	25	19	,	,	PUNCT
ap-2090	25	20	calling	call	VERB
ap-2090	25	21	it	it	PRON
ap-2090	25	22	a	a	DET
ap-2090	25	23	map	map	NOUN
ap-2090	25	24	.	.	PUNCT
ap-2090	26	1	if	if	SCONJ
ap-2090	26	2	,	,	PUNCT
ap-2090	26	3	for	for	ADP
ap-2090	26	4	some	some	DET
ap-2090	26	5	value	value	NOUN
ap-2090	26	6	of	of	ADP
ap-2090	26	7	xi	xi	PROPN
ap-2090	26	8	,	,	PUNCT
ap-2090	26	9	say	say	VERB
ap-2090	26	10	x∗	x∗	PROPN
ap-2090	26	11	,	,	PUNCT
ap-2090	26	12	it	it	PRON
ap-2090	26	13	goes	go	VERB
ap-2090	26	14	into	into	ADP
ap-2090	26	15	itself	itself	PRON
ap-2090	26	16	and	and	CCONJ
ap-2090	26	17	goes	go	VERB
ap-2090	26	18	nowhere	nowhere	ADV
ap-2090	26	19	else	else	ADV
ap-2090	26	20	,	,	PUNCT
ap-2090	26	21	it	it	PRON
ap-2090	26	22	is	be	AUX
ap-2090	26	23	termed	term	VERB
ap-2090	26	24	a	a	DET
ap-2090	26	25	fixed	fix	VERB
ap-2090	26	26	point	point	NOUN
ap-2090	26	27	of	of	ADP
ap-2090	26	28	f	f	PROPN
ap-2090	26	29	,	,	PUNCT
ap-2090	26	30	i.e.	i.e.	X
ap-2090	26	31	,	,	PUNCT
ap-2090	26	32	f(x∗	f(x∗	NOUN
ap-2090	26	33	)	)	PUNCT
ap-2090	27	1	=	=	PUNCT
ap-2090	27	2	x∗.	x∗.	ADJ
ap-2090	27	3	to	to	ADP
ap-2090	27	4	a	a	DET
ap-2090	27	5	fixed	fix	VERB
ap-2090	27	6	point	point	NOUN
ap-2090	27	7	,	,	PUNCT
ap-2090	27	8	f	f	PROPN
ap-2090	27	9	acts	act	VERB
ap-2090	27	10	like	like	ADP
ap-2090	27	11	an	an	DET
ap-2090	27	12	identity	identity	NOUN
ap-2090	27	13	operation	operation	NOUN
ap-2090	27	14	.	.	PUNCT
ap-2090	28	1	let	let	VERB
ap-2090	28	2	x1	x1	NUM
ap-2090	28	3	,	,	PUNCT
ap-2090	28	4	x2	x2	PROPN
ap-2090	28	5	,	,	PUNCT
ap-2090	28	6	x3	x3	ADJ
ap-2090	28	7	,	,	PUNCT
ap-2090	28	8	..	..	PUNCT
ap-2090	28	9	be	be	AUX
ap-2090	28	10	a	a	DET
ap-2090	28	11	set	set	NOUN
ap-2090	28	12	of	of	ADP
ap-2090	28	13	points	point	NOUN
ap-2090	28	14	in	in	ADP
ap-2090	28	15	(	(	PUNCT
ap-2090	28	16	0,1	0,1	NOUN
ap-2090	28	17	)	)	PUNCT
ap-2090	28	18	,	,	PUNCT
ap-2090	28	19	not	not	PART
ap-2090	28	20	necessarily	necessarily	ADV
ap-2090	28	21	ordered	order	VERB
ap-2090	28	22	.	.	PUNCT
ap-2090	29	1	by	by	ADP
ap-2090	29	2	the	the	DET
ap-2090	29	3	action	action	NOUN
ap-2090	29	4	of	of	ADP
ap-2090	29	5	f	f	PROPN
ap-2090	29	6	,	,	PUNCT
ap-2090	29	7	let	let	VERB
ap-2090	29	8	f(x1	f(x1	NOUN
ap-2090	29	9	)	)	PUNCT
ap-2090	29	10	=	=	SYM
ap-2090	29	11	x2	x2	PROPN
ap-2090	29	12	,	,	PUNCT
ap-2090	29	13	f(x2	f(x2	NOUN
ap-2090	29	14	)	)	PUNCT
ap-2090	29	15	=	=	SYM
ap-2090	29	16	x3	x3	ADJ
ap-2090	29	17	,	,	PUNCT
ap-2090	29	18	.	.	PUNCT
ap-2090	29	19	.	.	PUNCT
ap-2090	30	1	.	.	PUNCT
ap-2090	31	1	,	,	PUNCT
ap-2090	31	2	f(xn	f(xn	X
ap-2090	31	3	)	)	PUNCT
ap-2090	31	4	=	=	SYM
ap-2090	31	5	xn+1	xn+1	NUM
ap-2090	31	6	etc	etc	X
ap-2090	31	7	.	.	X
ap-2090	31	8	,	,	PUNCT
ap-2090	31	9	a	a	DET
ap-2090	31	10	process	process	NOUN
ap-2090	31	11	termed	term	VERB
ap-2090	31	12	an	an	DET
ap-2090	31	13	iteration	iteration	NOUN
ap-2090	31	14	.	.	PUNCT
ap-2090	32	1	suppose	suppose	VERB
ap-2090	32	2	f(x3	f(x3	VERB
ap-2090	32	3	)	)	PUNCT
ap-2090	33	1	=	=	SYM
ap-2090	33	2	x1	x1	PROPN
ap-2090	33	3	,	,	PUNCT
ap-2090	33	4	i.e.	i.e.	X
ap-2090	33	5	,	,	PUNCT
ap-2090	33	6	x4	x4	PROPN
ap-2090	33	7	=	=	SYM
ap-2090	33	8	x1	x1	PROPN
ap-2090	33	9	.	.	PUNCT
ap-2090	34	1	the	the	DET
ap-2090	34	2	iteration	iteration	NOUN
ap-2090	34	3	has	have	AUX
ap-2090	34	4	terminated	terminate	VERB
ap-2090	34	5	in	in	ADP
ap-2090	34	6	3	3	NUM
ap-2090	34	7	steps	step	NOUN
ap-2090	34	8	giving	give	VERB
ap-2090	34	9	the	the	DET
ap-2090	34	10	3	3	NUM
ap-2090	34	11	-	-	PUNCT
ap-2090	34	12	cycle	cycle	NOUN
ap-2090	34	13	condition	condition	NOUN
ap-2090	34	14	:	:	PUNCT
ap-2090	34	15	f(x1	f(x1	X
ap-2090	34	16	)	)	PUNCT
ap-2090	34	17	=	=	SYM
ap-2090	34	18	x2	x2	PROPN
ap-2090	34	19	,	,	PUNCT
ap-2090	34	20	f(x2	f(x2	NOUN
ap-2090	34	21	)	)	PUNCT
ap-2090	34	22	=	=	SYM
ap-2090	34	23	x3	x3	ADJ
ap-2090	34	24	,	,	PUNCT
ap-2090	34	25	and	and	CCONJ
ap-2090	34	26	f(x3	f(x3	NUM
ap-2090	34	27	)	)	PUNCT
ap-2090	35	1	=	=	SYM
ap-2090	35	2	x1	x1	PROPN
ap-2090	35	3	,	,	PUNCT
ap-2090	35	4	which	which	PRON
ap-2090	35	5	may	may	AUX
ap-2090	35	6	be	be	AUX
ap-2090	35	7	expressed	express	VERB
ap-2090	35	8	as	as	ADP
ap-2090	35	9	f(f(f(x)))≡f3(x	f(f(f(x)))≡f3(x	PROPN
ap-2090	35	10	)	)	PUNCT
ap-2090	35	11	=	=	SYM
ap-2090	36	1	x	x	NOUN
ap-2090	36	2	,	,	PUNCT
ap-2090	36	3	where	where	SCONJ
ap-2090	36	4	x	x	X
ap-2090	36	5	=	=	SYM
ap-2090	36	6	x1	x1	PROPN
ap-2090	36	7	,	,	PUNCT
ap-2090	36	8	x2	x2	PROPN
ap-2090	36	9	,	,	PUNCT
ap-2090	36	10	or	or	CCONJ
ap-2090	36	11	x3	x3	ADJ
ap-2090	36	12	,	,	PUNCT
ap-2090	36	13	and	and	CCONJ
ap-2090	36	14	x1	x1	NUM
ap-2090	36	15	6	6	NUM
ap-2090	36	16	=	=	SYM
ap-2090	36	17	x2	x2	PROPN
ap-2090	36	18	6	6	NUM
ap-2090	36	19	=	=	SYM
ap-2090	36	20	x3	x3	ADJ
ap-2090	36	21	.	.	PUNCT
ap-2090	37	1	these	these	DET
ap-2090	37	2	three	three	NUM
ap-2090	37	3	values	value	NOUN
ap-2090	37	4	are	be	AUX
ap-2090	37	5	fixed	fix	VERB
ap-2090	37	6	points	point	NOUN
ap-2090	37	7	of	of	ADP
ap-2090	37	8	f3	f3	NOUN
ap-2090	37	9	,	,	PUNCT
ap-2090	37	10	giving	give	VERB
ap-2090	37	11	3	3	NUM
ap-2090	37	12	-	-	PUNCT
ap-2090	37	13	cycle	cycle	NOUN
ap-2090	37	14	.	.	PUNCT
ap-2090	38	1	excluded	exclude	VERB
ap-2090	38	2	are	be	AUX
ap-2090	38	3	fixed	fix	VERB
ap-2090	38	4	points	point	NOUN
ap-2090	38	5	of	of	ADP
ap-2090	38	6	f	f	NOUN
ap-2090	38	7	,	,	PUNCT
ap-2090	38	8	which	which	PRON
ap-2090	38	9	are	be	AUX
ap-2090	38	10	also	also	ADV
ap-2090	38	11	fixed	fix	VERB
ap-2090	38	12	points	point	NOUN
ap-2090	38	13	of	of	ADP
ap-2090	38	14	f3	f3	NOUN
ap-2090	38	15	,	,	PUNCT
ap-2090	38	16	since	since	SCONJ
ap-2090	38	17	they	they	PRON
ap-2090	38	18	do	do	AUX
ap-2090	38	19	not	not	PART
ap-2090	38	20	satisfy	satisfy	VERB
ap-2090	38	21	the	the	DET
ap-2090	38	22	3	3	NUM
ap-2090	38	23	-	-	PUNCT
ap-2090	38	24	cycle	cycle	NOUN
ap-2090	38	25	condition	condition	NOUN
ap-2090	38	26	.	.	PUNCT
ap-2090	39	1	if	if	SCONJ
ap-2090	39	2	f(xn	f(xn	NOUN
ap-2090	39	3	)	)	PUNCT
ap-2090	40	1	=	=	SYM
ap-2090	40	2	x1	x1	PROPN
ap-2090	40	3	,	,	PUNCT
ap-2090	40	4	it	it	PRON
ap-2090	40	5	defines	define	VERB
ap-2090	40	6	n	n	PRON
ap-2090	40	7	-cycle	-cycle	NOUN
ap-2090	40	8	.	.	PUNCT
ap-2090	41	1	it	it	PRON
ap-2090	41	2	would	would	AUX
ap-2090	41	3	be	be	AUX
ap-2090	41	4	termed	term	VERB
ap-2090	41	5	periodic	periodic	ADJ
ap-2090	41	6	since	since	ADV
ap-2090	41	7	,	,	PUNCT
ap-2090	41	8	like	like	ADP
ap-2090	41	9	an	an	DET
ap-2090	41	10	orbital	orbital	ADJ
ap-2090	41	11	motion	motion	NOUN
ap-2090	41	12	,	,	PUNCT
ap-2090	41	13	it	it	PRON
ap-2090	41	14	returns	return	VERB
ap-2090	41	15	to	to	ADP
ap-2090	41	16	its	its	PRON
ap-2090	41	17	initial	initial	ADJ
ap-2090	41	18	point	point	NOUN
ap-2090	41	19	after	after	ADP
ap-2090	41	20	n	n	ADP
ap-2090	41	21	steps	step	NOUN
ap-2090	41	22	.	.	PUNCT
ap-2090	42	1	if	if	SCONJ
ap-2090	42	2	,	,	PUNCT
ap-2090	42	3	however	however	ADV
ap-2090	42	4	,	,	PUNCT
ap-2090	42	5	f(xn	f(xn	PROPN
ap-2090	42	6	)	)	PUNCT
ap-2090	42	7	=	=	SYM
ap-2090	42	8	xn+1	xn+1	PROPN
ap-2090	42	9	,	,	PUNCT
ap-2090	42	10	n	n	PRON
ap-2090	42	11	→∞	→∞	NOUN
ap-2090	42	12	,	,	PUNCT
ap-2090	42	13	never	never	ADV
ap-2090	42	14	returning	return	VERB
ap-2090	42	15	to	to	ADP
ap-2090	42	16	its	its	PRON
ap-2090	42	17	initial	initial	ADJ
ap-2090	42	18	point	point	NOUN
ap-2090	42	19	,	,	PUNCT
ap-2090	42	20	it	it	PRON
ap-2090	42	21	would	would	AUX
ap-2090	42	22	be	be	AUX
ap-2090	42	23	chaotic	chaotic	ADJ
ap-2090	42	24	,	,	PUNCT
ap-2090	42	25	like	like	ADP
ap-2090	42	26	a	a	DET
ap-2090	42	27	trajectory	trajectory	NOUN
ap-2090	42	28	which	which	PRON
ap-2090	42	29	never	never	ADV
ap-2090	42	30	closes	close	VERB
ap-2090	42	31	.	.	PUNCT
ap-2090	43	1	even	even	ADV
ap-2090	43	2	at	at	ADP
ap-2090	43	3	this	this	DET
ap-2090	43	4	stage	stage	NOUN
ap-2090	43	5	,	,	PUNCT
ap-2090	43	6	one	one	PRON
ap-2090	43	7	might	might	AUX
ap-2090	43	8	ask	ask	VERB
ap-2090	43	9	whether	whether	SCONJ
ap-2090	43	10	there	there	PRON
ap-2090	43	11	are	be	VERB
ap-2090	43	12	some	some	DET
ap-2090	43	13	properties	property	NOUN
ap-2090	43	14	of	of	ADP
ap-2090	43	15	the	the	DET
ap-2090	43	16	interval	interval	NOUN
ap-2090	43	17	of	of	ADP
ap-2090	43	18	points	point	NOUN
ap-2090	43	19	,	,	PUNCT
ap-2090	43	20	to	to	PART
ap-2090	43	21	which	which	PRON
ap-2090	43	22	the	the	DET
ap-2090	43	23	two	two	NUM
ap-2090	43	24	fundamentally	fundamentally	ADV
ap-2090	43	25	different	different	ADJ
ap-2090	43	26	kinds	kind	NOUN
ap-2090	43	27	of	of	ADP
ap-2090	43	28	trajectory	trajectory	NOUN
ap-2090	43	29	could	could	AUX
ap-2090	43	30	be	be	AUX
ap-2090	43	31	ascribed	ascribe	VERB
ap-2090	43	32	.	.	PUNCT
ap-2090	44	1	a	a	DET
ap-2090	44	2	periodic	periodic	ADJ
ap-2090	44	3	trajectory	trajectory	NOUN
ap-2090	44	4	has	have	VERB
ap-2090	44	5	a	a	DET
ap-2090	44	6	finite	finite	ADJ
ap-2090	44	7	set	set	NOUN
ap-2090	44	8	of	of	ADP
ap-2090	44	9	“	"	PUNCT
ap-2090	44	10	frequencies	frequency	NOUN
ap-2090	44	11	”	"	PUNCT
ap-2090	44	12	.	.	PUNCT
ap-2090	45	1	they	they	PRON
ap-2090	45	2	are	be	AUX
ap-2090	45	3	probably	probably	ADV
ap-2090	45	4	connected	connect	VERB
ap-2090	45	5	one	one	NUM
ap-2090	45	6	with	with	ADP
ap-2090	45	7	another	another	PRON
ap-2090	45	8	through	through	ADP
ap-2090	45	9	a	a	DET
ap-2090	45	10	sequence	sequence	NOUN
ap-2090	45	11	of	of	ADP
ap-2090	45	12	rational	rational	ADJ
ap-2090	45	13	numbers	number	NOUN
ap-2090	45	14	that	that	PRON
ap-2090	45	15	fill	fill	VERB
ap-2090	45	16	the	the	DET
ap-2090	45	17	interval	interval	NOUN
ap-2090	45	18	.	.	PUNCT
ap-2090	46	1	a	a	DET
ap-2090	46	2	chaotic	chaotic	ADJ
ap-2090	46	3	trajectory	trajectory	NOUN
ap-2090	46	4	goes	go	VERB
ap-2090	46	5	from	from	ADP
ap-2090	46	6	one	one	NUM
ap-2090	46	7	point	point	NOUN
ap-2090	46	8	to	to	ADP
ap-2090	46	9	another	another	PRON
ap-2090	46	10	in	in	ADP
ap-2090	46	11	a	a	DET
ap-2090	46	12	finite	finite	ADJ
ap-2090	46	13	interval	interval	NOUN
ap-2090	46	14	unpredictably	unpredictably	ADV
ap-2090	46	15	and	and	CCONJ
ap-2090	46	16	never	never	ADV
ap-2090	46	17	returning	return	VERB
ap-2090	46	18	.	.	PUNCT
ap-2090	47	1	this	this	DET
ap-2090	47	2	kind	kind	NOUN
ap-2090	47	3	of	of	ADP
ap-2090	47	4	behavior	behavior	NOUN
ap-2090	47	5	suggests	suggest	VERB
ap-2090	47	6	that	that	SCONJ
ap-2090	47	7	to	to	PART
ap-2090	47	8	describe	describe	VERB
ap-2090	47	9	its	its	PRON
ap-2090	47	10	movement	movement	NOUN
ap-2090	47	11	one	one	PRON
ap-2090	47	12	would	would	AUX
ap-2090	47	13	have	have	VERB
ap-2090	47	14	to	to	PART
ap-2090	47	15	bring	bring	VERB
ap-2090	47	16	in	in	ADP
ap-2090	47	17	the	the	DET
ap-2090	47	18	irrational	irrational	ADJ
ap-2090	47	19	numbers	number	NOUN
ap-2090	47	20	that	that	PRON
ap-2090	47	21	fill	fill	VERB
ap-2090	47	22	the	the	DET
ap-2090	47	23	interval	interval	NOUN
ap-2090	47	24	.	.	PUNCT
ap-2090	48	1	returning	return	VERB
ap-2090	48	2	to	to	ADP
ap-2090	48	3	the	the	DET
ap-2090	48	4	theorem	theorem	NOUN
ap-2090	48	5	of	of	ADP
ap-2090	48	6	sharkovskii	sharkovskii	PROPN
ap-2090	48	7	and	and	CCONJ
ap-2090	48	8	liyorke	liyorke	PROPN
ap-2090	48	9	,	,	PUNCT
ap-2090	48	10	what	what	PRON
ap-2090	48	11	is	be	AUX
ap-2090	48	12	required	require	VERB
ap-2090	48	13	is	be	AUX
ap-2090	48	14	to	to	PART
ap-2090	48	15	obtain	obtain	VERB
ap-2090	48	16	the	the	DET
ap-2090	48	17	3	3	NUM
ap-2090	48	18	fixed	fix	VERB
ap-2090	48	19	points	point	NOUN
ap-2090	48	20	of	of	ADP
ap-2090	48	21	3	3	NUM
ap-2090	48	22	-	-	PUNCT
ap-2090	48	23	cycle	cycle	NOUN
ap-2090	48	24	for	for	ADP
ap-2090	48	25	a	a	DET
ap-2090	48	26	particular	particular	ADJ
ap-2090	48	27	1d	1d	NUM
ap-2090	48	28	continuous	continuous	ADJ
ap-2090	48	29	map	map	NOUN
ap-2090	48	30	.	.	PUNCT
ap-2090	49	1	it	it	PRON
ap-2090	49	2	amounts	amount	VERB
ap-2090	49	3	to	to	ADP
ap-2090	49	4	solving	solve	VERB
ap-2090	49	5	for	for	ADP
ap-2090	49	6	three	three	NUM
ap-2090	49	7	roots	root	NOUN
ap-2090	49	8	of	of	ADP
ap-2090	49	9	f3(x	f3(x	NOUN
ap-2090	49	10	)	)	PUNCT
ap-2090	49	11	−	−	NOUN
ap-2090	50	1	x	x	SYM
ap-2090	51	1	=	=	SYM
ap-2090	51	2	0	0	NUM
ap-2090	51	3	,	,	PUNCT
ap-2090	51	4	probably	probably	ADV
ap-2090	51	5	a	a	DET
ap-2090	51	6	cubic	cubic	ADJ
ap-2090	51	7	equation	equation	NOUN
ap-2090	51	8	,	,	PUNCT
ap-2090	51	9	which	which	PRON
ap-2090	51	10	should	should	AUX
ap-2090	51	11	pose	pose	VERB
ap-2090	51	12	no	no	DET
ap-2090	51	13	special	special	ADJ
ap-2090	51	14	difficulty	difficulty	NOUN
ap-2090	51	15	.	.	PUNCT
ap-2090	52	1	had	have	VERB
ap-2090	52	2	the	the	DET
ap-2090	52	3	theorem	theorem	NOUN
ap-2090	52	4	called	call	VERB
ap-2090	52	5	for	for	ADP
ap-2090	52	6	4	4	NUM
ap-2090	52	7	-	-	PUNCT
ap-2090	52	8	cycle	cycle	NOUN
ap-2090	52	9	,	,	PUNCT
ap-2090	52	10	presenting	present	VERB
ap-2090	52	11	a	a	DET
ap-2090	52	12	quartic	quartic	ADJ
ap-2090	52	13	equation	equation	NOUN
ap-2090	52	14	,	,	PUNCT
ap-2090	52	15	it	it	PRON
ap-2090	52	16	would	would	AUX
ap-2090	52	17	still	still	ADV
ap-2090	52	18	be	be	AUX
ap-2090	52	19	no	no	DET
ap-2090	52	20	challenge	challenge	NOUN
ap-2090	52	21	.	.	PUNCT
ap-2090	53	1	had	have	AUX
ap-2090	53	2	it	it	PRON
ap-2090	53	3	asked	ask	VERB
ap-2090	53	4	for	for	ADP
ap-2090	53	5	5	5	NUM
ap-2090	53	6	-	-	PUNCT
ap-2090	53	7	cycle	cycle	NOUN
ap-2090	53	8	,	,	PUNCT
ap-2090	53	9	any	any	DET
ap-2090	53	10	attempt	attempt	NOUN
ap-2090	53	11	at	at	ADP
ap-2090	53	12	an	an	DET
ap-2090	53	13	analytical	analytical	ADJ
ap-2090	53	14	solution	solution	NOUN
ap-2090	53	15	would	would	AUX
ap-2090	53	16	have	have	VERB
ap-2090	53	17	to	to	PART
ap-2090	53	18	be	be	AUX
ap-2090	53	19	abandoned	abandon	VERB
ap-2090	53	20	.	.	PUNCT
ap-2090	54	1	fortuitously	fortuitously	ADV
ap-2090	54	2	it	it	PRON
ap-2090	54	3	is	be	AUX
ap-2090	54	4	3	3	NUM
ap-2090	54	5	-	-	PUNCT
ap-2090	54	6	cycle	cycle	NOUN
ap-2090	54	7	that	that	SCONJ
ap-2090	54	8	the	the	DET
ap-2090	54	9	theorem	theorem	NOUN
ap-2090	54	10	asks	ask	VERB
ap-2090	54	11	for	for	ADP
ap-2090	54	12	and	and	CCONJ
ap-2090	54	13	this	this	PRON
ap-2090	54	14	is	be	AUX
ap-2090	54	15	why	why	SCONJ
ap-2090	54	16	an	an	DET
ap-2090	54	17	analytical	analytical	ADJ
ap-2090	54	18	study	study	NOUN
ap-2090	54	19	seems	seem	VERB
ap-2090	54	20	feasible	feasible	ADJ
ap-2090	54	21	.	.	PUNCT
ap-2090	55	1	but	but	CCONJ
ap-2090	55	2	we	we	PRON
ap-2090	55	3	can	can	AUX
ap-2090	55	4	not	not	PART
ap-2090	55	5	know	know	VERB
ap-2090	55	6	for	for	ADP
ap-2090	55	7	sure	sure	ADJ
ap-2090	55	8	until	until	SCONJ
ap-2090	55	9	the	the	DET
ap-2090	55	10	3	3	NUM
ap-2090	55	11	-	-	PUNCT
ap-2090	55	12	cycle	cycle	NOUN
ap-2090	55	13	of	of	ADP
ap-2090	55	14	an	an	DET
ap-2090	55	15	applicable	applicable	ADJ
ap-2090	55	16	map	map	NOUN
ap-2090	55	17	is	be	AUX
ap-2090	55	18	constructed	construct	VERB
ap-2090	55	19	.	.	PUNCT
ap-2090	56	1	we	we	PRON
ap-2090	56	2	shall	shall	AUX
ap-2090	56	3	now	now	ADV
ap-2090	56	4	turn	turn	VERB
ap-2090	56	5	to	to	ADP
ap-2090	56	6	one	one	NUM
ap-2090	56	7	such	such	ADJ
ap-2090	56	8	map	map	NOUN
ap-2090	56	9	to	to	PART
ap-2090	56	10	see	see	VERB
ap-2090	56	11	what	what	PRON
ap-2090	56	12	is	be	AUX
ap-2090	56	13	to	to	PART
ap-2090	56	14	be	be	AUX
ap-2090	56	15	realized	realize	VERB
ap-2090	56	16	.	.	PUNCT
ap-2090	57	1	130	130	NUM
ap-2090	57	2	http://dx.doi.org/10.14311/ap.2014.54.0130	http://dx.doi.org/10.14311/ap.2014.54.0130	NOUN
ap-2090	57	3	http://ojs.cvut.cz/ojs/index.php/ap	http://ojs.cvut.cz/ojs/index.php/ap	PROPN
ap-2090	57	4	vol	vol	NOUN
ap-2090	57	5	.	.	PUNCT
ap-2090	58	1	54	54	NUM
ap-2090	58	2	no	no	NOUN
ap-2090	58	3	.	.	PUNCT
ap-2090	59	1	2/2014	2/2014	PROPN
ap-2090	59	2	can	can	AUX
ap-2090	59	3	one	one	NUM
ap-2090	59	4	really	really	ADV
ap-2090	59	5	study	study	VERB
ap-2090	59	6	chaos	chaos	NOUN
ap-2090	59	7	analytically	analytically	ADV
ap-2090	59	8	?	?	PUNCT
ap-2090	60	1	3	3	X
ap-2090	60	2	.	.	X
ap-2090	60	3	logistic	logistic	ADJ
ap-2090	60	4	map	map	NOUN
ap-2090	60	5	perhaps	perhaps	ADV
ap-2090	60	6	one	one	NUM
ap-2090	60	7	of	of	ADP
ap-2090	60	8	the	the	DET
ap-2090	60	9	simplest	simple	ADJ
ap-2090	60	10	and	and	CCONJ
ap-2090	60	11	no	no	ADV
ap-2090	60	12	doubt	doubt	ADV
ap-2090	60	13	the	the	DET
ap-2090	60	14	most	most	ADV
ap-2090	60	15	widely	widely	ADV
ap-2090	60	16	studied	study	VERB
ap-2090	60	17	among	among	ADP
ap-2090	60	18	all	all	DET
ap-2090	60	19	chaotic	chaotic	ADJ
ap-2090	60	20	maps	map	NOUN
ap-2090	60	21	is	be	AUX
ap-2090	60	22	the	the	DET
ap-2090	60	23	logistic	logistic	ADJ
ap-2090	60	24	map	map	NOUN
ap-2090	60	25	.	.	PUNCT
ap-2090	61	1	its	its	PRON
ap-2090	61	2	widespread	widespread	ADJ
ap-2090	61	3	appeal	appeal	NOUN
ap-2090	61	4	recalls	recall	VERB
ap-2090	61	5	the	the	DET
ap-2090	61	6	ising	ise	VERB
ap-2090	61	7	model	model	NOUN
ap-2090	61	8	of	of	ADP
ap-2090	61	9	magnetism	magnetism	NOUN
ap-2090	61	10	in	in	ADP
ap-2090	61	11	statistical	statistical	ADJ
ap-2090	61	12	mechanics	mechanic	NOUN
ap-2090	61	13	.	.	PUNCT
ap-2090	62	1	perhaps	perhaps	ADV
ap-2090	62	2	it	it	PRON
ap-2090	62	3	is	be	AUX
ap-2090	62	4	not	not	PART
ap-2090	62	5	too	too	ADV
ap-2090	62	6	far	far	ADV
ap-2090	62	7	-	-	PUNCT
ap-2090	62	8	fetched	fetched	ADJ
ap-2090	62	9	to	to	PART
ap-2090	62	10	call	call	VERB
ap-2090	62	11	the	the	DET
ap-2090	62	12	logistic	logistic	ADJ
ap-2090	62	13	map	map	NOUN
ap-2090	62	14	the	the	DET
ap-2090	62	15	ising	ise	VERB
ap-2090	62	16	model	model	NOUN
ap-2090	62	17	of	of	ADP
ap-2090	62	18	chaos	chaos	NOUN
ap-2090	62	19	.	.	PUNCT
ap-2090	63	1	the	the	DET
ap-2090	63	2	logistic	logistic	ADJ
ap-2090	63	3	map	map	NOUN
ap-2090	63	4	is	be	AUX
ap-2090	63	5	a	a	DET
ap-2090	63	6	one	one	NUM
ap-2090	63	7	-	-	PUNCT
ap-2090	63	8	dimensional	dimensional	ADJ
ap-2090	63	9	continuous	continuous	ADJ
ap-2090	63	10	map	map	NOUN
ap-2090	63	11	of	of	ADP
ap-2090	63	12	real	real	ADJ
ap-2090	63	13	numbers	number	NOUN
ap-2090	63	14	in	in	ADP
ap-2090	63	15	the	the	DET
ap-2090	63	16	interval	interval	NOUN
ap-2090	63	17	(	(	PUNCT
ap-2090	63	18	0	0	NUM
ap-2090	63	19	,	,	PUNCT
ap-2090	63	20	1	1	NUM
ap-2090	63	21	)	)	PUNCT
ap-2090	63	22	,	,	PUNCT
ap-2090	63	23	to	to	PART
ap-2090	63	24	which	which	PRON
ap-2090	63	25	the	the	DET
ap-2090	63	26	theorem	theorem	NOUN
ap-2090	63	27	of	of	ADP
ap-2090	63	28	sharkovskii	sharkovskii	PROPN
ap-2090	63	29	is	be	AUX
ap-2090	63	30	applicable	applicable	ADJ
ap-2090	63	31	.	.	PUNCT
ap-2090	64	1	the	the	DET
ap-2090	64	2	usual	usual	ADJ
ap-2090	64	3	way	way	NOUN
ap-2090	64	4	to	to	PART
ap-2090	64	5	define	define	VERB
ap-2090	64	6	the	the	DET
ap-2090	64	7	map	map	NOUN
ap-2090	64	8	is	be	AUX
ap-2090	64	9	as	as	ADP
ap-2090	64	10	x′	x′	PROPN
ap-2090	64	11	=	=	SYM
ap-2090	64	12	f(x	f(x	PROPN
ap-2090	64	13	)	)	PUNCT
ap-2090	64	14	,	,	PUNCT
ap-2090	64	15	where	where	SCONJ
ap-2090	64	16	f(x	f(x	NOUN
ap-2090	64	17	)	)	PUNCT
ap-2090	64	18	=	=	PUNCT
ap-2090	65	1	ax(1−	ax(1−	PROPN
ap-2090	65	2	x	x	NOUN
ap-2090	65	3	)	)	PUNCT
ap-2090	65	4	,	,	PUNCT
ap-2090	65	5	x	x	PUNCT
ap-2090	65	6	=	=	SYM
ap-2090	65	7	(	(	PUNCT
ap-2090	65	8	0	0	NUM
ap-2090	65	9	,	,	PUNCT
ap-2090	65	10	1	1	NUM
ap-2090	65	11	)	)	PUNCT
ap-2090	65	12	,	,	PUNCT
ap-2090	65	13	(	(	PUNCT
ap-2090	65	14	1	1	X
ap-2090	65	15	)	)	PUNCT
ap-2090	65	16	where	where	SCONJ
ap-2090	65	17	a	a	PRON
ap-2090	65	18	is	be	AUX
ap-2090	65	19	a	a	DET
ap-2090	65	20	control	control	NOUN
ap-2090	65	21	parameter	parameter	NOUN
ap-2090	65	22	,	,	PUNCT
ap-2090	65	23	0	0	NUM
ap-2090	65	24	≤	≤	NOUN
ap-2090	65	25	a	a	DET
ap-2090	65	26	≤	≤	NUM
ap-2090	65	27	4	4	NUM
ap-2090	65	28	.	.	PUNCT
ap-2090	66	1	the	the	DET
ap-2090	66	2	fixed	fix	VERB
ap-2090	66	3	points	point	NOUN
ap-2090	66	4	of	of	ADP
ap-2090	66	5	f	f	PROPN
ap-2090	66	6	and	and	CCONJ
ap-2090	66	7	f2	f2	PROPN
ap-2090	66	8	for	for	ADP
ap-2090	66	9	(	(	PUNCT
ap-2090	66	10	1	1	X
ap-2090	66	11	)	)	PUNCT
ap-2090	66	12	are	be	AUX
ap-2090	66	13	obtained	obtain	VERB
ap-2090	66	14	readily	readily	ADV
ap-2090	66	15	:	:	PUNCT
ap-2090	66	16	by	by	ADP
ap-2090	66	17	solving	solve	VERB
ap-2090	66	18	f(x	f(x	PROPN
ap-2090	66	19	)	)	PUNCT
ap-2090	67	1	−	−	NOUN
ap-2090	68	1	x	x	SYM
ap-2090	68	2	=	=	SYM
ap-2090	68	3	0	0	NUM
ap-2090	68	4	,	,	PUNCT
ap-2090	68	5	we	we	PRON
ap-2090	68	6	obtain	obtain	VERB
ap-2090	68	7	x	x	X
ap-2090	69	1	=	=	SYM
ap-2090	69	2	x0	x0	PUNCT
ap-2090	69	3	=	=	NOUN
ap-2090	69	4	1	1	NUM
ap-2090	69	5	−	−	PROPN
ap-2090	69	6	a−1	a−1	PROPN
ap-2090	69	7	,	,	PUNCT
ap-2090	69	8	a	a	DET
ap-2090	69	9	>	>	X
ap-2090	69	10	1	1	NUM
ap-2090	69	11	.	.	PUNCT
ap-2090	70	1	next	next	ADJ
ap-2090	70	2	by	by	ADP
ap-2090	70	3	solving	solve	VERB
ap-2090	70	4	(	(	PUNCT
ap-2090	70	5	f2	f2	PROPN
ap-2090	70	6	−	−	PROPN
ap-2090	70	7	x)/(f	x)/(f	PROPN
ap-2090	70	8	−	−	PROPN
ap-2090	70	9	x	x	NOUN
ap-2090	70	10	)	)	PUNCT
ap-2090	70	11	=	=	SYM
ap-2090	70	12	0	0	NUM
ap-2090	70	13	,	,	PUNCT
ap-2090	70	14	we	we	PRON
ap-2090	70	15	obtain	obtain	VERB
ap-2090	70	16	a	a	DET
ap-2090	70	17	quadratic	quadratic	ADJ
ap-2090	70	18	equation	equation	NOUN
ap-2090	70	19	,	,	PUNCT
ap-2090	70	20	from	from	ADP
ap-2090	70	21	which	which	PRON
ap-2090	70	22	x	x	X
ap-2090	70	23	=	=	PUNCT
ap-2090	71	1	x1,2	x1,2	PROPN
ap-2090	71	2	=	=	PUNCT
ap-2090	71	3	(	(	PUNCT
ap-2090	71	4	a	a	DET
ap-2090	71	5	+	+	NUM
ap-2090	71	6	1)/2a{1	1)/2a{1	NUM
ap-2090	71	7	±	±	NUM
ap-2090	71	8	√	√	PROPN
ap-2090	71	9	(	(	PUNCT
ap-2090	71	10	a−	a−	PROPN
ap-2090	71	11	3)/(a+	3)/(a+	NUM
ap-2090	71	12	1	1	NUM
ap-2090	71	13	)	)	PUNCT
ap-2090	71	14	}	}	PUNCT
ap-2090	71	15	,	,	PUNCT
ap-2090	71	16	a	a	DET
ap-2090	71	17	>	>	X
ap-2090	71	18	3	3	X
ap-2090	71	19	.	.	PUNCT
ap-2090	72	1	now	now	ADV
ap-2090	72	2	we	we	PRON
ap-2090	72	3	anticipate	anticipate	VERB
ap-2090	72	4	that	that	SCONJ
ap-2090	72	5	(	(	PUNCT
ap-2090	72	6	f3−x)/(f−x	f3−x)/(f−x	PROPN
ap-2090	72	7	)	)	PUNCT
ap-2090	72	8	=	=	SYM
ap-2090	72	9	0	0	NUM
ap-2090	72	10	is	be	AUX
ap-2090	72	11	a	a	DET
ap-2090	72	12	cubic	cubic	ADJ
ap-2090	72	13	equation	equation	NOUN
ap-2090	72	14	,	,	PUNCT
ap-2090	72	15	from	from	ADP
ap-2090	72	16	which	which	PRON
ap-2090	72	17	3	3	NUM
ap-2090	72	18	roots	root	NOUN
ap-2090	72	19	are	be	AUX
ap-2090	72	20	to	to	PART
ap-2090	72	21	be	be	AUX
ap-2090	72	22	found	find	VERB
ap-2090	72	23	.	.	PUNCT
ap-2090	73	1	we	we	PRON
ap-2090	73	2	now	now	ADV
ap-2090	73	3	construct	construct	VERB
ap-2090	73	4	q(x	q(x	NOUN
ap-2090	73	5	)	)	PUNCT
ap-2090	73	6	≡	≡	PROPN
ap-2090	73	7	(	(	PUNCT
ap-2090	73	8	f3	f3	PROPN
ap-2090	73	9	−	−	PROPN
ap-2090	73	10	x)/(f	x)/(f	PROPN
ap-2090	73	11	−	−	PROPN
ap-2090	73	12	x	x	NOUN
ap-2090	73	13	)	)	PUNCT
ap-2090	73	14	=	=	SYM
ap-2090	73	15	0	0	NUM
ap-2090	73	16	in	in	ADP
ap-2090	73	17	a	a	DET
ap-2090	73	18	new	new	ADJ
ap-2090	73	19	variable	variable	NOUN
ap-2090	73	20	t	t	NOUN
ap-2090	73	21	=	=	PUNCT
ap-2090	73	22	ax	ax	NOUN
ap-2090	73	23	for	for	ADP
ap-2090	73	24	simplification	simplification	NOUN
ap-2090	73	25	[	[	X
ap-2090	73	26	5	5	NUM
ap-2090	73	27	]	]	PUNCT
ap-2090	73	28	:	:	PUNCT
ap-2090	74	1	q(t	q(t	X
ap-2090	74	2	)	)	PUNCT
ap-2090	74	3	=	=	SYM
ap-2090	74	4	t6	t6	PROPN
ap-2090	74	5	−	−	PROPN
ap-2090	74	6	(	(	PUNCT
ap-2090	74	7	3a+	3a+	NUM
ap-2090	74	8	1)t5	1)t5	NUM
ap-2090	74	9	+	+	CCONJ
ap-2090	74	10	(	(	PUNCT
ap-2090	74	11	3a2	3a2	NUM
ap-2090	74	12	+	+	CCONJ
ap-2090	74	13	4a+	4a+	NUM
ap-2090	74	14	1)t4	1)t4	NUM
ap-2090	74	15	−	−	PROPN
ap-2090	74	16	(	(	PUNCT
ap-2090	74	17	a3	a3	NOUN
ap-2090	74	18	+	+	CCONJ
ap-2090	74	19	5a2	5a2	NUM
ap-2090	74	20	+	+	CCONJ
ap-2090	74	21	3a+	3a+	NUM
ap-2090	74	22	1)t3	1)t3	NUM
ap-2090	74	23	+	+	CCONJ
ap-2090	74	24	(	(	PUNCT
ap-2090	74	25	2a3	2a3	NUM
ap-2090	74	26	+	+	NUM
ap-2090	74	27	3a2	3a2	NUM
ap-2090	74	28	+	+	CCONJ
ap-2090	74	29	3a+	3a+	NUM
ap-2090	74	30	1)t2	1)t2	NUM
ap-2090	74	31	−	−	PROPN
ap-2090	74	32	(	(	PUNCT
ap-2090	74	33	a3	a3	NOUN
ap-2090	74	34	+	+	CCONJ
ap-2090	74	35	2a2	2a2	NUM
ap-2090	74	36	+	+	CCONJ
ap-2090	74	37	2a+	2a+	NUM
ap-2090	74	38	1)t+	1)t+	NUM
ap-2090	74	39	(	(	PUNCT
ap-2090	74	40	a2	a2	PROPN
ap-2090	74	41	+	+	CCONJ
ap-2090	74	42	a+	a+	SYM
ap-2090	74	43	1	1	NUM
ap-2090	74	44	)	)	PUNCT
ap-2090	74	45	.	.	PUNCT
ap-2090	75	1	(	(	PUNCT
ap-2090	75	2	2	2	X
ap-2090	75	3	)	)	PUNCT
ap-2090	75	4	to	to	ADP
ap-2090	75	5	our	our	PRON
ap-2090	75	6	surprise	surprise	NOUN
ap-2090	75	7	it	it	PRON
ap-2090	75	8	is	be	AUX
ap-2090	75	9	a	a	DET
ap-2090	75	10	sextic	sextic	ADJ
ap-2090	75	11	equation	equation	NOUN
ap-2090	75	12	.	.	PUNCT
ap-2090	76	1	since	since	SCONJ
ap-2090	76	2	there	there	PRON
ap-2090	76	3	are	be	VERB
ap-2090	76	4	no	no	DET
ap-2090	76	5	known	know	VERB
ap-2090	76	6	formulas	formula	NOUN
ap-2090	76	7	for	for	ADP
ap-2090	76	8	solving	solve	VERB
ap-2090	76	9	any	any	DET
ap-2090	76	10	equations	equation	NOUN
ap-2090	76	11	of	of	ADP
ap-2090	76	12	degrees	degree	NOUN
ap-2090	76	13	equal	equal	ADJ
ap-2090	76	14	to	to	ADP
ap-2090	76	15	or	or	CCONJ
ap-2090	76	16	higher	high	ADJ
ap-2090	76	17	than	than	ADP
ap-2090	76	18	5	5	NUM
ap-2090	76	19	,	,	PUNCT
ap-2090	76	20	at	at	ADP
ap-2090	76	21	a	a	DET
ap-2090	76	22	first	first	ADJ
ap-2090	76	23	glance	glance	NOUN
ap-2090	76	24	we	we	PRON
ap-2090	76	25	seem	seem	VERB
ap-2090	76	26	to	to	PART
ap-2090	76	27	have	have	AUX
ap-2090	76	28	arrived	arrive	VERB
ap-2090	76	29	at	at	ADP
ap-2090	76	30	a	a	DET
ap-2090	76	31	brick	brick	NOUN
ap-2090	76	32	wall	wall	NOUN
ap-2090	76	33	.	.	PUNCT
ap-2090	77	1	perhaps	perhaps	ADV
ap-2090	77	2	q(t	q(t	PROPN
ap-2090	77	3	)	)	PUNCT
ap-2090	77	4	is	be	AUX
ap-2090	77	5	not	not	PART
ap-2090	77	6	a	a	DET
ap-2090	77	7	general	general	ADJ
ap-2090	77	8	sextic	sextic	ADJ
ap-2090	77	9	equation	equation	NOUN
ap-2090	77	10	but	but	CCONJ
ap-2090	77	11	a	a	DET
ap-2090	77	12	special	special	ADJ
ap-2090	77	13	one	one	NUM
ap-2090	77	14	.	.	PUNCT
ap-2090	78	1	if	if	SCONJ
ap-2090	78	2	so	so	ADV
ap-2090	78	3	,	,	PUNCT
ap-2090	78	4	it	it	PRON
ap-2090	78	5	might	might	AUX
ap-2090	78	6	be	be	AUX
ap-2090	78	7	solvable	solvable	ADJ
ap-2090	78	8	in	in	ADP
ap-2090	78	9	some	some	DET
ap-2090	78	10	special	special	ADJ
ap-2090	78	11	way	way	NOUN
ap-2090	78	12	.	.	PUNCT
ap-2090	79	1	if	if	SCONJ
ap-2090	79	2	ek	ek	PROPN
ap-2090	79	3	denotes	denote	VERB
ap-2090	79	4	an	an	DET
ap-2090	79	5	equation	equation	NOUN
ap-2090	79	6	of	of	ADP
ap-2090	79	7	degree	degree	NOUN
ap-2090	79	8	k	k	PROPN
ap-2090	79	9	,	,	PUNCT
ap-2090	79	10	e6	e6	PROPN
ap-2090	79	11	would	would	AUX
ap-2090	79	12	be	be	AUX
ap-2090	79	13	solvable	solvable	ADJ
ap-2090	79	14	at	at	ADP
ap-2090	79	15	least	least	ADJ
ap-2090	79	16	in	in	ADP
ap-2090	79	17	principle	principle	NOUN
ap-2090	79	18	if	if	SCONJ
ap-2090	79	19	it	it	PRON
ap-2090	79	20	could	could	AUX
ap-2090	79	21	be	be	AUX
ap-2090	79	22	expressed	express	VERB
ap-2090	79	23	as	as	ADP
ap-2090	79	24	:	:	PUNCT
ap-2090	79	25	e2	e2	PROPN
ap-2090	79	26	×	×	PROPN
ap-2090	79	27	e2	e2	PROPN
ap-2090	79	28	×	×	PROPN
ap-2090	79	29	e2	e2	PROPN
ap-2090	79	30	or	or	CCONJ
ap-2090	79	31	e3	e3	VERB
ap-2090	79	32	×	×	NOUN
ap-2090	79	33	e3	e3	NOUN
ap-2090	79	34	.	.	PUNCT
ap-2090	80	1	looking	look	VERB
ap-2090	80	2	at	at	ADP
ap-2090	80	3	(	(	PUNCT
ap-2090	80	4	2	2	NUM
ap-2090	80	5	)	)	PUNCT
ap-2090	80	6	,	,	PUNCT
ap-2090	80	7	we	we	PRON
ap-2090	80	8	see	see	VERB
ap-2090	80	9	at	at	ADP
ap-2090	80	10	once	once	ADV
ap-2090	80	11	that	that	PRON
ap-2090	80	12	q(t	q(t	NOUN
ap-2090	80	13	)	)	PUNCT
ap-2090	80	14	must	must	AUX
ap-2090	80	15	be	be	AUX
ap-2090	80	16	real	real	ADJ
ap-2090	80	17	if	if	SCONJ
ap-2090	80	18	t	t	PROPN
ap-2090	80	19	is	be	AUX
ap-2090	80	20	real	real	ADJ
ap-2090	80	21	since	since	SCONJ
ap-2090	80	22	its	its	PRON
ap-2090	80	23	coefficients	coefficient	NOUN
ap-2090	80	24	are	be	AUX
ap-2090	80	25	all	all	ADV
ap-2090	80	26	real	real	ADJ
ap-2090	80	27	.	.	PUNCT
ap-2090	81	1	this	this	PRON
ap-2090	81	2	means	mean	VERB
ap-2090	81	3	that	that	SCONJ
ap-2090	81	4	the	the	DET
ap-2090	81	5	6	6	NUM
ap-2090	81	6	roots	root	NOUN
ap-2090	81	7	can	can	AUX
ap-2090	81	8	only	only	ADV
ap-2090	81	9	be	be	AUX
ap-2090	81	10	either	either	CCONJ
ap-2090	81	11	3	3	NUM
ap-2090	81	12	pairs	pair	NOUN
ap-2090	81	13	of	of	ADP
ap-2090	81	14	complex	complex	ADJ
ap-2090	81	15	conjugates	conjugate	NOUN
ap-2090	81	16	or	or	CCONJ
ap-2090	81	17	two	two	NUM
ap-2090	81	18	sets	set	NOUN
ap-2090	81	19	of	of	ADP
ap-2090	81	20	3	3	NUM
ap-2090	81	21	real	real	ADJ
ap-2090	81	22	roots	root	NOUN
ap-2090	81	23	,	,	PUNCT
ap-2090	81	24	corresponding	correspond	VERB
ap-2090	81	25	to	to	ADP
ap-2090	81	26	two	two	NUM
ap-2090	81	27	solvable	solvable	ADJ
ap-2090	81	28	forms	form	NOUN
ap-2090	81	29	for	for	ADP
ap-2090	81	30	q(t	q(t	NOUN
ap-2090	81	31	)	)	PUNCT
ap-2090	81	32	.	.	PUNCT
ap-2090	82	1	this	this	PRON
ap-2090	82	2	impels	impel	VERB
ap-2090	82	3	us	we	PRON
ap-2090	82	4	to	to	PART
ap-2090	82	5	look	look	VERB
ap-2090	82	6	for	for	ADP
ap-2090	82	7	an	an	DET
ap-2090	82	8	indication	indication	NOUN
ap-2090	82	9	that	that	SCONJ
ap-2090	82	10	q(t	q(t	NOUN
ap-2090	82	11	)	)	PUNCT
ap-2090	82	12	is	be	AUX
ap-2090	82	13	not	not	PART
ap-2090	82	14	general	general	ADJ
ap-2090	82	15	.	.	PUNCT
ap-2090	83	1	if	if	SCONJ
ap-2090	83	2	(	(	PUNCT
ap-2090	83	3	2	2	X
ap-2090	83	4	)	)	PUNCT
ap-2090	83	5	is	be	AUX
ap-2090	83	6	expressed	express	VERB
ap-2090	83	7	as	as	ADP
ap-2090	83	8	q(t	q(t	ADJ
ap-2090	83	9	)	)	PUNCT
ap-2090	84	1	=	=	SYM
ap-2090	84	2	6∑	6∑	NUM
ap-2090	84	3	k=0	k=0	PROPN
ap-2090	84	4	(	(	PUNCT
ap-2090	84	5	−)kδkt	−)kδkt	PROPN
ap-2090	84	6	k	k	PROPN
ap-2090	84	7	,	,	PUNCT
ap-2090	84	8	(	(	PUNCT
ap-2090	84	9	3	3	X
ap-2090	84	10	)	)	PUNCT
ap-2090	84	11	we	we	PRON
ap-2090	84	12	find	find	VERB
ap-2090	84	13	that	that	SCONJ
ap-2090	84	14	∆	∆	PROPN
ap-2090	84	15	≡	≡	PROPN
ap-2090	84	16	∑5	∑5	PROPN
ap-2090	84	17	k=0(−)kδk	k=0(−)kδk	PROPN
ap-2090	84	18	=	=	SYM
ap-2090	84	19	0	0	NUM
ap-2090	84	20	,	,	PUNCT
ap-2090	84	21	referred	refer	VERB
ap-2090	84	22	to	to	ADP
ap-2090	84	23	as	as	ADP
ap-2090	84	24	the	the	DET
ap-2090	84	25	delta	delta	NOUN
ap-2090	84	26	sum	sum	NOUN
ap-2090	84	27	rule	rule	NOUN
ap-2090	84	28	.	.	PUNCT
ap-2090	85	1	this	this	DET
ap-2090	85	2	sum	sum	NOUN
ap-2090	85	3	rule	rule	NOUN
ap-2090	85	4	is	be	AUX
ap-2090	85	5	a	a	DET
ap-2090	85	6	clear	clear	ADJ
ap-2090	85	7	indication	indication	NOUN
ap-2090	85	8	that	that	SCONJ
ap-2090	85	9	q(t	q(t	NOUN
ap-2090	85	10	)	)	PUNCT
ap-2090	85	11	is	be	AUX
ap-2090	85	12	not	not	PART
ap-2090	85	13	a	a	DET
ap-2090	85	14	general	general	ADJ
ap-2090	85	15	sextic	sextic	ADJ
ap-2090	85	16	equation	equation	NOUN
ap-2090	85	17	.	.	PUNCT
ap-2090	86	1	4	4	X
ap-2090	86	2	.	.	X
ap-2090	86	3	solving	solve	VERB
ap-2090	86	4	the	the	DET
ap-2090	86	5	sextic	sextic	ADJ
ap-2090	86	6	equation	equation	NOUN
ap-2090	86	7	:	:	PUNCT
ap-2090	86	8	preliminaries	preliminary	NOUN
ap-2090	86	9	if	if	SCONJ
ap-2090	86	10	a	a	DET
ap-2090	86	11	=	=	NOUN
ap-2090	86	12	0	0	NUM
ap-2090	86	13	in	in	ADP
ap-2090	86	14	(	(	PUNCT
ap-2090	86	15	2	2	NUM
ap-2090	86	16	)	)	PUNCT
ap-2090	86	17	,	,	PUNCT
ap-2090	86	18	which	which	PRON
ap-2090	86	19	still	still	ADV
ap-2090	86	20	satisfies	satisfy	VERB
ap-2090	86	21	the	the	DET
ap-2090	86	22	delta	delta	NOUN
ap-2090	86	23	sum	sum	NOUN
ap-2090	86	24	rule	rule	NOUN
ap-2090	86	25	,	,	PUNCT
ap-2090	86	26	the	the	DET
ap-2090	86	27	roots	root	NOUN
ap-2090	86	28	are	be	AUX
ap-2090	86	29	:	:	PUNCT
ap-2090	86	30	exp(±iπ/7	exp(±iπ/7	NOUN
ap-2090	86	31	)	)	PUNCT
ap-2090	86	32	,	,	PUNCT
ap-2090	86	33	exp(±i3π/7	exp(±i3π/7	VERB
ap-2090	86	34	)	)	PUNCT
ap-2090	86	35	and	and	CCONJ
ap-2090	86	36	exp(±i5π/7	exp(±i5π/7	NOUN
ap-2090	86	37	)	)	PUNCT
ap-2090	86	38	,	,	PUNCT
ap-2090	86	39	3	3	NUM
ap-2090	86	40	pairs	pair	NOUN
ap-2090	86	41	of	of	ADP
ap-2090	86	42	complex	complex	ADJ
ap-2090	86	43	conjugates	conjugate	NOUN
ap-2090	86	44	lying	lie	VERB
ap-2090	86	45	on	on	ADP
ap-2090	86	46	the	the	DET
ap-2090	86	47	unit	unit	NOUN
ap-2090	86	48	circle	circle	NOUN
ap-2090	86	49	.	.	PUNCT
ap-2090	87	1	if	if	SCONJ
ap-2090	87	2	the	the	DET
ap-2090	87	3	roots	root	NOUN
ap-2090	87	4	are	be	AUX
ap-2090	87	5	to	to	PART
ap-2090	87	6	be	be	AUX
ap-2090	87	7	real	real	ADJ
ap-2090	87	8	,	,	PUNCT
ap-2090	87	9	none	none	NOUN
ap-2090	87	10	of	of	ADP
ap-2090	87	11	them	they	PRON
ap-2090	87	12	may	may	AUX
ap-2090	87	13	lie	lie	VERB
ap-2090	87	14	on	on	ADP
ap-2090	87	15	the	the	DET
ap-2090	87	16	negative	negative	ADJ
ap-2090	87	17	real	real	ADJ
ap-2090	87	18	axis	axis	NOUN
ap-2090	87	19	of	of	ADP
ap-2090	87	20	t	t	PROPN
ap-2090	87	21	since	since	SCONJ
ap-2090	87	22	the	the	DET
ap-2090	87	23	coefficients	coefficient	NOUN
ap-2090	87	24	of	of	ADP
ap-2090	87	25	t	t	PROPN
ap-2090	87	26	in	in	ADP
ap-2090	87	27	odd	odd	ADJ
ap-2090	87	28	power	power	NOUN
ap-2090	87	29	have	have	VERB
ap-2090	87	30	‘	'	PUNCT
ap-2090	87	31	−	−	NOUN
ap-2090	87	32	’	'	PUNCT
ap-2090	87	33	signs	sign	NOUN
ap-2090	87	34	.	.	PUNCT
ap-2090	88	1	the	the	DET
ap-2090	88	2	six	six	NUM
ap-2090	88	3	roots	root	NOUN
ap-2090	88	4	must	must	AUX
ap-2090	88	5	all	all	PRON
ap-2090	88	6	lie	lie	VERB
ap-2090	88	7	on	on	ADP
ap-2090	88	8	the	the	DET
ap-2090	88	9	positive	positive	ADJ
ap-2090	88	10	real	real	ADJ
ap-2090	88	11	axis	axis	NOUN
ap-2090	88	12	.	.	PUNCT
ap-2090	89	1	if	if	SCONJ
ap-2090	89	2	a→∞	a→∞	NUM
ap-2090	89	3	,	,	PUNCT
ap-2090	89	4	one	one	NUM
ap-2090	89	5	of	of	ADP
ap-2090	89	6	the	the	DET
ap-2090	89	7	roots	root	NOUN
ap-2090	89	8	goes	go	VERB
ap-2090	89	9	as	as	ADP
ap-2090	89	10	1	1	NUM
ap-2090	89	11	/	/	SYM
ap-2090	89	12	a	a	DET
ap-2090	89	13	asymptotically	asymptotically	NOUN
ap-2090	89	14	.	.	PUNCT
ap-2090	90	1	thus	thus	ADV
ap-2090	90	2	as	as	ADP
ap-2090	90	3	a	a	DET
ap-2090	90	4	increases	increase	NOUN
ap-2090	90	5	from	from	ADP
ap-2090	90	6	zero	zero	NUM
ap-2090	90	7	,	,	PUNCT
ap-2090	90	8	the	the	DET
ap-2090	90	9	three	three	NUM
ap-2090	90	10	pairs	pair	NOUN
ap-2090	90	11	of	of	ADP
ap-2090	90	12	complex	complex	ADJ
ap-2090	90	13	roots	root	NOUN
ap-2090	90	14	must	must	AUX
ap-2090	90	15	move	move	VERB
ap-2090	90	16	toward	toward	ADP
ap-2090	90	17	the	the	DET
ap-2090	90	18	positive	positive	ADJ
ap-2090	90	19	real	real	ADJ
ap-2090	90	20	axis	axis	NOUN
ap-2090	90	21	.	.	PUNCT
ap-2090	91	1	at	at	ADP
ap-2090	91	2	some	some	DET
ap-2090	91	3	value	value	NOUN
ap-2090	91	4	of	of	ADP
ap-2090	91	5	a	a	DET
ap-2090	91	6	=	=	PUNCT
ap-2090	91	7	ã	ã	PROPN
ap-2090	91	8	say	say	VERB
ap-2090	91	9	,	,	PUNCT
ap-2090	91	10	each	each	DET
ap-2090	91	11	complex	complex	ADJ
ap-2090	91	12	-	-	PUNCT
ap-2090	91	13	conjugate	conjugate	ADJ
ap-2090	91	14	pair	pair	NOUN
ap-2090	91	15	becomes	become	VERB
ap-2090	91	16	two	two	NUM
ap-2090	91	17	identical	identical	ADJ
ap-2090	91	18	real	real	ADJ
ap-2090	91	19	roots	root	NOUN
ap-2090	91	20	simultaneously	simultaneously	ADV
ap-2090	91	21	.	.	PUNCT
ap-2090	92	1	thereafter	thereafter	ADV
ap-2090	92	2	each	each	DET
ap-2090	92	3	degenerate	degenerate	ADJ
ap-2090	92	4	real	real	ADJ
ap-2090	92	5	root	root	NOUN
ap-2090	92	6	splits	split	VERB
ap-2090	92	7	into	into	ADP
ap-2090	92	8	two	two	NUM
ap-2090	92	9	real	real	ADJ
ap-2090	92	10	roots	root	NOUN
ap-2090	92	11	.	.	PUNCT
ap-2090	93	1	this	this	DET
ap-2090	93	2	overall	overall	ADJ
ap-2090	93	3	behavior	behavior	NOUN
ap-2090	93	4	is	be	AUX
ap-2090	93	5	consistent	consistent	ADJ
ap-2090	93	6	with	with	ADP
ap-2090	93	7	the	the	DET
ap-2090	93	8	two	two	NUM
ap-2090	93	9	solvable	solvable	ADJ
ap-2090	93	10	forms	form	NOUN
ap-2090	93	11	for	for	ADP
ap-2090	93	12	q(t	q(t	NOUN
ap-2090	93	13	)	)	PUNCT
ap-2090	93	14	.	.	PUNCT
ap-2090	94	1	we	we	PRON
ap-2090	94	2	can	can	AUX
ap-2090	94	3	not	not	PART
ap-2090	94	4	yet	yet	ADV
ap-2090	94	5	know	know	VERB
ap-2090	94	6	however	however	ADV
ap-2090	94	7	whether	whether	SCONJ
ap-2090	94	8	ã	ã	PROPN
ap-2090	94	9	<	<	X
ap-2090	94	10	4	4	NUM
ap-2090	94	11	or	or	CCONJ
ap-2090	94	12	>	>	ADP
ap-2090	94	13	4	4	NUM
ap-2090	94	14	.	.	PUNCT
ap-2090	95	1	the	the	DET
ap-2090	95	2	existence	existence	NOUN
ap-2090	95	3	of	of	ADP
ap-2090	95	4	3	3	NUM
ap-2090	95	5	-	-	PUNCT
ap-2090	95	6	cycle	cycle	NOUN
ap-2090	95	7	in	in	ADP
ap-2090	95	8	the	the	DET
ap-2090	95	9	logistic	logistic	ADJ
ap-2090	95	10	map	map	NOUN
ap-2090	95	11	requires	require	VERB
ap-2090	95	12	that	that	SCONJ
ap-2090	95	13	ã	ã	PROPN
ap-2090	95	14	be	be	AUX
ap-2090	95	15	less	less	ADJ
ap-2090	95	16	than	than	ADP
ap-2090	95	17	4	4	NUM
ap-2090	95	18	and	and	CCONJ
ap-2090	95	19	also	also	ADV
ap-2090	95	20	that	that	SCONJ
ap-2090	95	21	at	at	ADV
ap-2090	95	22	least	least	ADV
ap-2090	95	23	one	one	NUM
ap-2090	95	24	set	set	NOUN
ap-2090	95	25	of	of	ADP
ap-2090	95	26	real	real	ADJ
ap-2090	95	27	roots	root	NOUN
ap-2090	95	28	lies	lie	VERB
ap-2090	95	29	in	in	ADP
ap-2090	95	30	the	the	DET
ap-2090	95	31	interval	interval	NOUN
ap-2090	95	32	t	t	NOUN
ap-2090	95	33	=	=	SYM
ap-2090	95	34	(	(	PUNCT
ap-2090	95	35	0	0	NUM
ap-2090	95	36	,	,	PUNCT
ap-2090	95	37	a	a	NOUN
ap-2090	95	38	)	)	PUNCT
ap-2090	95	39	or	or	CCONJ
ap-2090	95	40	x	x	X
ap-2090	95	41	=	=	SYM
ap-2090	95	42	(	(	PUNCT
ap-2090	95	43	0	0	NUM
ap-2090	95	44	,	,	PUNCT
ap-2090	95	45	1	1	NUM
ap-2090	95	46	)	)	PUNCT
ap-2090	95	47	.	.	PUNCT
ap-2090	96	1	if	if	SCONJ
ap-2090	96	2	otherwise	otherwise	ADV
ap-2090	96	3	,	,	PUNCT
ap-2090	96	4	3	3	NUM
ap-2090	96	5	-	-	PUNCT
ap-2090	96	6	cycle	cycle	NOUN
ap-2090	96	7	does	do	AUX
ap-2090	96	8	not	not	PART
ap-2090	96	9	exist	exist	VERB
ap-2090	96	10	in	in	ADP
ap-2090	96	11	it	it	PRON
ap-2090	96	12	.	.	PUNCT
ap-2090	97	1	if	if	SCONJ
ap-2090	97	2	a	a	DET
ap-2090	97	3	>	>	X
ap-2090	97	4	ã	ã	PROPN
ap-2090	97	5	,	,	PUNCT
ap-2090	97	6	there	there	PRON
ap-2090	97	7	must	must	AUX
ap-2090	97	8	be	be	AUX
ap-2090	97	9	two	two	NUM
ap-2090	97	10	sets	set	NOUN
ap-2090	97	11	of	of	ADP
ap-2090	97	12	3	3	NUM
ap-2090	97	13	positive	positive	ADJ
ap-2090	97	14	real	real	ADJ
ap-2090	97	15	roots	root	NOUN
ap-2090	97	16	.	.	PUNCT
ap-2090	98	1	hence	hence	ADV
ap-2090	98	2	,	,	PUNCT
ap-2090	98	3	we	we	PRON
ap-2090	98	4	shall	shall	AUX
ap-2090	98	5	write	write	VERB
ap-2090	98	6	q(t	q(t	PROPN
ap-2090	98	7	)	)	PUNCT
ap-2090	98	8	=	=	PUNCT
ap-2090	98	9	q(t)×	q(t)×	PROPN
ap-2090	98	10	q′(t	q′(t	PROPN
ap-2090	98	11	)	)	PUNCT
ap-2090	98	12	,	,	PUNCT
ap-2090	98	13	where	where	SCONJ
ap-2090	98	14	q	q	NOUN
ap-2090	98	15	and	and	CCONJ
ap-2090	98	16	q′	q′	NOUN
ap-2090	98	17	are	be	AUX
ap-2090	98	18	cubic	cubic	ADJ
ap-2090	98	19	equations	equation	NOUN
ap-2090	98	20	with	with	ADP
ap-2090	98	21	(	(	PUNCT
ap-2090	98	22	as	as	ADV
ap-2090	98	23	yet	yet	ADV
ap-2090	98	24	undetermined	undetermined	ADJ
ap-2090	98	25	)	)	PUNCT
ap-2090	98	26	roots	root	NOUN
ap-2090	98	27	,	,	PUNCT
ap-2090	98	28	resp	resp	NOUN
ap-2090	98	29	.	.	PUNCT
ap-2090	98	30	,	,	PUNCT
ap-2090	98	31	tk	tk	PROPN
ap-2090	98	32	and	and	CCONJ
ap-2090	98	33	t′k	t′k	PROPN
ap-2090	98	34	,	,	PUNCT
ap-2090	98	35	k	k	PROPN
ap-2090	98	36	=	=	SYM
ap-2090	98	37	1	1	NUM
ap-2090	98	38	,	,	PUNCT
ap-2090	98	39	2	2	NUM
ap-2090	98	40	,	,	PUNCT
ap-2090	98	41	3	3	NUM
ap-2090	98	42	.	.	PUNCT
ap-2090	99	1	thus	thus	ADV
ap-2090	99	2	one	one	NUM
ap-2090	99	3	can	can	AUX
ap-2090	99	4	express	express	VERB
ap-2090	99	5	q(t	q(t	NOUN
ap-2090	99	6	)	)	PUNCT
ap-2090	99	7	as	as	ADP
ap-2090	99	8	q(t	q(t	NOUN
ap-2090	99	9	)	)	PUNCT
ap-2090	99	10	=	=	SYM
ap-2090	99	11	t3	t3	NOUN
ap-2090	100	1	−	−	PROPN
ap-2090	100	2	αt2	αt2	NOUN
ap-2090	100	3	+	+	CCONJ
ap-2090	100	4	βt−	βt−	PUNCT
ap-2090	100	5	γ	γ	X
ap-2090	100	6	,	,	PUNCT
ap-2090	100	7	(	(	PUNCT
ap-2090	100	8	4	4	NUM
ap-2090	100	9	)	)	PUNCT
ap-2090	100	10	where	where	SCONJ
ap-2090	100	11	α	α	PROPN
ap-2090	100	12	=	=	SYM
ap-2090	100	13	t1+t2+t3	t1+t2+t3	PROPN
ap-2090	100	14	,	,	PUNCT
ap-2090	100	15	β	β	X
ap-2090	100	16	=	=	SYM
ap-2090	100	17	t1t2+t2t3+t3t1	t1t2+t2t3+t3t1	PROPN
ap-2090	100	18	,	,	PUNCT
ap-2090	100	19	γ	γ	X
ap-2090	100	20	=	=	SYM
ap-2090	100	21	t1t2t3	t1t2t3	NOUN
ap-2090	100	22	,	,	PUNCT
ap-2090	100	23	and	and	CCONJ
ap-2090	100	24	q′(t	q′(t	NOUN
ap-2090	100	25	)	)	PUNCT
ap-2090	100	26	similarly	similarly	ADV
ap-2090	100	27	with	with	ADP
ap-2090	100	28	primes	prime	NOUN
ap-2090	100	29	written	write	VERB
ap-2090	100	30	on	on	ADP
ap-2090	100	31	.	.	PUNCT
ap-2090	101	1	they	they	PRON
ap-2090	101	2	(	(	PUNCT
ap-2090	101	3	αβγ	αβγ	NOUN
ap-2090	101	4	)	)	PUNCT
ap-2090	101	5	will	will	AUX
ap-2090	101	6	be	be	AUX
ap-2090	101	7	referred	refer	VERB
ap-2090	101	8	to	to	ADP
ap-2090	101	9	as	as	ADP
ap-2090	101	10	trigonals	trigonal	NOUN
ap-2090	101	11	.	.	PUNCT
ap-2090	102	1	to	to	PART
ap-2090	102	2	understand	understand	VERB
ap-2090	102	3	why	why	SCONJ
ap-2090	102	4	q	q	NOUN
ap-2090	102	5	is	be	AUX
ap-2090	102	6	special	special	ADJ
ap-2090	102	7	we	we	PRON
ap-2090	102	8	look	look	VERB
ap-2090	102	9	for	for	ADP
ap-2090	102	10	some	some	DET
ap-2090	102	11	structural	structural	ADJ
ap-2090	102	12	relations	relation	NOUN
ap-2090	102	13	between	between	ADP
ap-2090	102	14	the	the	DET
ap-2090	102	15	trigonals	trigonal	NOUN
ap-2090	102	16	of	of	ADP
ap-2090	102	17	the	the	DET
ap-2090	102	18	same	same	ADJ
ap-2090	102	19	kind	kind	NOUN
ap-2090	102	20	contained	contain	VERB
ap-2090	102	21	in	in	ADP
ap-2090	102	22	the	the	DET
ap-2090	102	23	structure	structure	NOUN
ap-2090	102	24	of	of	ADP
ap-2090	102	25	the	the	DET
ap-2090	102	26	logistic	logistic	ADJ
ap-2090	102	27	map	map	NOUN
ap-2090	102	28	.	.	PUNCT
ap-2090	103	1	indeed	indeed	ADV
ap-2090	103	2	we	we	PRON
ap-2090	103	3	find	find	VERB
ap-2090	103	4	that	that	SCONJ
ap-2090	104	1	[	[	X
ap-2090	104	2	7	7	NUM
ap-2090	104	3	]	]	NUM
ap-2090	104	4	:	:	PUNCT
ap-2090	104	5	α−β+γ	α−β+γ	SYM
ap-2090	104	6	=	=	SYM
ap-2090	104	7	0	0	NUM
ap-2090	104	8	and	and	CCONJ
ap-2090	104	9	α′−β′+γ′	α′−β′+γ′	PROPN
ap-2090	104	10	=	=	SYM
ap-2090	104	11	0	0	PROPN
ap-2090	104	12	,	,	PUNCT
ap-2090	104	13	together	together	ADV
ap-2090	104	14	called	call	VERB
ap-2090	104	15	the	the	DET
ap-2090	104	16	trigonal	trigonal	ADJ
ap-2090	104	17	relation	relation	NOUN
ap-2090	104	18	.	.	PUNCT
ap-2090	105	1	to	to	ADP
ap-2090	105	2	no	no	DET
ap-2090	105	3	surprise	surprise	NOUN
ap-2090	105	4	the	the	DET
ap-2090	105	5	trigonal	trigonal	ADJ
ap-2090	105	6	relation	relation	NOUN
ap-2090	105	7	implies	imply	VERB
ap-2090	105	8	the	the	DET
ap-2090	105	9	delta	delta	NOUN
ap-2090	105	10	sum	sum	NOUN
ap-2090	105	11	rule	rule	NOUN
ap-2090	105	12	.	.	PUNCT
ap-2090	106	1	in	in	ADP
ap-2090	106	2	addition	addition	NOUN
ap-2090	106	3	,	,	PUNCT
ap-2090	106	4	β	β	X
ap-2090	106	5	=	=	SYM
ap-2090	106	6	(	(	PUNCT
ap-2090	106	7	a+1)α−(a2	a+1)α−(a2	X
ap-2090	106	8	+	+	NOUN
ap-2090	106	9	a+1	a+1	ADJ
ap-2090	106	10	)	)	PUNCT
ap-2090	106	11	and	and	CCONJ
ap-2090	106	12	γ	γ	X
ap-2090	106	13	=	=	SYM
ap-2090	106	14	aα−(a2	aα−(a2	NOUN
ap-2090	106	15	+	+	CCONJ
ap-2090	106	16	a	a	DET
ap-2090	106	17	+	+	NUM
ap-2090	106	18	1	1	NUM
ap-2090	106	19	)	)	PUNCT
ap-2090	106	20	,	,	PUNCT
ap-2090	106	21	leaving	leave	VERB
ap-2090	106	22	only	only	ADV
ap-2090	106	23	α	α	NOUN
ap-2090	106	24	and	and	CCONJ
ap-2090	106	25	α′	α′	NOUN
ap-2090	106	26	independent	independent	ADJ
ap-2090	106	27	,	,	PUNCT
ap-2090	106	28	leaving	leave	VERB
ap-2090	106	29	only	only	ADV
ap-2090	106	30	two	two	NUM
ap-2090	106	31	unknowns	unknown	NOUN
ap-2090	106	32	to	to	ADP
ap-2090	106	33	the	the	DET
ap-2090	106	34	problem	problem	NOUN
ap-2090	106	35	.	.	PUNCT
ap-2090	107	1	the	the	DET
ap-2090	107	2	transition	transition	NOUN
ap-2090	107	3	condition	condition	NOUN
ap-2090	107	4	e2×e2×e2	e2×e2×e2	PART
ap-2090	107	5	=	=	SYM
ap-2090	107	6	e3×e3	e3×e3	PROPN
ap-2090	107	7	,	,	PUNCT
ap-2090	107	8	together	together	ADV
ap-2090	107	9	with	with	ADP
ap-2090	107	10	the	the	DET
ap-2090	107	11	trigonal	trigonal	ADJ
ap-2090	107	12	relation	relation	NOUN
ap-2090	107	13	yields	yield	VERB
ap-2090	107	14	σ	σ	NOUN
ap-2090	107	15	=	=	SYM
ap-2090	107	16	0	0	PROPN
ap-2090	107	17	,	,	PUNCT
ap-2090	107	18	where	where	SCONJ
ap-2090	107	19	σ	σ	NOUN
ap-2090	107	20	=	=	SYM
ap-2090	107	21	(	(	PUNCT
ap-2090	107	22	a2	a2	PROPN
ap-2090	107	23	−	−	PROPN
ap-2090	107	24	2a−	2a−	PROPN
ap-2090	107	25	7)1/2	7)1/2	NUM
ap-2090	107	26	.	.	PUNCT
ap-2090	108	1	hence	hence	ADV
ap-2090	108	2	,	,	PUNCT
ap-2090	108	3	ã	ã	PROPN
ap-2090	108	4	=	=	SYM
ap-2090	108	5	1	1	NUM
ap-2090	108	6	+	+	CCONJ
ap-2090	108	7	√	√	NUM
ap-2090	108	8	8	8	NUM
ap-2090	108	9	=	=	SYM
ap-2090	108	10	3.872281323	3.872281323	NUM
ap-2090	108	11	·	·	PUNCT
ap-2090	108	12	·	·	PUNCT
ap-2090	108	13	·	·	PUNCT
ap-2090	109	1	as	as	ADP
ap-2090	109	2	the	the	DET
ap-2090	109	3	transition	transition	NOUN
ap-2090	109	4	value	value	NOUN
ap-2090	109	5	.	.	PUNCT
ap-2090	110	1	since	since	SCONJ
ap-2090	110	2	ã	ã	PROPN
ap-2090	110	3	<	<	X
ap-2090	110	4	4	4	NUM
ap-2090	110	5	,	,	PUNCT
ap-2090	110	6	3	3	NUM
ap-2090	110	7	-	-	PUNCT
ap-2090	110	8	cycle	cycle	NOUN
ap-2090	110	9	can	can	AUX
ap-2090	110	10	exist	exist	VERB
ap-2090	110	11	in	in	ADP
ap-2090	110	12	the	the	DET
ap-2090	110	13	logistic	logistic	ADJ
ap-2090	110	14	map	map	NOUN
ap-2090	110	15	.	.	PUNCT
ap-2090	111	1	there	there	PRON
ap-2090	111	2	are	be	VERB
ap-2090	111	3	two	two	NUM
ap-2090	111	4	sets	set	NOUN
ap-2090	111	5	of	of	ADP
ap-2090	111	6	3	3	NUM
ap-2090	111	7	positive	positive	ADJ
ap-2090	111	8	real	real	ADJ
ap-2090	111	9	roots	root	NOUN
ap-2090	111	10	,	,	PUNCT
ap-2090	111	11	or	or	CCONJ
ap-2090	111	12	two	two	NUM
ap-2090	111	13	forms	form	NOUN
ap-2090	111	14	of	of	ADP
ap-2090	111	15	3	3	NUM
ap-2090	111	16	-	-	PUNCT
ap-2090	111	17	cycle	cycle	NOUN
ap-2090	111	18	,	,	PUNCT
ap-2090	111	19	that	that	PRON
ap-2090	111	20	had	have	AUX
ap-2090	111	21	not	not	PART
ap-2090	111	22	been	be	AUX
ap-2090	111	23	implied	imply	VERB
ap-2090	111	24	by	by	ADP
ap-2090	111	25	the	the	DET
ap-2090	111	26	theorem	theorem	NOUN
ap-2090	111	27	of	of	ADP
ap-2090	111	28	sharkovskii	sharkovskii	PROPN
ap-2090	111	29	,	,	PUNCT
ap-2090	111	30	but	but	CCONJ
ap-2090	111	31	that	that	PRON
ap-2090	111	32	evidently	evidently	ADV
ap-2090	111	33	exist	exist	VERB
ap-2090	111	34	in	in	ADP
ap-2090	111	35	the	the	DET
ap-2090	111	36	logistic	logistic	ADJ
ap-2090	111	37	map	map	NOUN
ap-2090	111	38	.	.	PUNCT
ap-2090	112	1	unless	unless	SCONJ
ap-2090	112	2	their	their	PRON
ap-2090	112	3	relationship	relationship	NOUN
ap-2090	112	4	is	be	AUX
ap-2090	112	5	found	find	VERB
ap-2090	112	6	,	,	PUNCT
ap-2090	112	7	the	the	DET
ap-2090	112	8	two	two	NUM
ap-2090	112	9	cubic	cubic	ADJ
ap-2090	112	10	equations	equation	NOUN
ap-2090	112	11	are	be	AUX
ap-2090	112	12	still	still	ADV
ap-2090	112	13	left	leave	VERB
ap-2090	112	14	with	with	ADP
ap-2090	112	15	one	one	NUM
ap-2090	112	16	unknown	unknown	ADJ
ap-2090	112	17	each	each	PRON
ap-2090	112	18	,	,	PUNCT
ap-2090	112	19	α	α	NOUN
ap-2090	112	20	or	or	CCONJ
ap-2090	112	21	α′.	α′.	NUM
ap-2090	112	22	how	how	SCONJ
ap-2090	112	23	to	to	PART
ap-2090	112	24	distinguish	distinguish	VERB
ap-2090	112	25	the	the	DET
ap-2090	112	26	two	two	NUM
ap-2090	112	27	remains	remain	VERB
ap-2090	112	28	an	an	DET
ap-2090	112	29	obstacle	obstacle	NOUN
ap-2090	112	30	to	to	ADP
ap-2090	112	31	the	the	DET
ap-2090	112	32	final	final	ADJ
ap-2090	112	33	solution	solution	NOUN
ap-2090	112	34	.	.	PUNCT
ap-2090	113	1	5	5	X
ap-2090	113	2	.	.	X
ap-2090	113	3	solving	solve	VERB
ap-2090	113	4	the	the	DET
ap-2090	113	5	sextic	sextic	ADJ
ap-2090	113	6	equation	equation	NOUN
ap-2090	113	7	:	:	PUNCT
ap-2090	113	8	an	an	DET
ap-2090	113	9	internal	internal	ADJ
ap-2090	113	10	degree	degree	NOUN
ap-2090	113	11	of	of	ADP
ap-2090	113	12	freedom	freedom	NOUN
ap-2090	113	13	if	if	SCONJ
ap-2090	113	14	there	there	PRON
ap-2090	113	15	are	be	VERB
ap-2090	113	16	two	two	NUM
ap-2090	113	17	forms	form	NOUN
ap-2090	113	18	of	of	ADP
ap-2090	113	19	3	3	NUM
ap-2090	113	20	-	-	PUNCT
ap-2090	113	21	cycle	cycle	NOUN
ap-2090	113	22	,	,	PUNCT
ap-2090	113	23	they	they	PRON
ap-2090	113	24	must	must	AUX
ap-2090	113	25	be	be	AUX
ap-2090	113	26	distinguishable	distinguishable	ADJ
ap-2090	113	27	by	by	ADP
ap-2090	113	28	some	some	DET
ap-2090	113	29	internal	internal	ADJ
ap-2090	113	30	degree	degree	NOUN
ap-2090	113	31	of	of	ADP
ap-2090	113	32	freedom	freedom	NOUN
ap-2090	113	33	in	in	ADP
ap-2090	113	34	q	q	PROPN
ap-2090	114	1	and	and	CCONJ
ap-2090	114	2	q′.	q′.	ADV
ap-2090	114	3	it	it	PRON
ap-2090	114	4	can	can	AUX
ap-2090	114	5	not	not	PART
ap-2090	114	6	appear	appear	VERB
ap-2090	114	7	in	in	ADP
ap-2090	114	8	q	q	NOUN
ap-2090	114	9	,	,	PUNCT
ap-2090	114	10	meaning	mean	VERB
ap-2090	114	11	that	that	SCONJ
ap-2090	114	12	it	it	PRON
ap-2090	114	13	can	can	AUX
ap-2090	114	14	not	not	PART
ap-2090	114	15	appear	appear	VERB
ap-2090	114	16	in	in	ADP
ap-2090	114	17	δ′s	δ′s	ADJ
ap-2090	114	18	,	,	PUNCT
ap-2090	114	19	the	the	DET
ap-2090	114	20	coefficients	coefficient	NOUN
ap-2090	114	21	of	of	ADP
ap-2090	114	22	t	t	PROPN
ap-2090	114	23	in	in	ADP
ap-2090	114	24	q	q	NOUN
ap-2090	114	25	,	,	PUNCT
ap-2090	114	26	e.g.	e.g.	ADV
ap-2090	114	27	δ5	δ5	NOUN
ap-2090	114	28	=	=	SYM
ap-2090	114	29	3a	3a	NUM
ap-2090	114	30	+	+	SYM
ap-2090	114	31	1	1	NUM
ap-2090	114	32	=	=	SYM
ap-2090	114	33	α	α	NOUN
ap-2090	114	34	+	+	NOUN
ap-2090	114	35	α′.	α′.	NOUN
ap-2090	114	36	since	since	SCONJ
ap-2090	114	37	the	the	DET
ap-2090	114	38	two	two	NUM
ap-2090	114	39	forms	form	NOUN
ap-2090	114	40	must	must	AUX
ap-2090	114	41	be	be	AUX
ap-2090	114	42	equivalent	equivalent	ADJ
ap-2090	114	43	,	,	PUNCT
ap-2090	114	44	they	they	PRON
ap-2090	114	45	are	be	AUX
ap-2090	114	46	like	like	ADP
ap-2090	114	47	two	two	NUM
ap-2090	114	48	states	state	NOUN
ap-2090	114	49	of	of	ADP
ap-2090	114	50	parity	parity	NOUN
ap-2090	114	51	,	,	PUNCT
ap-2090	114	52	131	131	NUM
ap-2090	114	53	m.	m.	NOUN
ap-2090	114	54	howard	howard	PROPN
ap-2090	114	55	lee	lee	PROPN
ap-2090	114	56	acta	acta	PROPN
ap-2090	114	57	polytechnica	polytechnica	PROPN
ap-2090	114	58	which	which	PRON
ap-2090	114	59	could	could	AUX
ap-2090	114	60	be	be	AUX
ap-2090	114	61	represented	represent	VERB
ap-2090	114	62	by	by	ADP
ap-2090	114	63	a	a	DET
ap-2090	114	64	double	double	ADV
ap-2090	114	65	-	-	PUNCT
ap-2090	114	66	valued	value	VERB
ap-2090	114	67	function	function	NOUN
ap-2090	114	68	.	.	PUNCT
ap-2090	115	1	for	for	ADP
ap-2090	115	2	3	3	NUM
ap-2090	115	3	-	-	PUNCT
ap-2090	115	4	cycle	cycle	NOUN
ap-2090	115	5	in	in	ADP
ap-2090	115	6	the	the	DET
ap-2090	115	7	logistic	logistic	ADJ
ap-2090	115	8	map	map	NOUN
ap-2090	115	9	it	it	PRON
ap-2090	115	10	happens	happen	VERB
ap-2090	115	11	to	to	PART
ap-2090	115	12	be	be	AUX
ap-2090	115	13	σ	σ	NOUN
ap-2090	115	14	=	=	SYM
ap-2090	115	15	(	(	PUNCT
ap-2090	115	16	a2	a2	PROPN
ap-2090	115	17	−	−	PROPN
ap-2090	116	1	2a−	2a−	NUM
ap-2090	116	2	7)1/2	7)1/2	NUM
ap-2090	117	1	[	[	X
ap-2090	117	2	7	7	NUM
ap-2090	117	3	]	]	PUNCT
ap-2090	117	4	.	.	PUNCT
ap-2090	118	1	thus	thus	ADV
ap-2090	118	2	,	,	PUNCT
ap-2090	118	3	we	we	PRON
ap-2090	118	4	shall	shall	AUX
ap-2090	118	5	assume	assume	VERB
ap-2090	118	6	α	α	NOUN
ap-2090	118	7	=	=	NOUN
ap-2090	118	8	1	1	NUM
ap-2090	118	9	2δ5	2δ5	NUM
ap-2090	119	1	+	+	NOUN
ap-2090	119	2	kσ	kσ	PROPN
ap-2090	119	3	,	,	PUNCT
ap-2090	119	4	(	(	PUNCT
ap-2090	119	5	5	5	NUM
ap-2090	119	6	)	)	PUNCT
ap-2090	119	7	α′	α′	NUM
ap-2090	120	1	=	=	SYM
ap-2090	120	2	1	1	NUM
ap-2090	120	3	2δ5	2δ5	NUM
ap-2090	120	4	−kσ	−kσ	NOUN
ap-2090	120	5	,	,	PUNCT
ap-2090	120	6	(	(	PUNCT
ap-2090	120	7	6	6	NUM
ap-2090	120	8	)	)	PUNCT
ap-2090	120	9	where	where	SCONJ
ap-2090	120	10	if	if	SCONJ
ap-2090	120	11	k	k	PROPN
ap-2090	120	12	=	=	SYM
ap-2090	120	13	1/2	1/2	NUM
ap-2090	120	14	it	it	PRON
ap-2090	120	15	yields	yield	VERB
ap-2090	120	16	all	all	DET
ap-2090	120	17	the	the	DET
ap-2090	120	18	coefficients	coefficient	NOUN
ap-2090	120	19	(	(	PUNCT
ap-2090	120	20	δ′s	δ′	NOUN
ap-2090	120	21	)	)	PUNCT
ap-2090	120	22	exactly	exactly	ADV
ap-2090	120	23	.	.	PUNCT
ap-2090	121	1	therewith	therewith	NOUN
ap-2090	121	2	we	we	PRON
ap-2090	121	3	finally	finally	ADV
ap-2090	121	4	obtain	obtain	VERB
ap-2090	121	5	:	:	PUNCT
ap-2090	121	6	α	α	NOUN
ap-2090	121	7	=	=	SYM
ap-2090	121	8	1	1	NUM
ap-2090	121	9	2(3a+	2(3a+	NUM
ap-2090	121	10	1	1	NUM
ap-2090	121	11	+	+	NUM
ap-2090	121	12	σ	σ	NOUN
ap-2090	121	13	)	)	PUNCT
ap-2090	121	14	,	,	PUNCT
ap-2090	121	15	(	(	PUNCT
ap-2090	121	16	7	7	X
ap-2090	121	17	)	)	PUNCT
ap-2090	121	18	β	β	NOUN
ap-2090	121	19	=	=	NOUN
ap-2090	121	20	1	1	NUM
ap-2090	121	21	2(a2	2(a2	NUM
ap-2090	121	22	+	+	CCONJ
ap-2090	121	23	2a−	2a−	NUM
ap-2090	121	24	1	1	NUM
ap-2090	121	25	+	+	CCONJ
ap-2090	121	26	(	(	PUNCT
ap-2090	121	27	a+	a+	X
ap-2090	121	28	1)σ	1)σ	NUM
ap-2090	121	29	)	)	PUNCT
ap-2090	121	30	,	,	PUNCT
ap-2090	121	31	(	(	PUNCT
ap-2090	121	32	8)	8)	NUM
ap-2090	121	33	γ	γ	X
ap-2090	121	34	=	=	SYM
ap-2090	121	35	1	1	NUM
ap-2090	121	36	2(a2	2(a2	NUM
ap-2090	121	37	−	−	NOUN
ap-2090	121	38	a−	a−	PROPN
ap-2090	121	39	2	2	NUM
ap-2090	121	40	+	+	CCONJ
ap-2090	121	41	aσ	aσ	NOUN
ap-2090	121	42	)	)	PUNCT
ap-2090	121	43	,	,	PUNCT
ap-2090	121	44	(	(	PUNCT
ap-2090	121	45	9	9	NUM
ap-2090	121	46	)	)	PUNCT
ap-2090	121	47	and	and	CCONJ
ap-2090	121	48	α′	α′	NUM
ap-2090	121	49	,	,	PUNCT
ap-2090	121	50	β′	β′	NUM
ap-2090	121	51	,	,	PUNCT
ap-2090	121	52	γ′	γ′	PUNCT
ap-2090	121	53	by	by	ADP
ap-2090	121	54	taking	take	VERB
ap-2090	121	55	−σ	−σ	NOUN
ap-2090	121	56	in	in	ADP
ap-2090	121	57	(	(	PUNCT
ap-2090	121	58	7)–(9	7)–(9	NOUN
ap-2090	121	59	)	)	PUNCT
ap-2090	121	60	.	.	PUNCT
ap-2090	122	1	hence	hence	ADV
ap-2090	122	2	if	if	SCONJ
ap-2090	122	3	q	q	X
ap-2090	122	4	=	=	SYM
ap-2090	122	5	q(σ	q(σ	NUM
ap-2090	122	6	)	)	PUNCT
ap-2090	122	7	,	,	PUNCT
ap-2090	122	8	q′	q′	NOUN
ap-2090	122	9	=	=	PUNCT
ap-2090	122	10	q(−σ	q(−σ	NOUN
ap-2090	122	11	)	)	PUNCT
ap-2090	122	12	.	.	PUNCT
ap-2090	123	1	it	it	PRON
ap-2090	123	2	is	be	AUX
ap-2090	123	3	sufficient	sufficient	ADJ
ap-2090	123	4	to	to	PART
ap-2090	123	5	solve	solve	VERB
ap-2090	123	6	only	only	ADV
ap-2090	123	7	one	one	NUM
ap-2090	123	8	of	of	ADP
ap-2090	123	9	them	they	PRON
ap-2090	123	10	.	.	PUNCT
ap-2090	124	1	even	even	ADV
ap-2090	124	2	beforehand	beforehand	ADV
ap-2090	124	3	,	,	PUNCT
ap-2090	124	4	it	it	PRON
ap-2090	124	5	is	be	AUX
ap-2090	124	6	possible	possible	ADJ
ap-2090	124	7	to	to	PART
ap-2090	124	8	prove	prove	VERB
ap-2090	124	9	that	that	SCONJ
ap-2090	124	10	tk	tk	PROPN
ap-2090	124	11	and	and	CCONJ
ap-2090	124	12	t′k	t′k	PROPN
ap-2090	124	13	,	,	PUNCT
ap-2090	124	14	k	k	PROPN
ap-2090	124	15	=	=	SYM
ap-2090	124	16	1	1	NUM
ap-2090	124	17	,	,	PUNCT
ap-2090	124	18	2	2	NUM
ap-2090	124	19	,	,	PUNCT
ap-2090	124	20	3	3	NUM
ap-2090	124	21	,	,	PUNCT
ap-2090	124	22	all	all	PRON
ap-2090	124	23	lie	lie	VERB
ap-2090	124	24	in	in	ADP
ap-2090	124	25	the	the	DET
ap-2090	124	26	interval	interval	NOUN
ap-2090	124	27	(	(	PUNCT
ap-2090	124	28	0	0	NUM
ap-2090	124	29	,	,	PUNCT
ap-2090	124	30	a	a	DET
ap-2090	124	31	)	)	PUNCT
ap-2090	124	32	.	.	PUNCT
ap-2090	125	1	the	the	DET
ap-2090	125	2	proof	proof	NOUN
ap-2090	125	3	goes	go	VERB
ap-2090	125	4	as	as	SCONJ
ap-2090	125	5	follows	follow	VERB
ap-2090	125	6	:	:	PUNCT
ap-2090	125	7	let	let	VERB
ap-2090	125	8	tk	tk	PROPN
ap-2090	125	9	=	=	SYM
ap-2090	125	10	a−	a−	PROPN
ap-2090	125	11	ε	ε	PROPN
ap-2090	125	12	.	.	PUNCT
ap-2090	126	1	if	if	SCONJ
ap-2090	126	2	ã	ã	PROPN
ap-2090	126	3	≤	≤	VERB
ap-2090	126	4	a	a	DET
ap-2090	126	5	≤	≤	NUM
ap-2090	126	6	4	4	NUM
ap-2090	126	7	,	,	PUNCT
ap-2090	126	8	tk	tk	PROPN
ap-2090	126	9	>	>	X
ap-2090	126	10	0	0	PUNCT
ap-2090	127	1	as	as	SCONJ
ap-2090	127	2	already	already	ADV
ap-2090	127	3	proved	prove	VERB
ap-2090	127	4	.	.	PUNCT
ap-2090	128	1	hence	hence	ADV
ap-2090	128	2	a−	a−	PROPN
ap-2090	128	3	ε	ε	PROPN
ap-2090	128	4	>	>	X
ap-2090	128	5	0	0	PROPN
ap-2090	128	6	.	.	PUNCT
ap-2090	129	1	this	this	PRON
ap-2090	129	2	gives	give	VERB
ap-2090	129	3	ε	ε	PROPN
ap-2090	129	4	<	<	X
ap-2090	129	5	a	a	PROPN
ap-2090	129	6	,	,	PUNCT
ap-2090	129	7	an	an	DET
ap-2090	129	8	upper	upper	ADJ
ap-2090	129	9	bound	bind	VERB
ap-2090	129	10	on	on	ADP
ap-2090	129	11	ε	ε	PROPN
ap-2090	129	12	.	.	PUNCT
ap-2090	129	13	to	to	PART
ap-2090	129	14	determine	determine	VERB
ap-2090	129	15	a	a	DET
ap-2090	129	16	lower	low	ADJ
ap-2090	129	17	bound	bind	VERB
ap-2090	129	18	on	on	ADP
ap-2090	129	19	ε	ε	PROPN
ap-2090	129	20	,	,	PUNCT
ap-2090	129	21	we	we	PRON
ap-2090	129	22	consider	consider	VERB
ap-2090	129	23	the	the	DET
ap-2090	129	24	trigonal	trigonal	ADJ
ap-2090	129	25	relation	relation	NOUN
ap-2090	129	26	with	with	ADP
ap-2090	129	27	tk	tk	PROPN
ap-2090	129	28	=	=	SYM
ap-2090	129	29	a−	a−	PROPN
ap-2090	129	30	ε	ε	PROPN
ap-2090	129	31	for	for	ADP
ap-2090	129	32	any	any	DET
ap-2090	129	33	k	k	NOUN
ap-2090	129	34	=	=	SYM
ap-2090	129	35	1	1	NUM
ap-2090	129	36	,	,	PUNCT
ap-2090	129	37	2	2	NUM
ap-2090	129	38	or	or	CCONJ
ap-2090	129	39	3	3	NUM
ap-2090	129	40	.	.	PUNCT
ap-2090	130	1	by	by	ADP
ap-2090	130	2	(	(	PUNCT
ap-2090	130	3	7)–(9	7)–(9	NOUN
ap-2090	130	4	)	)	PUNCT
ap-2090	130	5	we	we	PRON
ap-2090	130	6	find	find	VERB
ap-2090	130	7	that	that	SCONJ
ap-2090	130	8	α−	α−	ADP
ap-2090	130	9	β	β	X
ap-2090	130	10	+	+	CCONJ
ap-2090	130	11	γ	γ	X
ap-2090	130	12	=	=	SYM
ap-2090	130	13	−r(ε)/(a−	−r(ε)/(a−	PROPN
ap-2090	130	14	ε	ε	PROPN
ap-2090	130	15	)	)	PUNCT
ap-2090	130	16	,	,	PUNCT
ap-2090	130	17	(	(	PUNCT
ap-2090	130	18	10	10	NUM
ap-2090	130	19	)	)	PUNCT
ap-2090	130	20	where	where	SCONJ
ap-2090	130	21	r(ε	r(ε	NOUN
ap-2090	130	22	)	)	PUNCT
ap-2090	130	23	=	=	PRON
ap-2090	130	24	ε3	ε3	VERB
ap-2090	130	25	−aε2	−aε2	ADP
ap-2090	130	26	+	+	NOUN
ap-2090	130	27	bε−	bε−	PROPN
ap-2090	130	28	1	1	NUM
ap-2090	130	29	,	,	PUNCT
ap-2090	130	30	(	(	PUNCT
ap-2090	130	31	11	11	NUM
ap-2090	130	32	)	)	PUNCT
ap-2090	130	33	where	where	SCONJ
ap-2090	130	34	a	a	PRON
ap-2090	130	35	and	and	CCONJ
ap-2090	130	36	b	b	NOUN
ap-2090	130	37	are	be	AUX
ap-2090	130	38	real	real	ADJ
ap-2090	130	39	and	and	CCONJ
ap-2090	130	40	positive	positive	ADJ
ap-2090	130	41	if	if	SCONJ
ap-2090	130	42	ã	ã	PROPN
ap-2090	130	43	≤	≤	VERB
ap-2090	130	44	a	a	DET
ap-2090	130	45	≤	≤	NUM
ap-2090	130	46	4	4	NUM
ap-2090	130	47	.	.	PUNCT
ap-2090	131	1	the	the	DET
ap-2090	131	2	trigonal	trigonal	PROPN
ap-2090	131	3	relation	relation	NOUN
ap-2090	131	4	requires	require	VERB
ap-2090	131	5	that	that	SCONJ
ap-2090	131	6	r(ε	r(ε	VERB
ap-2090	131	7	)	)	PUNCT
ap-2090	131	8	=	=	SYM
ap-2090	132	1	0	0	X
ap-2090	132	2	.	.	PUNCT
ap-2090	133	1	this	this	PRON
ap-2090	133	2	is	be	AUX
ap-2090	133	3	possible	possible	ADJ
ap-2090	133	4	if	if	SCONJ
ap-2090	133	5	and	and	CCONJ
ap-2090	133	6	only	only	ADV
ap-2090	133	7	if	if	SCONJ
ap-2090	133	8	ε	ε	PROPN
ap-2090	133	9	>	>	X
ap-2090	133	10	0	0	PROPN
ap-2090	133	11	,	,	PUNCT
ap-2090	133	12	which	which	PRON
ap-2090	133	13	gives	give	VERB
ap-2090	133	14	a	a	DET
ap-2090	133	15	lower	low	ADJ
ap-2090	133	16	bound	bind	VERB
ap-2090	133	17	on	on	ADP
ap-2090	133	18	ε	ε	PROPN
ap-2090	133	19	.	.	PUNCT
ap-2090	134	1	thus	thus	ADV
ap-2090	134	2	,	,	PUNCT
ap-2090	134	3	0	0	PUNCT
ap-2090	134	4	<	<	X
ap-2090	134	5	ε	ε	X
ap-2090	134	6	<	<	X
ap-2090	134	7	a	a	PROPN
ap-2090	134	8	and	and	CCONJ
ap-2090	134	9	this	this	PRON
ap-2090	134	10	implies	imply	VERB
ap-2090	135	1	that	that	SCONJ
ap-2090	135	2	tk	tk	PROPN
ap-2090	135	3	=	=	PUNCT
ap-2090	135	4	a	a	DET
ap-2090	135	5	−	−	PROPN
ap-2090	135	6	ε	ε	PROPN
ap-2090	135	7	must	must	AUX
ap-2090	135	8	lie	lie	VERB
ap-2090	135	9	in	in	ADP
ap-2090	135	10	the	the	DET
ap-2090	135	11	interval	interval	NOUN
ap-2090	135	12	t	t	NOUN
ap-2090	135	13	=	=	SYM
ap-2090	135	14	(	(	PUNCT
ap-2090	135	15	0	0	NUM
ap-2090	135	16	,	,	PUNCT
ap-2090	135	17	a	a	PRON
ap-2090	135	18	)	)	PUNCT
ap-2090	135	19	.	.	PUNCT
ap-2090	136	1	this	this	DET
ap-2090	136	2	proof	proof	NOUN
ap-2090	136	3	agrees	agree	VERB
ap-2090	136	4	with	with	ADP
ap-2090	136	5	the	the	DET
ap-2090	136	6	cubic	cubic	ADJ
ap-2090	136	7	solutions	solution	NOUN
ap-2090	136	8	obtained	obtain	VERB
ap-2090	136	9	from	from	ADP
ap-2090	136	10	q	q	PROPN
ap-2090	136	11	=	=	SYM
ap-2090	136	12	0	0	NUM
ap-2090	136	13	and	and	CCONJ
ap-2090	136	14	q′	q′	NOUN
ap-2090	136	15	=	=	SYM
ap-2090	137	1	0	0	X
ap-2090	137	2	.	.	PUNCT
ap-2090	138	1	we	we	PRON
ap-2090	138	2	conclude	conclude	VERB
ap-2090	138	3	that	that	SCONJ
ap-2090	138	4	the	the	DET
ap-2090	138	5	3	3	NUM
ap-2090	138	6	-	-	PUNCT
ap-2090	138	7	cycle	cycle	NOUN
ap-2090	138	8	exists	exist	VERB
ap-2090	138	9	continuously	continuously	ADV
ap-2090	138	10	in	in	ADP
ap-2090	138	11	the	the	DET
ap-2090	138	12	logistic	logistic	ADJ
ap-2090	138	13	map	map	NOUN
ap-2090	138	14	from	from	ADP
ap-2090	138	15	a	a	DET
ap-2090	138	16	=	=	SYM
ap-2090	138	17	ã	ã	PROPN
ap-2090	138	18	to	to	ADP
ap-2090	138	19	a	a	DET
ap-2090	138	20	=	=	NOUN
ap-2090	138	21	4	4	NUM
ap-2090	138	22	.	.	PUNCT
ap-2090	139	1	throughout	throughout	ADP
ap-2090	139	2	this	this	DET
ap-2090	139	3	domain	domain	NOUN
ap-2090	139	4	there	there	PRON
ap-2090	139	5	exists	exist	VERB
ap-2090	139	6	chaos	chaos	NOUN
ap-2090	139	7	because	because	SCONJ
ap-2090	139	8	there	there	PRON
ap-2090	139	9	are	be	VERB
ap-2090	139	10	,	,	PUNCT
ap-2090	139	11	according	accord	VERB
ap-2090	139	12	to	to	ADP
ap-2090	139	13	the	the	DET
ap-2090	139	14	theorem	theorem	NOUN
ap-2090	139	15	of	of	ADP
ap-2090	139	16	sharkovskii	sharkovskii	PROPN
ap-2090	139	17	,	,	PUNCT
ap-2090	139	18	infinitely	infinitely	ADV
ap-2090	139	19	many	many	ADJ
ap-2090	139	20	cycles	cycle	NOUN
ap-2090	139	21	of	of	ADP
ap-2090	139	22	all	all	DET
ap-2090	139	23	kinds	kind	NOUN
ap-2090	139	24	.	.	PUNCT
ap-2090	140	1	6	6	X
ap-2090	140	2	.	.	X
ap-2090	140	3	conclusion	conclusion	NOUN
ap-2090	140	4	:	:	PUNCT
ap-2090	140	5	aleph	aleph	NOUN
ap-2090	140	6	cycle	cycle	NOUN
ap-2090	140	7	and	and	CCONJ
ap-2090	140	8	chaotic	chaotic	ADJ
ap-2090	140	9	trajectories	trajectory	NOUN
ap-2090	140	10	the	the	DET
ap-2090	140	11	theorem	theorem	NOUN
ap-2090	140	12	of	of	ADP
ap-2090	140	13	sharkovskii	sharkovskii	PROPN
ap-2090	140	14	asserts	assert	VERB
ap-2090	140	15	that	that	SCONJ
ap-2090	140	16	if	if	SCONJ
ap-2090	140	17	3	3	NUM
ap-2090	140	18	-	-	PUNCT
ap-2090	140	19	cycle	cycle	NOUN
ap-2090	140	20	exists	exist	VERB
ap-2090	140	21	in	in	ADP
ap-2090	140	22	a	a	DET
ap-2090	140	23	domain	domain	NOUN
ap-2090	140	24	,	,	PUNCT
ap-2090	140	25	all	all	DET
ap-2090	140	26	other	other	ADJ
ap-2090	140	27	cycles	cycle	NOUN
ap-2090	140	28	exist	exist	VERB
ap-2090	140	29	in	in	ADP
ap-2090	140	30	that	that	DET
ap-2090	140	31	domain	domain	NOUN
ap-2090	140	32	.	.	PUNCT
ap-2090	141	1	for	for	ADP
ap-2090	141	2	the	the	DET
ap-2090	141	3	logistic	logistic	ADJ
ap-2090	141	4	map	map	NOUN
ap-2090	141	5	we	we	PRON
ap-2090	141	6	take	take	VERB
ap-2090	141	7	this	this	PRON
ap-2090	141	8	to	to	PART
ap-2090	141	9	mean	mean	VERB
ap-2090	141	10	that	that	SCONJ
ap-2090	141	11	there	there	PRON
ap-2090	141	12	are	be	VERB
ap-2090	141	13	infinitely	infinitely	ADV
ap-2090	141	14	many	many	ADJ
ap-2090	141	15	cycles	cycle	NOUN
ap-2090	141	16	of	of	ADP
ap-2090	141	17	all	all	DET
ap-2090	141	18	varieties	variety	NOUN
ap-2090	141	19	in	in	ADP
ap-2090	141	20	the	the	DET
ap-2090	141	21	domain	domain	NOUN
ap-2090	141	22	ã	ã	PROPN
ap-2090	141	23	≤	≤	VERB
ap-2090	141	24	a	a	DET
ap-2090	141	25	≤	≤	NUM
ap-2090	141	26	4	4	NUM
ap-2090	141	27	.	.	PUNCT
ap-2090	142	1	thus	thus	ADV
ap-2090	142	2	,	,	PUNCT
ap-2090	142	3	in	in	ADP
ap-2090	142	4	this	this	DET
ap-2090	142	5	domain	domain	NOUN
ap-2090	142	6	the	the	DET
ap-2090	142	7	fixed	fix	VERB
ap-2090	142	8	points	point	NOUN
ap-2090	142	9	form	form	VERB
ap-2090	142	10	a	a	DET
ap-2090	142	11	dense	dense	ADJ
ap-2090	142	12	set	set	NOUN
ap-2090	142	13	of	of	ADP
ap-2090	142	14	points	point	NOUN
ap-2090	142	15	in	in	ADP
ap-2090	142	16	x	x	X
ap-2090	142	17	=	=	SYM
ap-2090	142	18	(	(	PUNCT
ap-2090	142	19	0	0	NUM
ap-2090	142	20	,	,	PUNCT
ap-2090	142	21	1	1	NUM
ap-2090	142	22	)	)	PUNCT
ap-2090	142	23	,	,	PUNCT
ap-2090	142	24	a	a	DET
ap-2090	142	25	strip	strip	NOUN
ap-2090	142	26	made	make	VERB
ap-2090	142	27	up	up	ADP
ap-2090	142	28	of	of	ADP
ap-2090	142	29	irrational	irrational	ADJ
ap-2090	142	30	points	point	NOUN
ap-2090	142	31	of	of	ADP
ap-2090	142	32	all	all	DET
ap-2090	142	33	different	different	ADJ
ap-2090	142	34	kinds	kind	NOUN
ap-2090	142	35	.	.	PUNCT
ap-2090	143	1	it	it	PRON
ap-2090	143	2	has	have	VERB
ap-2090	143	3	a	a	DET
ap-2090	143	4	finite	finite	ADJ
ap-2090	143	5	measure	measure	NOUN
ap-2090	143	6	.	.	PUNCT
ap-2090	144	1	if	if	SCONJ
ap-2090	144	2	x1	x1	PROPN
ap-2090	144	3	is	be	AUX
ap-2090	144	4	an	an	DET
ap-2090	144	5	initial	initial	ADJ
ap-2090	144	6	value	value	NOUN
ap-2090	144	7	that	that	PRON
ap-2090	144	8	belongs	belong	VERB
ap-2090	144	9	to	to	ADP
ap-2090	144	10	this	this	DET
ap-2090	144	11	set	set	NOUN
ap-2090	144	12	of	of	ADP
ap-2090	144	13	points	point	NOUN
ap-2090	144	14	of	of	ADP
ap-2090	144	15	finite	finite	ADJ
ap-2090	144	16	measure	measure	NOUN
ap-2090	144	17	,	,	PUNCT
ap-2090	144	18	fn	fn	PROPN
ap-2090	144	19	(	(	PUNCT
ap-2090	144	20	x1	x1	PROPN
ap-2090	144	21	)	)	PUNCT
ap-2090	144	22	=	=	SYM
ap-2090	144	23	xn+1	xn+1	PROPN
ap-2090	144	24	,	,	PUNCT
ap-2090	144	25	n	n	X
ap-2090	144	26	→	→	SYM
ap-2090	144	27	∞	∞	PROPN
ap-2090	144	28	,	,	PUNCT
ap-2090	144	29	termed	term	VERB
ap-2090	144	30	an	an	DET
ap-2090	144	31	aleph	aleph	NOUN
ap-2090	144	32	cycle	cycle	NOUN
ap-2090	144	33	.	.	PUNCT
ap-2090	145	1	evidently	evidently	ADV
ap-2090	145	2	an	an	DET
ap-2090	145	3	aleph	aleph	NOUN
ap-2090	145	4	cycle	cycle	NOUN
ap-2090	145	5	describes	describe	VERB
ap-2090	145	6	a	a	DET
ap-2090	145	7	chaotic	chaotic	ADJ
ap-2090	145	8	trajectory	trajectory	NOUN
ap-2090	145	9	.	.	PUNCT
ap-2090	146	1	if	if	SCONJ
ap-2090	146	2	x0	x0	PROPN
ap-2090	146	3	is	be	AUX
ap-2090	146	4	an	an	DET
ap-2090	146	5	initial	initial	ADJ
ap-2090	146	6	value	value	NOUN
ap-2090	146	7	that	that	PRON
ap-2090	146	8	does	do	AUX
ap-2090	146	9	not	not	PART
ap-2090	146	10	belong	belong	VERB
ap-2090	146	11	to	to	ADP
ap-2090	146	12	this	this	DET
ap-2090	146	13	set	set	NOUN
ap-2090	146	14	,	,	PUNCT
ap-2090	146	15	thus	thus	ADV
ap-2090	146	16	belongs	belong	VERB
ap-2090	146	17	to	to	ADP
ap-2090	146	18	a	a	DET
ap-2090	146	19	set	set	NOUN
ap-2090	146	20	of	of	ADP
ap-2090	146	21	points	point	NOUN
ap-2090	146	22	of	of	ADP
ap-2090	146	23	0	0	NUM
ap-2090	146	24	measure	measure	NOUN
ap-2090	146	25	,	,	PUNCT
ap-2090	146	26	fm	fm	PROPN
ap-2090	146	27	(	(	PUNCT
ap-2090	146	28	x0	x0	PROPN
ap-2090	146	29	)	)	PUNCT
ap-2090	147	1	=	=	SYM
ap-2090	147	2	x0	x0	PROPN
ap-2090	147	3	,	,	PUNCT
ap-2090	147	4	m	m	VERB
ap-2090	147	5	<	<	X
ap-2090	147	6	∞	∞	PROPN
ap-2090	147	7	,	,	PUNCT
ap-2090	147	8	a	a	DET
ap-2090	147	9	finite	finite	ADJ
ap-2090	147	10	cycle	cycle	NOUN
ap-2090	147	11	,	,	PUNCT
ap-2090	147	12	giving	give	VERB
ap-2090	147	13	a	a	DET
ap-2090	147	14	periodic	periodic	ADJ
ap-2090	147	15	trajectory	trajectory	NOUN
ap-2090	147	16	.	.	PUNCT
ap-2090	148	1	this	this	DET
ap-2090	148	2	assertion	assertion	NOUN
ap-2090	148	3	can	can	AUX
ap-2090	148	4	be	be	AUX
ap-2090	148	5	justified	justify	VERB
ap-2090	148	6	for	for	ADP
ap-2090	148	7	some	some	DET
ap-2090	148	8	special	special	ADJ
ap-2090	148	9	values	value	NOUN
ap-2090	148	10	of	of	ADP
ap-2090	148	11	a	a	DET
ap-2090	148	12	such	such	ADJ
ap-2090	148	13	as	as	ADP
ap-2090	148	14	a	a	DET
ap-2090	148	15	=	=	SYM
ap-2090	148	16	4	4	NUM
ap-2090	148	17	,	,	PUNCT
ap-2090	148	18	for	for	ADP
ap-2090	148	19	which	which	PRON
ap-2090	148	20	one	one	PRON
ap-2090	148	21	can	can	AUX
ap-2090	148	22	obtain	obtain	VERB
ap-2090	148	23	a	a	DET
ap-2090	148	24	large	large	ADJ
ap-2090	148	25	number	number	NOUN
ap-2090	148	26	of	of	ADP
ap-2090	148	27	fixed	fix	VERB
ap-2090	148	28	points	point	NOUN
ap-2090	148	29	from	from	ADP
ap-2090	148	30	which	which	PRON
ap-2090	148	31	a	a	DET
ap-2090	148	32	set	set	NOUN
ap-2090	148	33	of	of	ADP
ap-2090	148	34	points	point	NOUN
ap-2090	148	35	of	of	ADP
ap-2090	148	36	finite	finite	ADJ
ap-2090	148	37	measure	measure	NOUN
ap-2090	148	38	can	can	AUX
ap-2090	148	39	be	be	AUX
ap-2090	148	40	constructed	construct	VERB
ap-2090	148	41	[	[	PUNCT
ap-2090	148	42	6	6	NUM
ap-2090	148	43	]	]	PUNCT
ap-2090	148	44	.	.	PUNCT
ap-2090	149	1	an	an	DET
ap-2090	149	2	aleph	aleph	PROPN
ap-2090	149	3	cycle	cycle	NOUN
ap-2090	149	4	has	have	VERB
ap-2090	149	5	a	a	DET
ap-2090	149	6	very	very	ADV
ap-2090	149	7	close	close	ADJ
ap-2090	149	8	connection	connection	NOUN
ap-2090	149	9	to	to	ADP
ap-2090	149	10	an	an	DET
ap-2090	149	11	ergodic	ergodic	ADJ
ap-2090	149	12	trajectory	trajectory	NOUN
ap-2090	149	13	in	in	ADP
ap-2090	149	14	statistical	statistical	ADJ
ap-2090	149	15	mechanics	mechanic	NOUN
ap-2090	149	16	according	accord	VERB
ap-2090	149	17	to	to	ADP
ap-2090	149	18	the	the	DET
ap-2090	149	19	ergometic	ergometic	ADJ
ap-2090	149	20	theory	theory	NOUN
ap-2090	149	21	of	of	ADP
ap-2090	149	22	the	the	DET
ap-2090	149	23	ergodic	ergodic	ADJ
ap-2090	149	24	hypothesis	hypothesis	NOUN
ap-2090	149	25	[	[	X
ap-2090	149	26	8	8	NUM
ap-2090	149	27	]	]	PUNCT
ap-2090	149	28	.	.	PUNCT
ap-2090	150	1	it	it	PRON
ap-2090	150	2	is	be	AUX
ap-2090	150	3	thus	thus	ADV
ap-2090	150	4	possible	possible	ADJ
ap-2090	150	5	to	to	PART
ap-2090	150	6	gain	gain	VERB
ap-2090	150	7	via	via	ADP
ap-2090	150	8	an	an	DET
ap-2090	150	9	aleph	aleph	PROPN
ap-2090	150	10	cycle	cycle	NOUN
ap-2090	150	11	an	an	DET
ap-2090	150	12	insight	insight	NOUN
ap-2090	150	13	into	into	ADP
ap-2090	150	14	the	the	DET
ap-2090	150	15	ergodic	ergodic	ADJ
ap-2090	150	16	theory	theory	NOUN
ap-2090	150	17	of	of	ADP
ap-2090	150	18	chaos	chaos	NOUN
ap-2090	150	19	that	that	PRON
ap-2090	150	20	has	have	AUX
ap-2090	150	21	been	be	AUX
ap-2090	150	22	proposed	propose	VERB
ap-2090	150	23	[	[	PUNCT
ap-2090	150	24	9	9	NUM
ap-2090	150	25	]	]	PUNCT
ap-2090	150	26	.	.	PUNCT
ap-2090	151	1	as	as	SCONJ
ap-2090	151	2	we	we	PRON
ap-2090	151	3	have	have	AUX
ap-2090	151	4	stated	state	VERB
ap-2090	151	5	at	at	ADP
ap-2090	151	6	the	the	DET
ap-2090	151	7	outset	outset	NOUN
ap-2090	151	8	,	,	PUNCT
ap-2090	151	9	while	while	SCONJ
ap-2090	151	10	it	it	PRON
ap-2090	151	11	would	would	AUX
ap-2090	151	12	be	be	AUX
ap-2090	151	13	nearly	nearly	ADV
ap-2090	151	14	impossible	impossible	ADJ
ap-2090	151	15	to	to	PART
ap-2090	151	16	study	study	VERB
ap-2090	151	17	chaos	chaos	NOUN
ap-2090	151	18	analytically	analytically	ADV
ap-2090	151	19	,	,	PUNCT
ap-2090	151	20	the	the	DET
ap-2090	151	21	theorem	theorem	NOUN
ap-2090	151	22	of	of	ADP
ap-2090	151	23	sharkovskii	sharkovskii	PROPN
ap-2090	151	24	gives	give	VERB
ap-2090	151	25	a	a	DET
ap-2090	151	26	window	window	NOUN
ap-2090	151	27	through	through	ADP
ap-2090	151	28	which	which	PRON
ap-2090	151	29	something	something	PRON
ap-2090	151	30	can	can	AUX
ap-2090	151	31	be	be	AUX
ap-2090	151	32	done	do	VERB
ap-2090	151	33	analytically	analytically	ADV
ap-2090	151	34	,	,	PUNCT
ap-2090	151	35	as	as	SCONJ
ap-2090	151	36	we	we	PRON
ap-2090	151	37	have	have	AUX
ap-2090	151	38	described	describe	VERB
ap-2090	151	39	here	here	ADV
ap-2090	151	40	in	in	ADP
ap-2090	151	41	this	this	DET
ap-2090	151	42	short	short	ADJ
ap-2090	151	43	note	note	NOUN
ap-2090	151	44	.	.	PUNCT
ap-2090	152	1	acknowledgements	acknowledgement	NOUN
ap-2090	152	2	i	i	PRON
ap-2090	152	3	thank	thank	VERB
ap-2090	152	4	dr	dr	PROPN
ap-2090	152	5	miloslav	miloslav	PROPN
ap-2090	152	6	znojil	znojil	NOUN
ap-2090	152	7	of	of	ADP
ap-2090	152	8	the	the	DET
ap-2090	152	9	doppler	doppler	NOUN
ap-2090	152	10	institute	institute	NOUN
ap-2090	152	11	for	for	ADP
ap-2090	152	12	mathematical	mathematical	ADJ
ap-2090	152	13	physics	physics	NOUN
ap-2090	152	14	and	and	CCONJ
ap-2090	152	15	applied	apply	VERB
ap-2090	152	16	mathematics	mathematic	NOUN
ap-2090	152	17	,	,	PUNCT
ap-2090	152	18	rez	rez	NOUN
ap-2090	152	19	,	,	PUNCT
ap-2090	152	20	czech	czech	ADJ
ap-2090	152	21	republic	republic	NOUN
ap-2090	152	22	for	for	ADP
ap-2090	152	23	suggesting	suggest	VERB
ap-2090	152	24	that	that	SCONJ
ap-2090	152	25	i	i	PRON
ap-2090	152	26	write	write	VERB
ap-2090	152	27	this	this	DET
ap-2090	152	28	article	article	NOUN
ap-2090	152	29	based	base	VERB
ap-2090	152	30	on	on	ADP
ap-2090	152	31	a	a	DET
ap-2090	152	32	talk	talk	NOUN
ap-2090	152	33	presented	present	VERB
ap-2090	152	34	at	at	ADP
ap-2090	152	35	vila	vila	PROPN
ap-2090	152	36	lanna	lanna	PROPN
ap-2090	152	37	,	,	PUNCT
ap-2090	152	38	prague	prague	NOUN
ap-2090	152	39	in	in	ADP
ap-2090	152	40	october	october	PROPN
ap-2090	152	41	2013	2013	NUM
ap-2090	152	42	.	.	PUNCT
ap-2090	153	1	references	reference	NOUN
ap-2090	153	2	[	[	X
ap-2090	153	3	1	1	NUM
ap-2090	153	4	]	]	PUNCT
ap-2090	153	5	b.	b.	PROPN
ap-2090	153	6	hu	hu	PROPN
ap-2090	153	7	,	,	PUNCT
ap-2090	153	8	phys	phys	PROPN
ap-2090	153	9	.	.	PUNCT
ap-2090	154	1	rep	rep	PROPN
ap-2090	154	2	.	.	PROPN
ap-2090	154	3	91	91	NUM
ap-2090	154	4	,	,	PUNCT
ap-2090	154	5	234	234	NUM
ap-2090	154	6	(	(	PUNCT
ap-2090	154	7	1982	1982	NUM
ap-2090	154	8	)	)	PUNCT
ap-2090	154	9	doi	doi	NOUN
ap-2090	154	10	:	:	PUNCT
ap-2090	154	11	10.1016/0370	10.1016/0370	NUM
ap-2090	154	12	-	-	SYM
ap-2090	154	13	1573(82)90057	1573(82)90057	NUM
ap-2090	154	14	-	-	PUNCT
ap-2090	154	15	6	6	NUM
ap-2090	154	16	[	[	SYM
ap-2090	154	17	2	2	NUM
ap-2090	154	18	]	]	PUNCT
ap-2090	154	19	d.	d.	NOUN
ap-2090	154	20	gulick	gulick	PROPN
ap-2090	154	21	,	,	PUNCT
ap-2090	154	22	encounters	encounter	VERB
ap-2090	154	23	with	with	ADP
ap-2090	154	24	chaos	chaos	NOUN
ap-2090	154	25	(	(	PUNCT
ap-2090	154	26	mcgraw	mcgraw	NOUN
ap-2090	154	27	-	-	PUNCT
ap-2090	154	28	hill	hill	PROPN
ap-2090	154	29	,	,	PUNCT
ap-2090	154	30	ny	ny	PROPN
ap-2090	154	31	1992	1992	NUM
ap-2090	154	32	)	)	PUNCT
ap-2090	154	33	.	.	PUNCT
ap-2090	155	1	[	[	X
ap-2090	155	2	3	3	X
ap-2090	155	3	]	]	X
ap-2090	155	4	a.n	a.n	PROPN
ap-2090	155	5	.	.	PROPN
ap-2090	155	6	sharkovskii	sharkovskii	PROPN
ap-2090	155	7	,	,	PUNCT
ap-2090	155	8	ukr	ukr	PROPN
ap-2090	155	9	.	.	PROPN
ap-2090	155	10	math	math	PROPN
ap-2090	155	11	.	.	PUNCT
ap-2090	156	1	z.	z.	PROPN
ap-2090	156	2	16	16	NUM
ap-2090	156	3	.	.	PUNCT
ap-2090	156	4	61	61	NUM
ap-2090	156	5	(	(	PUNCT
ap-2090	156	6	1964	1964	NUM
ap-2090	156	7	)	)	PUNCT
ap-2090	156	8	.	.	PUNCT
ap-2090	157	1	[	[	X
ap-2090	157	2	4	4	NUM
ap-2090	157	3	]	]	X
ap-2090	157	4	t	t	PROPN
ap-2090	157	5	-	-	PUNCT
ap-2090	157	6	y	y	PROPN
ap-2090	157	7	li	li	PROPN
ap-2090	157	8	and	and	CCONJ
ap-2090	157	9	j.a	j.a	PROPN
ap-2090	157	10	.	.	PROPN
ap-2090	157	11	yorke	yorke	PROPN
ap-2090	157	12	,	,	PUNCT
ap-2090	157	13	am	be	AUX
ap-2090	157	14	.	.	PUNCT
ap-2090	157	15	math	math	NOUN
ap-2090	157	16	.	.	PUNCT
ap-2090	158	1	monthly	monthly	ADJ
ap-2090	158	2	82	82	NUM
ap-2090	158	3	,	,	PUNCT
ap-2090	158	4	985	985	NUM
ap-2090	158	5	(	(	PUNCT
ap-2090	158	6	1975	1975	NUM
ap-2090	158	7	)	)	PUNCT
ap-2090	158	8	.	.	PUNCT
ap-2090	159	1	[	[	X
ap-2090	159	2	5	5	X
ap-2090	159	3	]	]	X
ap-2090	159	4	m.h	m.h	PROPN
ap-2090	159	5	.	.	PROPN
ap-2090	159	6	lee	lee	PROPN
ap-2090	159	7	,	,	PUNCT
ap-2090	159	8	acta	acta	PROPN
ap-2090	159	9	phys	phys	PROPN
ap-2090	159	10	.	.	PUNCT
ap-2090	160	1	pol	pol	PROPN
ap-2090	160	2	.	.	PUNCT
ap-2090	161	1	b	b	PROPN
ap-2090	161	2	42	42	NUM
ap-2090	161	3	,	,	PUNCT
ap-2090	161	4	1071	1071	NUM
ap-2090	161	5	(	(	PUNCT
ap-2090	161	6	2011	2011	NUM
ap-2090	161	7	)	)	PUNCT
ap-2090	161	8	.	.	PUNCT
ap-2090	162	1	[	[	X
ap-2090	162	2	6	6	NUM
ap-2090	162	3	]	]	X
ap-2090	162	4	m.h	m.h	PROPN
ap-2090	162	5	.	.	PROPN
ap-2090	162	6	lee	lee	PROPN
ap-2090	162	7	,	,	PUNCT
ap-2090	162	8	acta	acta	PROPN
ap-2090	162	9	phys	phys	PROPN
ap-2090	162	10	.	.	PUNCT
ap-2090	163	1	pol	pol	PROPN
ap-2090	163	2	.	.	PUNCT
ap-2090	164	1	b	b	PROPN
ap-2090	164	2	43,1053	43,1053	NUM
ap-2090	164	3	(	(	PUNCT
ap-2090	164	4	2012	2012	NUM
ap-2090	164	5	)	)	PUNCT
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ap-2090	165	5	.	.	PROPN
ap-2090	165	6	lee	lee	PROPN
ap-2090	165	7	,	,	PUNCT
ap-2090	165	8	acta	acta	PROPN
ap-2090	165	9	phys	phys	PROPN
ap-2090	165	10	.	.	PUNCT
ap-2090	166	1	pol	pol	PROPN
ap-2090	166	2	.	.	PUNCT
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ap-2090	167	3	,	,	PUNCT
ap-2090	167	4	925	925	NUM
ap-2090	167	5	(	(	PUNCT
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ap-2090	167	7	)	)	PUNCT
ap-2090	167	8	.	.	PUNCT
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ap-2090	168	5	.	.	PROPN
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ap-2090	168	7	,	,	PUNCT
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ap-2090	169	4	.	.	PUNCT
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ap-2090	170	2	,	,	PUNCT
ap-2090	170	3	250601	250601	NUM
ap-2090	170	4	(	(	PUNCT
ap-2090	170	5	2001	2001	NUM
ap-2090	170	6	)	)	PUNCT
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ap-2090	170	8	:	:	PUNCT
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ap-2090	171	2	.	.	PROPN
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ap-2090	172	2	,	,	PUNCT
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ap-2090	172	4	(	(	PUNCT
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ap-2090	172	10	/	/	SYM
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ap-2090	172	12	[	[	X
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ap-2090	172	16	.	.	PUNCT
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ap-2090	173	3	d.	d.	PROPN
ap-2090	173	4	rulle	rulle	PROPN
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ap-2090	176	6	)	)	PUNCT
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ap-2090	176	8	:	:	PUNCT
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ap-2090	176	10	/	/	SYM
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ap-2090	176	16	http://dx.doi.org/10.1103/revmodphys.57.617	http://dx.doi.org/10.1103/revmodphys.57.617	PROPN
ap-2090	176	17	acta	acta	PROPN
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ap-2090	176	22	1	1	NUM
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ap-2090	176	24	2	2	NUM
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ap-2090	176	39	4	4	NUM
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ap-2090	176	42	sextic	sextic	ADJ
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ap-2090	176	53	internal	internal	ADJ
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ap-2090	176	57	6	6	NUM
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ap-2090	176	59	:	:	PUNCT
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ap-2090	176	64	trajectories	trajectory	NOUN
ap-2090	176	65	acknowledgements	acknowledgement	NOUN
ap-2090	176	66	references	reference	NOUN