id	sid	tid	token	lemma	pos
cana-578	1	1	communications	communication	NOUN
cana-578	1	2	on	on	ADP
cana-578	1	3	applied	apply	VERB
cana-578	1	4	nonlinear	nonlinear	ADJ
cana-578	1	5	analysis	analysis	NOUN
cana-578	1	6	issn	issn	NOUN
cana-578	1	7	:	:	PUNCT
cana-578	1	8	1074	1074	NUM
cana-578	1	9	-	-	PUNCT
cana-578	1	10	133x	133x	NUM
cana-578	1	11	vol	vol	NOUN
cana-578	1	12	31	31	NUM
cana-578	1	13	no	no	NOUN
cana-578	1	14	.	.	PUNCT
cana-578	2	1	1s	1s	NUM
cana-578	2	2	(	(	PUNCT
cana-578	2	3	2024	2024	NUM
cana-578	2	4	)	)	PUNCT
cana-578	2	5	187	187	NUM
cana-578	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-578	2	7	periodic	periodic	ADJ
cana-578	2	8	behaviour	behaviour	NOUN
cana-578	2	9	of	of	ADP
cana-578	2	10	general	general	ADJ
cana-578	2	11	systems	systems	PROPN
cana-578	2	12	l.	l.	PROPN
cana-578	2	13	praveen	praveen	PROPN
cana-578	2	14	kumar1	kumar1	PROPN
cana-578	2	15	,	,	PUNCT
cana-578	2	16	vajha	vajha	VERB
cana-578	2	17	srinivasa	srinivasa	PROPN
cana-578	2	18	kumar2	kumar2	PROPN
cana-578	3	1	1research	1research	NUM
cana-578	3	2	scholar	scholar	NOUN
cana-578	3	3	,	,	PUNCT
cana-578	3	4	department	department	NOUN
cana-578	3	5	of	of	ADP
cana-578	3	6	mathematics	mathematic	NOUN
cana-578	3	7	,	,	PUNCT
cana-578	3	8	jntuh	jntuh	PROPN
cana-578	3	9	college	college	NOUN
cana-578	3	10	of	of	ADP
cana-578	3	11	engineering	engineering	PROPN
cana-578	3	12	,	,	PUNCT
cana-578	3	13	jntu	jntu	PROPN
cana-578	3	14	,	,	PUNCT
cana-578	3	15	kukatpalle	kukatpalle	PROPN
cana-578	3	16	,	,	PUNCT
cana-578	3	17	hyderabad	hyderabad	PROPN
cana-578	3	18	500085	500085	NUM
cana-578	3	19	telangana	telangana	PROPN
cana-578	3	20	state	state	PROPN
cana-578	3	21	,	,	PUNCT
cana-578	3	22	india	india	PROPN
cana-578	3	23	.	.	PUNCT
cana-578	3	24	2sr	2sr	PROPN
cana-578	3	25	.	.	PUNCT
cana-578	4	1	assistant	assistant	PROPN
cana-578	4	2	professor	professor	NOUN
cana-578	4	3	,	,	PUNCT
cana-578	4	4	department	department	NOUN
cana-578	4	5	of	of	ADP
cana-578	4	6	mathematics	mathematic	NOUN
cana-578	4	7	,	,	PUNCT
cana-578	4	8	jntuh	jntuh	PROPN
cana-578	4	9	college	college	NOUN
cana-578	4	10	of	of	ADP
cana-578	4	11	engineering	engineering	PROPN
cana-578	4	12	,	,	PUNCT
cana-578	4	13	hyderabad	hyderabad	PROPN
cana-578	4	14	,	,	PUNCT
cana-578	4	15	jntu	jntu	PROPN
cana-578	4	16	,	,	PUNCT
cana-578	4	17	kukatpalle	kukatpalle	PROPN
cana-578	4	18	,	,	PUNCT
cana-578	4	19	hyderabad	hyderabad	PROPN
cana-578	4	20	500085	500085	NUM
cana-578	4	21	,	,	PUNCT
cana-578	4	22	telangana	telangana	PROPN
cana-578	4	23	state	state	PROPN
cana-578	4	24	,	,	PUNCT
cana-578	4	25	india	india	PROPN
cana-578	4	26	.	.	PUNCT
cana-578	5	1	1email	1email	NUM
cana-578	5	2	:	:	PUNCT
cana-578	5	3	praveenkumarl55@gmail.com	praveenkumarl55@gmail.com	X
cana-578	5	4	2email	2email	NUM
cana-578	5	5	:	:	PUNCT
cana-578	5	6	vajhasrinu@gmail.com	vajhasrinu@gmail.com	X
cana-578	5	7	article	article	NOUN
cana-578	5	8	history	history	NOUN
cana-578	5	9	:	:	PUNCT
cana-578	5	10	received	receive	VERB
cana-578	5	11	:	:	PUNCT
cana-578	5	12	15	15	NUM
cana-578	5	13	-	-	SYM
cana-578	5	14	02	02	NUM
cana-578	5	15	-	-	PUNCT
cana-578	5	16	2024	2024	NUM
cana-578	5	17	revised	revise	VERB
cana-578	5	18	:	:	PUNCT
cana-578	5	19	09	09	NUM
cana-578	5	20	-	-	PUNCT
cana-578	5	21	04	04	NUM
cana-578	5	22	-	-	PUNCT
cana-578	5	23	2024	2024	NUM
cana-578	5	24	accepted	accept	VERB
cana-578	5	25	:	:	PUNCT
cana-578	5	26	24	24	NUM
cana-578	5	27	-	-	PUNCT
cana-578	5	28	04	04	NUM
cana-578	5	29	-	-	PUNCT
cana-578	5	30	2024	2024	NUM
cana-578	5	31	abstract	abstract	NOUN
cana-578	5	32	:	:	PUNCT
cana-578	5	33	a	a	DET
cana-578	5	34	continuous	continuous	ADJ
cana-578	5	35	function	function	NOUN
cana-578	5	36	on	on	ADP
cana-578	5	37	the	the	DET
cana-578	5	38	product	product	NOUN
cana-578	5	39	of	of	ADP
cana-578	5	40	compact	compact	ADJ
cana-578	5	41	metric	metric	ADJ
cana-578	5	42	spaces	space	NOUN
cana-578	5	43	to	to	ADP
cana-578	5	44	itself	itself	PRON
cana-578	5	45	and	and	CCONJ
cana-578	5	46	back	back	ADV
cana-578	5	47	to	to	ADP
cana-578	5	48	the	the	DET
cana-578	5	49	same	same	ADJ
cana-578	5	50	space	space	NOUN
cana-578	5	51	is	be	AUX
cana-578	5	52	known	know	VERB
cana-578	5	53	as	as	ADP
cana-578	5	54	a	a	DET
cana-578	5	55	genera	genera	NOUN
cana-578	5	56	system	system	NOUN
cana-578	5	57	.	.	PUNCT
cana-578	6	1	where	where	SCONJ
cana-578	6	2	each	each	DET
cana-578	6	3	element	element	NOUN
cana-578	6	4	's	's	PART
cana-578	6	5	orbit	orbit	NOUN
cana-578	6	6	is	be	AUX
cana-578	6	7	an	an	DET
cana-578	6	8	infinite	infinite	ADJ
cana-578	6	9	sequence	sequence	NOUN
cana-578	6	10	and	and	CCONJ
cana-578	6	11	where	where	SCONJ
cana-578	6	12	the	the	DET
cana-578	6	13	first	first	ADJ
cana-578	6	14	two	two	NUM
cana-578	6	15	elements	element	NOUN
cana-578	6	16	are	be	AUX
cana-578	6	17	the	the	DET
cana-578	6	18	same	same	ADJ
cana-578	6	19	as	as	SCONJ
cana-578	6	20	given	give	VERB
cana-578	6	21	and	and	CCONJ
cana-578	6	22	next	next	ADJ
cana-578	6	23	to	to	ADP
cana-578	6	24	it	it	PRON
cana-578	6	25	depends	depend	VERB
cana-578	6	26	on	on	ADP
cana-578	6	27	the	the	DET
cana-578	6	28	two	two	NUM
cana-578	6	29	elements	element	NOUN
cana-578	6	30	prior	prior	ADV
cana-578	6	31	to	to	ADP
cana-578	6	32	it	it	PRON
cana-578	6	33	form	form	VERB
cana-578	6	34	stronger	strong	ADJ
cana-578	6	35	conditions	condition	NOUN
cana-578	6	36	for	for	ADP
cana-578	6	37	the	the	DET
cana-578	6	38	orbit	orbit	NOUN
cana-578	6	39	.	.	PUNCT
cana-578	7	1	we	we	PRON
cana-578	7	2	could	could	AUX
cana-578	7	3	define	define	VERB
cana-578	7	4	an	an	DET
cana-578	7	5	m	m	ADJ
cana-578	7	6	-	-	PUNCT
cana-578	7	7	step	step	NOUN
cana-578	7	8	dynamical	dynamical	ADJ
cana-578	7	9	system	system	NOUN
cana-578	7	10	by	by	ADP
cana-578	7	11	extending	extend	VERB
cana-578	7	12	the	the	DET
cana-578	7	13	definition	definition	NOUN
cana-578	7	14	of	of	ADP
cana-578	7	15	a	a	DET
cana-578	7	16	compact	compact	ADJ
cana-578	7	17	metric	metric	ADJ
cana-578	7	18	space	space	NOUN
cana-578	7	19	to	to	ADP
cana-578	7	20	its	its	PRON
cana-578	7	21	m	m	NOUN
cana-578	7	22	-	-	PUNCT
cana-578	7	23	times	time	NOUN
cana-578	7	24	product	product	NOUN
cana-578	7	25	.	.	PUNCT
cana-578	8	1	because	because	SCONJ
cana-578	8	2	the	the	DET
cana-578	8	3	system	system	NOUN
cana-578	8	4	's	's	PART
cana-578	8	5	current	current	ADJ
cana-578	8	6	state	state	NOUN
cana-578	8	7	frequently	frequently	ADV
cana-578	8	8	depends	depend	VERB
cana-578	8	9	directly	directly	ADV
cana-578	8	10	on	on	ADP
cana-578	8	11	the	the	DET
cana-578	8	12	conditions	condition	NOUN
cana-578	8	13	of	of	ADP
cana-578	8	14	previous	previous	ADJ
cana-578	8	15	terms	term	NOUN
cana-578	8	16	,	,	PUNCT
cana-578	8	17	this	this	DET
cana-578	8	18	system	system	NOUN
cana-578	8	19	appears	appear	VERB
cana-578	8	20	more	more	ADV
cana-578	8	21	realistic	realistic	ADJ
cana-578	8	22	.	.	PUNCT
cana-578	9	1	the	the	DET
cana-578	9	2	basic	basic	ADJ
cana-578	9	3	theorems	theorem	NOUN
cana-578	9	4	regarding	regard	VERB
cana-578	9	5	periodic	periodic	ADJ
cana-578	9	6	points	point	NOUN
cana-578	9	7	and	and	CCONJ
cana-578	9	8	their	their	PRON
cana-578	9	9	related	relate	VERB
cana-578	9	10	points	point	NOUN
cana-578	9	11	,	,	PUNCT
cana-578	9	12	such	such	ADJ
cana-578	9	13	as	as	ADP
cana-578	9	14	fixed	fix	VERB
cana-578	9	15	point	point	NOUN
cana-578	9	16	,	,	PUNCT
cana-578	9	17	limit	limit	NOUN
cana-578	9	18	points	point	NOUN
cana-578	9	19	,	,	PUNCT
cana-578	9	20	recurrent	recurrent	ADJ
cana-578	9	21	points	point	NOUN
cana-578	9	22	,	,	PUNCT
cana-578	9	23	will	will	AUX
cana-578	9	24	be	be	AUX
cana-578	9	25	proved	prove	VERB
cana-578	9	26	in	in	ADP
cana-578	9	27	this	this	DET
cana-578	9	28	paper	paper	NOUN
cana-578	9	29	.	.	PUNCT
cana-578	10	1	we	we	PRON
cana-578	10	2	also	also	ADV
cana-578	10	3	define	define	VERB
cana-578	10	4	the	the	DET
cana-578	10	5	topological	topological	ADJ
cana-578	10	6	transitivity	transitivity	NOUN
cana-578	10	7	and	and	CCONJ
cana-578	10	8	its	its	PRON
cana-578	10	9	properties	property	NOUN
cana-578	10	10	.	.	PUNCT
cana-578	11	1	in	in	ADP
cana-578	11	2	the	the	DET
cana-578	11	3	end	end	NOUN
cana-578	11	4	we	we	PRON
cana-578	11	5	find	find	VERB
cana-578	11	6	periodic	periodic	ADJ
cana-578	11	7	points	point	NOUN
cana-578	11	8	with	with	ADP
cana-578	11	9	periods	period	NOUN
cana-578	11	10	one	one	NUM
cana-578	11	11	and	and	CCONJ
cana-578	11	12	two	two	NUM
cana-578	11	13	for	for	ADP
cana-578	11	14	affine	affine	NOUN
cana-578	11	15	maps	map	NOUN
cana-578	11	16	and	and	CCONJ
cana-578	11	17	periodicity	periodicity	NOUN
cana-578	11	18	of	of	ADP
cana-578	11	19	tent	tent	NOUN
cana-578	11	20	map	map	NOUN
cana-578	11	21	in	in	ADP
cana-578	11	22	dynamical	dynamical	ADJ
cana-578	11	23	system	system	NOUN
cana-578	11	24	and	and	CCONJ
cana-578	11	25	generalised	generalise	VERB
cana-578	11	26	dynamical	dynamical	ADJ
cana-578	11	27	system	system	NOUN
cana-578	11	28	.	.	PUNCT
cana-578	12	1	keywords	keyword	NOUN
cana-578	12	2	:	:	PUNCT
cana-578	12	3	generalized	generalize	VERB
cana-578	12	4	dynamical	dynamical	ADJ
cana-578	12	5	systems	system	NOUN
cana-578	12	6	,	,	PUNCT
cana-578	12	7	periods	period	NOUN
cana-578	12	8	and	and	CCONJ
cana-578	12	9	periodic	periodic	ADJ
cana-578	12	10	points	point	NOUN
cana-578	12	11	.	.	PUNCT
cana-578	13	1	1	1	X
cana-578	13	2	.	.	X
cana-578	13	3	introduction	introduction	NOUN
cana-578	13	4	topological	topological	ADJ
cana-578	13	5	dynamics	dynamic	NOUN
cana-578	13	6	is	be	AUX
cana-578	13	7	an	an	DET
cana-578	13	8	intriguing	intriguing	ADJ
cana-578	13	9	field	field	NOUN
cana-578	13	10	of	of	ADP
cana-578	13	11	mathematics	mathematic	NOUN
cana-578	13	12	.	.	PUNCT
cana-578	14	1	in	in	ADP
cana-578	14	2	that	that	DET
cana-578	14	3	periodicity	periodicity	NOUN
cana-578	14	4	is	be	AUX
cana-578	14	5	interesting	interesting	ADJ
cana-578	14	6	as	as	SCONJ
cana-578	14	7	it	it	PRON
cana-578	14	8	is	be	AUX
cana-578	14	9	related	relate	VERB
cana-578	14	10	to	to	ADP
cana-578	14	11	real	real	ADJ
cana-578	14	12	life	life	NOUN
cana-578	14	13	situations	situation	NOUN
cana-578	14	14	like	like	ADP
cana-578	14	15	the	the	DET
cana-578	14	16	path	path	NOUN
cana-578	14	17	of	of	ADP
cana-578	14	18	orbit	orbit	NOUN
cana-578	14	19	of	of	ADP
cana-578	14	20	planets	planet	NOUN
cana-578	14	21	etc	etc	X
cana-578	14	22	.	.	X
cana-578	15	1	a	a	DET
cana-578	15	2	number	number	NOUN
cana-578	15	3	of	of	ADP
cana-578	15	4	mathematicians	mathematician	NOUN
cana-578	15	5	have	have	AUX
cana-578	15	6	explained	explain	VERB
cana-578	15	7	the	the	DET
cana-578	15	8	periodic	periodic	ADJ
cana-578	15	9	behaviour	behaviour	NOUN
cana-578	15	10	on	on	ADP
cana-578	15	11	various	various	ADJ
cana-578	15	12	dynamical	dynamical	ADJ
cana-578	15	13	systems	system	NOUN
cana-578	15	14	,	,	PUNCT
cana-578	15	15	such	such	ADJ
cana-578	15	16	as	as	ADP
cana-578	15	17	linear	linear	PROPN
cana-578	15	18	operators[1	operators[1	NOUN
cana-578	15	19	]	]	PUNCT
cana-578	15	20	and	and	CCONJ
cana-578	15	21	operators	operator	NOUN
cana-578	15	22	on	on	ADP
cana-578	15	23	hilbert	hilbert	PROPN
cana-578	15	24	spaces[2	spaces[2	PROPN
cana-578	15	25	]	]	PUNCT
cana-578	15	26	and	and	CCONJ
cana-578	15	27	periods	period	NOUN
cana-578	15	28	and	and	CCONJ
cana-578	15	29	periodic	periodic	ADJ
cana-578	15	30	points	point	NOUN
cana-578	15	31	on	on	ADP
cana-578	15	32	linear	linear	ADJ
cana-578	15	33	cellular	cellular	ADJ
cana-578	15	34	automata	automata	NOUN
cana-578	15	35	in[3	in[3	PROPN
cana-578	15	36	]	]	PUNCT
cana-578	15	37	.	.	PUNCT
cana-578	16	1	the	the	DET
cana-578	16	2	generalized	generalized	ADJ
cana-578	16	3	systems	system	NOUN
cana-578	16	4	that	that	PRON
cana-578	16	5	define	define	VERB
cana-578	16	6	x×x	x×x	PUNCT
cana-578	16	7	→x	→x	PUNCT
cana-578	16	8	unlike	unlike	ADP
cana-578	16	9	x→x	x→x	PROPN
cana-578	16	10	as	as	SCONJ
cana-578	16	11	described	describe	VERB
cana-578	16	12	in	in	ADP
cana-578	16	13	dynamical	dynamical	ADJ
cana-578	16	14	systems	system	NOUN
cana-578	16	15	.	.	PUNCT
cana-578	17	1	this	this	DET
cana-578	17	2	new	new	ADJ
cana-578	17	3	concept	concept	NOUN
cana-578	17	4	was	be	AUX
cana-578	17	5	defined[4	defined[4	NOUN
cana-578	17	6	]	]	PUNCT
cana-578	17	7	in	in	ADP
cana-578	17	8	the	the	DET
cana-578	17	9	year	year	NOUN
cana-578	17	10	2008	2008	NUM
cana-578	17	11	.	.	PUNCT
cana-578	18	1	in	in	ADP
cana-578	18	2	this	this	DET
cana-578	18	3	instance	instance	NOUN
cana-578	18	4	,	,	PUNCT
cana-578	18	5	the	the	DET
cana-578	18	6	author	author	NOUN
cana-578	18	7	views	view	VERB
cana-578	18	8	x	x	PUNCT
cana-578	18	9	as	as	ADP
cana-578	18	10	a	a	DET
cana-578	18	11	complete	complete	ADJ
cana-578	18	12	metric	metric	ADJ
cana-578	18	13	space	space	NOUN
cana-578	18	14	.	.	PUNCT
cana-578	19	1	dumitru[5	dumitru[5	X
cana-578	19	2	]	]	X
cana-578	19	3	defined	define	VERB
cana-578	19	4	the	the	DET
cana-578	19	5	topological	topological	ADJ
cana-578	19	6	version	version	NOUN
cana-578	19	7	of	of	ADP
cana-578	19	8	generalized	generalized	ADJ
cana-578	19	9	iterated	iterate	VERB
cana-578	19	10	functions	function	NOUN
cana-578	19	11	based	base	VERB
cana-578	19	12	on	on	ADP
cana-578	19	13	this	this	DET
cana-578	19	14	novel	novel	ADJ
cana-578	19	15	idea	idea	NOUN
cana-578	19	16	.	.	PUNCT
cana-578	20	1	by	by	ADP
cana-578	20	2	creating	create	VERB
cana-578	20	3	new	new	ADJ
cana-578	20	4	concepts	concept	NOUN
cana-578	20	5	that	that	PRON
cana-578	20	6	work	work	VERB
cana-578	20	7	for	for	ADP
cana-578	20	8	generalized	generalized	ADJ
cana-578	20	9	systems[6	systems[6	PROPN
cana-578	20	10	]	]	PUNCT
cana-578	20	11	explored	explore	VERB
cana-578	20	12	chaos	chaos	NOUN
cana-578	20	13	and	and	CCONJ
cana-578	20	14	shadowing	shadow	VERB
cana-578	20	15	properties	property	NOUN
cana-578	20	16	in	in	ADP
cana-578	20	17	generalized	generalized	ADJ
cana-578	20	18	dynamical	dynamical	ADJ
cana-578	20	19	systems	system	NOUN
cana-578	20	20	in	in	ADP
cana-578	20	21	2023[7	2023[7	NUM
cana-578	20	22	]	]	PUNCT
cana-578	20	23	defined	define	VERB
cana-578	20	24	the	the	DET
cana-578	20	25	generalized	generalize	VERB
cana-578	20	26	function	function	NOUN
cana-578	20	27	systems	system	NOUN
cana-578	20	28	on	on	ADP
cana-578	20	29	metric	metric	ADJ
cana-578	20	30	spaces	space	NOUN
cana-578	20	31	.	.	PUNCT
cana-578	21	1	the	the	DET
cana-578	21	2	basic	basic	ADJ
cana-578	21	3	ideas	idea	NOUN
cana-578	21	4	of	of	ADP
cana-578	21	5	periodic	periodic	ADJ
cana-578	21	6	points	point	NOUN
cana-578	21	7	,	,	PUNCT
cana-578	21	8	such	such	ADJ
cana-578	21	9	as	as	ADP
cana-578	21	10	fixed	fix	VERB
cana-578	21	11	points	point	NOUN
cana-578	21	12	,	,	PUNCT
cana-578	21	13	periodic	periodic	ADJ
cana-578	21	14	points	point	NOUN
cana-578	21	15	that	that	PRON
cana-578	21	16	repeat	repeat	VERB
cana-578	21	17	,	,	PUNCT
cana-578	21	18	non	non	ADJ
cana-578	21	19	-	-	ADJ
cana-578	21	20	wandering	wandering	ADJ
cana-578	21	21	points	point	NOUN
cana-578	21	22	,	,	PUNCT
cana-578	21	23	and	and	CCONJ
cana-578	21	24	the	the	DET
cana-578	21	25	relationships	relationship	NOUN
cana-578	21	26	between	between	ADP
cana-578	21	27	them	they	PRON
cana-578	21	28	,	,	PUNCT
cana-578	21	29	will	will	AUX
cana-578	21	30	be	be	AUX
cana-578	21	31	covered	cover	VERB
cana-578	21	32	in	in	ADP
cana-578	21	33	this	this	DET
cana-578	21	34	work	work	NOUN
cana-578	21	35	.	.	PUNCT
cana-578	22	1	the	the	DET
cana-578	22	2	powerful	powerful	ADJ
cana-578	22	3	character	character	NOUN
cana-578	22	4	of	of	ADP
cana-578	22	5	a	a	DET
cana-578	22	6	point	point	NOUN
cana-578	22	7	and	and	CCONJ
cana-578	22	8	the	the	DET
cana-578	22	9	fact	fact	NOUN
cana-578	22	10	that	that	SCONJ
cana-578	22	11	it	it	PRON
cana-578	22	12	depends	depend	VERB
cana-578	22	13	on	on	ADP
cana-578	22	14	the	the	DET
cana-578	22	15	two	two	NUM
cana-578	22	16	or	or	CCONJ
cana-578	22	17	m	m	NOUN
cana-578	22	18	elements	element	NOUN
cana-578	22	19	(	(	PUNCT
cana-578	22	20	as	as	ADP
cana-578	22	21	in	in	ADP
cana-578	22	22	m	m	NOUN
cana-578	22	23	step	step	NOUN
cana-578	22	24	dynamical	dynamical	ADJ
cana-578	22	25	system	system	NOUN
cana-578	22	26	)	)	PUNCT
cana-578	22	27	before	before	ADP
cana-578	22	28	it	it	PRON
cana-578	22	29	in	in	ADP
cana-578	22	30	that	that	DET
cana-578	22	31	element	element	NOUN
cana-578	22	32	's	's	PART
cana-578	22	33	orbit	orbit	NOUN
cana-578	22	34	present	present	VERB
cana-578	22	35	the	the	DET
cana-578	22	36	biggest	big	ADJ
cana-578	22	37	challenge	challenge	NOUN
cana-578	22	38	in	in	ADP
cana-578	22	39	solving	solve	VERB
cana-578	22	40	this	this	DET
cana-578	22	41	system	system	NOUN
cana-578	22	42	.	.	PUNCT
cana-578	23	1	most	most	ADJ
cana-578	23	2	of	of	ADP
cana-578	23	3	the	the	DET
cana-578	23	4	statements	statement	NOUN
cana-578	23	5	that	that	PRON
cana-578	23	6	are	be	AUX
cana-578	23	7	true	true	ADJ
cana-578	23	8	for	for	ADP
cana-578	23	9	dynamical	dynamical	ADJ
cana-578	23	10	systems	system	NOUN
cana-578	23	11	are	be	AUX
cana-578	23	12	not	not	PART
cana-578	23	13	valid	valid	ADJ
cana-578	23	14	in	in	ADP
cana-578	23	15	the	the	DET
cana-578	23	16	present	present	ADJ
cana-578	23	17	system	system	NOUN
cana-578	23	18	mailto:vajhasrinu@gmail.com	mailto:vajhasrinu@gmail.com	NOUN
cana-578	23	19	communications	communication	NOUN
cana-578	23	20	on	on	ADP
cana-578	23	21	applied	apply	VERB
cana-578	23	22	nonlinear	nonlinear	ADJ
cana-578	23	23	analysis	analysis	NOUN
cana-578	23	24	issn	issn	NOUN
cana-578	23	25	:	:	PUNCT
cana-578	23	26	1074	1074	NUM
cana-578	23	27	-	-	PUNCT
cana-578	23	28	133x	133x	NUM
cana-578	23	29	vol	vol	NOUN
cana-578	23	30	31	31	NUM
cana-578	23	31	no	no	NOUN
cana-578	23	32	.	.	PUNCT
cana-578	24	1	1s	1s	NUM
cana-578	24	2	(	(	PUNCT
cana-578	24	3	2024	2024	NUM
cana-578	24	4	)	)	PUNCT
cana-578	24	5	188	188	NUM
cana-578	24	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-578	24	7	like	like	ADP
cana-578	24	8	the	the	DET
cana-578	24	9	fundamental	fundamental	ADJ
cana-578	24	10	elements	element	NOUN
cana-578	24	11	in	in	ADP
cana-578	24	12	a	a	DET
cana-578	24	13	periodic	periodic	ADJ
cana-578	24	14	orbit	orbit	NOUN
cana-578	24	15	are	be	AUX
cana-578	24	16	periodic	periodic	ADJ
cana-578	24	17	in	in	ADP
cana-578	24	18	dynamical	dynamical	ADJ
cana-578	24	19	systems	system	NOUN
cana-578	24	20	but	but	CCONJ
cana-578	24	21	not	not	PART
cana-578	24	22	true	true	ADJ
cana-578	24	23	in	in	ADP
cana-578	24	24	our	our	PRON
cana-578	24	25	case	case	NOUN
cana-578	24	26	.	.	PUNCT
cana-578	25	1	2	2	X
cana-578	25	2	.	.	X
cana-578	25	3	preliminaries	preliminary	NOUN
cana-578	25	4	in	in	ADP
cana-578	25	5	this	this	DET
cana-578	25	6	paper	paper	NOUN
cana-578	25	7	we	we	PRON
cana-578	25	8	refer	refer	VERB
cana-578	25	9	(	(	PUNCT
cana-578	25	10	𝑋	𝑋	NOUN
cana-578	25	11	,	,	PUNCT
cana-578	25	12	𝑑	𝑑	NOUN
cana-578	25	13	)	)	PUNCT
cana-578	25	14	as	as	ADP
cana-578	25	15	compact	compact	ADJ
cana-578	25	16	metric	metric	ADJ
cana-578	25	17	space	space	NOUN
cana-578	25	18	throughout	throughout	NOUN
cana-578	25	19	.	.	PUNCT
cana-578	26	1	the	the	DET
cana-578	26	2	continuous	continuous	ADJ
cana-578	26	3	map	map	NOUN
cana-578	26	4	𝑓	𝑓	NOUN
cana-578	26	5	:	:	PUNCT
cana-578	26	6	𝑋	𝑋	PROPN
cana-578	26	7	×	×	NOUN
cana-578	26	8	𝑋	𝑋	PROPN
cana-578	26	9	→	→	SYM
cana-578	26	10	𝑋	𝑋	PROPN
cana-578	26	11	is	be	AUX
cana-578	26	12	generalised	generalised	ADJ
cana-578	26	13	system	system	NOUN
cana-578	26	14	.	.	PUNCT
cana-578	27	1	in	in	ADP
cana-578	27	2	this	this	DET
cana-578	27	3	paper	paper	NOUN
cana-578	27	4	we	we	PRON
cana-578	27	5	analyse	analyse	VERB
cana-578	27	6	the	the	DET
cana-578	27	7	preliminary	preliminary	ADJ
cana-578	27	8	data	datum	NOUN
cana-578	27	9	and	and	CCONJ
cana-578	27	10	extending	extend	VERB
cana-578	27	11	domain	domain	NOUN
cana-578	27	12	to	to	ADP
cana-578	27	13	,	,	PUNCT
cana-578	27	14	𝑓	𝑓	X
cana-578	27	15	:	:	PUNCT
cana-578	27	16	𝑋	𝑋	NOUN
cana-578	27	17	×	×	NOUN
cana-578	27	18	𝑋	𝑋	PROPN
cana-578	27	19	×	×	NOUN
cana-578	27	20	…	…	PUNCT
cana-578	27	21	.	.	PUNCT
cana-578	28	1	𝑋(𝑚	𝑋(𝑚	PRON
cana-578	28	2	𝑡𝑖𝑚𝑒𝑠	𝑡𝑖𝑚𝑒𝑠	NOUN
cana-578	28	3	)	)	PUNCT
cana-578	28	4	→	→	PUNCT
cana-578	28	5	𝑋	𝑋	NOUN
cana-578	28	6	𝑜𝑟	𝑜𝑟	VERB
cana-578	28	7	𝑓	𝑓	PRON
cana-578	28	8	:	:	PUNCT
cana-578	28	9	𝑋𝑚	𝑋𝑚	PROPN
cana-578	28	10	⟶	⟶	NOUN
cana-578	28	11	𝑋	𝑋	NOUN
cana-578	28	12	(	(	PUNCT
cana-578	28	13	2.1	2.1	NUM
cana-578	28	14	)	)	PUNCT
cana-578	28	15	defining	define	VERB
cana-578	28	16	as	as	ADP
cana-578	28	17	𝑚	𝑚	ADP
cana-578	28	18	step	step	NOUN
cana-578	28	19	generalized	generalize	VERB
cana-578	28	20	system	system	NOUN
cana-578	28	21	considering	consider	VERB
cana-578	28	22	the	the	DET
cana-578	28	23	same	same	ADJ
cana-578	28	24	preliminaries	preliminary	NOUN
cana-578	28	25	.	.	PUNCT
cana-578	29	1	this	this	DET
cana-578	29	2	assumption	assumption	NOUN
cana-578	29	3	makes	make	VERB
cana-578	29	4	us	we	PRON
cana-578	29	5	calling	call	VERB
cana-578	29	6	the	the	DET
cana-578	29	7	regular	regular	ADJ
cana-578	29	8	dynamical	dynamical	ADJ
cana-578	29	9	system	system	NOUN
cana-578	29	10	as	as	ADP
cana-578	29	11	1	1	NUM
cana-578	29	12	-	-	PUNCT
cana-578	29	13	step	step	NOUN
cana-578	29	14	dynamical	dynamical	ADJ
cana-578	29	15	system	system	NOUN
cana-578	29	16	and	and	CCONJ
cana-578	29	17	generalized	generalized	ADJ
cana-578	29	18	system	system	NOUN
cana-578	29	19	as	as	ADP
cana-578	29	20	2	2	NUM
cana-578	29	21	-	-	PUNCT
cana-578	29	22	step	step	NOUN
cana-578	29	23	dynamical	dynamical	ADJ
cana-578	29	24	system	system	NOUN
cana-578	29	25	.	.	PUNCT
cana-578	30	1	the	the	DET
cana-578	30	2	orbit	orbit	NOUN
cana-578	30	3	of	of	ADP
cana-578	30	4	any	any	DET
cana-578	30	5	𝑥	𝑥	PRON
cana-578	30	6	∈	∈	NOUN
cana-578	30	7	𝑋	𝑋	NOUN
cana-578	30	8	is	be	AUX
cana-578	30	9	the	the	DET
cana-578	30	10	infinite	infinite	ADJ
cana-578	30	11	sequence	sequence	NOUN
cana-578	30	12	𝑂(𝑥	𝑂(𝑥	NUM
cana-578	30	13	)	)	PUNCT
cana-578	30	14	=	=	PRON
cana-578	30	15	{	{	PUNCT
cana-578	30	16	𝑥𝑛}0	𝑥𝑛}0	NOUN
cana-578	30	17	∞	∞	NUM
cana-578	30	18	where	where	SCONJ
cana-578	30	19	𝑓	𝑓	X
cana-578	30	20	:	:	PUNCT
cana-578	30	21	𝑋𝑚	𝑋𝑚	PROPN
cana-578	30	22	⟶	⟶	NOUN
cana-578	30	23	𝑋	𝑋	NOUN
cana-578	30	24	is	be	AUX
cana-578	30	25	a	a	DET
cana-578	30	26	continuous	continuous	ADJ
cana-578	30	27	map	map	NOUN
cana-578	30	28	and	and	CCONJ
cana-578	30	29	,	,	PUNCT
cana-578	30	30	𝑥0	𝑥0	PROPN
cana-578	30	31	=	=	SYM
cana-578	30	32	𝑥1	𝑥1	NOUN
cana-578	30	33	=	=	SYM
cana-578	30	34	⋯	⋯	PROPN
cana-578	30	35	=	=	SYM
cana-578	30	36	𝑥𝑚	𝑥𝑚	NOUN
cana-578	30	37	=	=	SYM
cana-578	30	38	𝑥	𝑥	PROPN
cana-578	30	39	and	and	CCONJ
cana-578	30	40	𝑓(𝑥𝑛−𝑚	𝑓(𝑥𝑛−𝑚	ADJ
cana-578	30	41	,	,	PUNCT
cana-578	30	42	𝑥𝑛−𝑚+1	𝑥𝑛−𝑚+1	NOUN
cana-578	30	43	,	,	PUNCT
cana-578	30	44	…	…	PUNCT
cana-578	30	45	.	.	PUNCT
cana-578	31	1	,	,	PUNCT
cana-578	31	2	𝑥𝑛−1	𝑥𝑛−1	NOUN
cana-578	31	3	)	)	PUNCT
cana-578	31	4	=	=	SYM
cana-578	31	5	𝑥𝑛	𝑥𝑛	PROPN
cana-578	31	6	,	,	PUNCT
cana-578	31	7	𝑛	𝑛	PRON
cana-578	31	8	≥	≥	NOUN
cana-578	31	9	𝑚	𝑚	X
cana-578	31	10	(	(	PUNCT
cana-578	31	11	2.2	2.2	NUM
cana-578	31	12	)	)	PUNCT
cana-578	31	13	for	for	ADP
cana-578	31	14	𝑚	𝑚	NOUN
cana-578	31	15	=	=	SYM
cana-578	31	16	2	2	NUM
cana-578	31	17	it	it	PRON
cana-578	31	18	becomes	become	VERB
cana-578	31	19	𝑥0	𝑥0	NOUN
cana-578	31	20	=	=	SYM
cana-578	31	21	𝑥1	𝑥1	NOUN
cana-578	31	22	=	=	SYM
cana-578	31	23	𝑥	𝑥	PROPN
cana-578	31	24	and	and	CCONJ
cana-578	31	25	𝑥𝑛	𝑥𝑛	PROPN
cana-578	31	26	=	=	SYM
cana-578	31	27	𝑓(𝑥𝑛−1	𝑓(𝑥𝑛−1	PROPN
cana-578	31	28	,	,	PUNCT
cana-578	31	29	𝑥𝑛−2	𝑥𝑛−2	NOUN
cana-578	31	30	)	)	PUNCT
cana-578	31	31	,	,	PUNCT
cana-578	31	32	𝑛	𝑛	DET
cana-578	31	33	≥	≥	NOUN
cana-578	31	34	2	2	NUM
cana-578	31	35	here	here	ADV
cana-578	31	36	𝑥𝑛	𝑥𝑛	AUX
cana-578	31	37	denotes	denote	VERB
cana-578	31	38	𝑛𝑡ℎ	𝑛𝑡ℎ	PRON
cana-578	31	39	term	term	NOUN
cana-578	31	40	in	in	ADP
cana-578	31	41	the	the	DET
cana-578	31	42	orbit	orbit	NOUN
cana-578	31	43	𝑂(𝑥	𝑂(𝑥	NUM
cana-578	31	44	)	)	PUNCT
cana-578	32	1	=	=	PRON
cana-578	32	2	{	{	PUNCT
cana-578	32	3	𝑥𝑛}0	𝑥𝑛}0	NOUN
cana-578	32	4	∞.	∞.	PROPN
cana-578	32	5	for	for	ADP
cana-578	32	6	the	the	DET
cana-578	32	7	convenience	convenience	NOUN
cana-578	32	8	we	we	PRON
cana-578	32	9	consider	consider	VERB
cana-578	32	10	𝑚	𝑚	NOUN
cana-578	32	11	=	=	SYM
cana-578	32	12	2	2	NUM
cana-578	32	13	and	and	CCONJ
cana-578	32	14	define	define	VERB
cana-578	32	15	the	the	DET
cana-578	32	16	preliminaries	preliminary	NOUN
cana-578	32	17	.	.	PUNCT
cana-578	33	1	we	we	PRON
cana-578	33	2	say	say	VERB
cana-578	33	3	that	that	SCONJ
cana-578	33	4	𝑥	𝑥	PROPN
cana-578	33	5	∈	∈	PROPN
cana-578	33	6	𝑋	𝑋	NOUN
cana-578	33	7	is	be	AUX
cana-578	33	8	fixed	fix	VERB
cana-578	33	9	if	if	SCONJ
cana-578	33	10	𝑓(𝑥	𝑓(𝑥	NOUN
cana-578	33	11	,	,	PUNCT
cana-578	33	12	𝑥	𝑥	NOUN
cana-578	33	13	)	)	PUNCT
cana-578	33	14	=	=	VERB
cana-578	34	1	𝑥.	𝑥.	NOUN
cana-578	34	2	(	(	PUNCT
cana-578	34	3	2.3	2.3	NUM
cana-578	34	4	)	)	PUNCT
cana-578	34	5	we	we	PRON
cana-578	34	6	say	say	VERB
cana-578	34	7	that	that	SCONJ
cana-578	34	8	𝑥	𝑥	PROPN
cana-578	34	9	∈	∈	PROPN
cana-578	34	10	𝑋	𝑋	NOUN
cana-578	34	11	is	be	AUX
cana-578	34	12	periodic	periodic	ADJ
cana-578	34	13	of	of	ADP
cana-578	34	14	periodic	periodic	NOUN
cana-578	34	15	of	of	ADP
cana-578	34	16	period	period	NOUN
cana-578	35	1	𝑛	𝑛	PRON
cana-578	35	2	if	if	SCONJ
cana-578	35	3	𝑥𝑘𝑛+𝑖	𝑥𝑘𝑛+𝑖	NOUN
cana-578	35	4	=	=	SYM
cana-578	35	5	𝑥𝑖	𝑥𝑖	PROPN
cana-578	35	6	for	for	ADP
cana-578	35	7	all	all	DET
cana-578	35	8	𝑘	𝑘	DET
cana-578	35	9	∈	∈	PROPN
cana-578	35	10	ℕ	ℕ	PROPN
cana-578	35	11	and	and	CCONJ
cana-578	35	12	0	0	NUM
cana-578	35	13	≤	≤	NUM
cana-578	35	14	𝑖	𝑖	X
cana-578	35	15	<	<	X
cana-578	35	16	𝑛.	𝑛.	NOUN
cana-578	35	17	we	we	PRON
cana-578	35	18	may	may	AUX
cana-578	35	19	call	call	VERB
cana-578	35	20	periodic	periodic	ADJ
cana-578	35	21	point	point	NOUN
cana-578	35	22	of	of	ADP
cana-578	35	23	period	period	NOUN
cana-578	35	24	one	one	NUM
cana-578	35	25	is	be	AUX
cana-578	35	26	fixed	fix	VERB
cana-578	35	27	point	point	NOUN
cana-578	35	28	.	.	PUNCT
cana-578	36	1	(	(	PUNCT
cana-578	36	2	2.4	2.4	NUM
cana-578	36	3	)	)	PUNCT
cana-578	36	4	we	we	PRON
cana-578	36	5	say	say	VERB
cana-578	36	6	that	that	SCONJ
cana-578	36	7	𝑥	𝑥	PROPN
cana-578	36	8	∈	∈	PROPN
cana-578	36	9	𝑋	𝑋	NOUN
cana-578	36	10	is	be	AUX
cana-578	36	11	strongly	strongly	ADV
cana-578	36	12	periodic	periodic	ADJ
cana-578	36	13	if	if	SCONJ
cana-578	36	14	every	every	DET
cana-578	36	15	element	element	NOUN
cana-578	36	16	of	of	ADP
cana-578	36	17	the	the	DET
cana-578	36	18	orbit	orbit	NOUN
cana-578	36	19	is	be	AUX
cana-578	36	20	periodic	periodic	ADJ
cana-578	36	21	.	.	PUNCT
cana-578	37	1	unlike	unlike	ADP
cana-578	37	2	in	in	ADP
cana-578	37	3	1step	1step	NUM
cana-578	37	4	dynamical	dynamical	ADJ
cana-578	37	5	systems	system	NOUN
cana-578	37	6	in	in	ADP
cana-578	37	7	general	general	ADJ
cana-578	37	8	systems	system	NOUN
cana-578	37	9	every	every	DET
cana-578	37	10	element	element	NOUN
cana-578	37	11	in	in	ADP
cana-578	37	12	periodic	periodic	ADJ
cana-578	37	13	orbit	orbit	NOUN
cana-578	37	14	need	need	AUX
cana-578	37	15	not	not	PART
cana-578	37	16	be	be	AUX
cana-578	37	17	periodic	periodic	ADJ
cana-578	37	18	.	.	PUNCT
cana-578	38	1	(	(	PUNCT
cana-578	38	2	2.5	2.5	NUM
cana-578	38	3	)	)	PUNCT
cana-578	38	4	the	the	DET
cana-578	38	5	recurrent	recurrent	ADJ
cana-578	38	6	point	point	NOUN
cana-578	38	7	is	be	AUX
cana-578	38	8	𝑥	𝑥	DET
cana-578	38	9	∈	∈	PROPN
cana-578	38	10	𝑋	𝑋	NOUN
cana-578	38	11	and	and	CCONJ
cana-578	38	12	𝑥	𝑥	PRON
cana-578	38	13	∈	∈	PROPN
cana-578	38	14	𝜔(𝑥	𝜔(𝑥	PROPN
cana-578	38	15	,	,	PUNCT
cana-578	38	16	𝑓	𝑓	X
cana-578	38	17	)	)	PUNCT
cana-578	38	18	that	that	PRON
cana-578	38	19	is	be	AUX
cana-578	38	20	for	for	ADP
cana-578	38	21	each	each	PRON
cana-578	38	22	𝜖	𝜖	X
cana-578	38	23	>	>	X
cana-578	38	24	0	0	PUNCT
cana-578	39	1	there	there	PRON
cana-578	39	2	is	be	VERB
cana-578	39	3	𝑛	𝑛	DET
cana-578	39	4	∈	∈	PROPN
cana-578	39	5	ℕ	ℕ	PROPN
cana-578	39	6	such	such	ADJ
cana-578	39	7	that	that	SCONJ
cana-578	39	8	𝑑(𝑥𝑛	𝑑(𝑥𝑛	PROPN
cana-578	39	9	,	,	PUNCT
cana-578	39	10	𝑥	𝑥	NOUN
cana-578	39	11	)	)	PUNCT
cana-578	39	12	<	<	X
cana-578	39	13	𝜖.	𝜖.	X
cana-578	39	14	(	(	PUNCT
cana-578	39	15	2.6	2.6	NUM
cana-578	39	16	)	)	PUNCT
cana-578	39	17	the	the	DET
cana-578	39	18	non	non	ADJ
cana-578	39	19	-	-	ADJ
cana-578	39	20	wandering	wandering	ADJ
cana-578	39	21	is	be	AUX
cana-578	39	22	𝑥	𝑥	DET
cana-578	39	23	∈	∈	PROPN
cana-578	39	24	𝑋	𝑋	NOUN
cana-578	39	25	and	and	CCONJ
cana-578	39	26	for	for	ADP
cana-578	39	27	each	each	PRON
cana-578	39	28	>	>	X
cana-578	39	29	0	0	PROPN
cana-578	39	30	,	,	PUNCT
cana-578	39	31	𝛿	𝛿	NOUN
cana-578	39	32	>	>	X
cana-578	39	33	0	0	PUNCT
cana-578	40	1	there	there	PRON
cana-578	40	2	is	be	VERB
cana-578	40	3	𝑛	𝑛	DET
cana-578	40	4	∈	∈	PROPN
cana-578	40	5	ℕ	ℕ	PROPN
cana-578	40	6	,	,	PUNCT
cana-578	40	7	𝑧	𝑧	PRON
cana-578	40	8	∈	∈	PROPN
cana-578	40	9	𝐵𝛿(𝑥	𝐵𝛿(𝑥	NOUN
cana-578	40	10	)	)	PUNCT
cana-578	40	11	implies	imply	VERB
cana-578	40	12	𝑑(𝑧𝑛	𝑑(𝑧𝑛	ADV
cana-578	40	13	,	,	PUNCT
cana-578	40	14	𝑥	𝑥	NOUN
cana-578	40	15	)	)	PUNCT
cana-578	40	16	<	<	X
cana-578	40	17	𝜖.	𝜖.	NOUN
cana-578	40	18	(	(	PUNCT
cana-578	40	19	2.7	2.7	NUM
cana-578	40	20	)	)	PUNCT
cana-578	40	21	we	we	PRON
cana-578	40	22	also	also	ADV
cana-578	40	23	define	define	VERB
cana-578	40	24	strong	strong	ADJ
cana-578	40	25	non	non	ADJ
cana-578	40	26	wandering	wander	VERB
cana-578	40	27	if	if	SCONJ
cana-578	40	28	𝑥	𝑥	PROPN
cana-578	40	29	is	be	AUX
cana-578	40	30	non	non	ADJ
cana-578	40	31	-	-	ADJ
cana-578	40	32	wandering	wandering	ADJ
cana-578	40	33	and	and	CCONJ
cana-578	40	34	every	every	DET
cana-578	40	35	element	element	NOUN
cana-578	40	36	of	of	ADP
cana-578	40	37	the	the	DET
cana-578	40	38	orbit	orbit	NOUN
cana-578	40	39	is	be	AUX
cana-578	40	40	also	also	ADV
cana-578	40	41	non	non	ADJ
cana-578	40	42	wandering	wander	VERB
cana-578	40	43	.	.	PUNCT
cana-578	41	1	(	(	PUNCT
cana-578	41	2	2.8	2.8	NUM
cana-578	41	3	)	)	PUNCT
cana-578	41	4	a	a	DET
cana-578	41	5	subset	subset	NOUN
cana-578	41	6	𝐵	𝐵	NOUN
cana-578	41	7	⊆	⊆	PROPN
cana-578	41	8	𝑋	𝑋	PROPN
cana-578	41	9	is	be	AUX
cana-578	41	10	said	say	VERB
cana-578	41	11	to	to	PART
cana-578	41	12	be	be	AUX
cana-578	41	13	invariant	invariant	ADJ
cana-578	41	14	if	if	SCONJ
cana-578	41	15	{	{	PUNCT
cana-578	41	16	𝑥𝑛}0	𝑥𝑛}0	NOUN
cana-578	41	17	∞	∞	NUM
cana-578	41	18	⊆	⊆	NUM
cana-578	41	19	𝐵	𝐵	NOUN
cana-578	41	20	for	for	ADP
cana-578	41	21	every	every	DET
cana-578	41	22	𝑥	𝑥	PROPN
cana-578	41	23	∈	∈	PROPN
cana-578	41	24	𝐵.	𝐵.	PROPN
cana-578	41	25	(	(	PUNCT
cana-578	41	26	2.9	2.9	NUM
cana-578	41	27	)	)	PUNCT
cana-578	41	28	we	we	PRON
cana-578	41	29	denote	denote	VERB
cana-578	41	30	the	the	DET
cana-578	41	31	following	following	ADJ
cana-578	41	32	notions	notion	NOUN
cana-578	41	33	for	for	ADP
cana-578	41	34	generalized	generalized	ADJ
cana-578	41	35	system	system	NOUN
cana-578	41	36	.	.	PUNCT
cana-578	42	1	𝑓𝑖𝑥(𝑓	𝑓𝑖𝑥(𝑓	NOUN
cana-578	42	2	)	)	PUNCT
cana-578	42	3	=	=	SYM
cana-578	42	4	fixed	fix	VERB
cana-578	42	5	points	point	NOUN
cana-578	42	6	in	in	ADP
cana-578	42	7	the	the	DET
cana-578	42	8	space	space	NOUN
cana-578	42	9	𝑋.	𝑋.	PROPN
cana-578	42	10	𝑃𝑒𝑟(𝑓)=periodic	𝑃𝑒𝑟(𝑓)=periodic	ADJ
cana-578	42	11	points	point	NOUN
cana-578	42	12	in	in	ADP
cana-578	42	13	𝑋.	𝑋.	PROPN
cana-578	42	14	ℛ(𝑓	ℛ(𝑓	PROPN
cana-578	42	15	)	)	PUNCT
cana-578	42	16	=	=	SYM
cana-578	42	17	recurrent	recurrent	ADJ
cana-578	42	18	points	point	NOUN
cana-578	42	19	in	in	ADP
cana-578	42	20	𝑋.	𝑋.	PROPN
cana-578	42	21	ω(𝑓	ω(𝑓	PROPN
cana-578	42	22	)	)	PUNCT
cana-578	43	1	=	=	SYM
cana-578	43	2	non	non	ADJ
cana-578	43	3	-	-	ADJ
cana-578	43	4	wandering	wandering	ADJ
cana-578	43	5	points	point	NOUN
cana-578	43	6	in	in	ADP
cana-578	43	7	𝑋.	𝑋.	PROPN
cana-578	43	8	communications	communication	NOUN
cana-578	43	9	on	on	ADP
cana-578	43	10	applied	apply	VERB
cana-578	43	11	nonlinear	nonlinear	ADJ
cana-578	43	12	analysis	analysis	NOUN
cana-578	43	13	issn	issn	NOUN
cana-578	43	14	:	:	PUNCT
cana-578	43	15	1074	1074	NUM
cana-578	43	16	-	-	PUNCT
cana-578	43	17	133x	133x	NUM
cana-578	43	18	vol	vol	NOUN
cana-578	43	19	31	31	NUM
cana-578	43	20	no	no	NOUN
cana-578	43	21	.	.	PUNCT
cana-578	44	1	1s	1s	NUM
cana-578	44	2	(	(	PUNCT
cana-578	44	3	2024	2024	NUM
cana-578	44	4	)	)	PUNCT
cana-578	44	5	189	189	NUM
cana-578	44	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-578	44	7	3	3	X
cana-578	44	8	.	.	NOUN
cana-578	44	9	results	result	NOUN
cana-578	44	10	since	since	SCONJ
cana-578	44	11	the	the	DET
cana-578	44	12	results	result	NOUN
cana-578	44	13	do	do	AUX
cana-578	44	14	not	not	PART
cana-578	44	15	hold	hold	VERB
cana-578	44	16	true	true	ADJ
cana-578	44	17	for	for	ADP
cana-578	44	18	generalized	generalized	ADJ
cana-578	44	19	systems	system	NOUN
cana-578	44	20	,	,	PUNCT
cana-578	44	21	we	we	PRON
cana-578	44	22	must	must	AUX
cana-578	44	23	demonstrate	demonstrate	VERB
cana-578	44	24	all	all	PRON
cana-578	44	25	of	of	ADP
cana-578	44	26	the	the	DET
cana-578	44	27	results	result	NOUN
cana-578	44	28	that	that	PRON
cana-578	44	29	hold	hold	VERB
cana-578	44	30	true	true	ADJ
cana-578	44	31	for	for	ADP
cana-578	44	32	1	1	NUM
cana-578	44	33	-	-	PUNCT
cana-578	44	34	step	step	NOUN
cana-578	44	35	dynamical	dynamical	ADJ
cana-578	44	36	systems	system	NOUN
cana-578	44	37	.	.	PUNCT
cana-578	45	1	theorem	theorem	VERB
cana-578	45	2	3.1	3.1	NUM
cana-578	45	3	.	.	PUNCT
cana-578	46	1	for	for	ADP
cana-578	46	2	any	any	DET
cana-578	46	3	generalized	generalized	ADJ
cana-578	46	4	system	system	NOUN
cana-578	46	5	defined	define	VERB
cana-578	46	6	in	in	ADP
cana-578	46	7	eq.(1	eq.(1	ADJ
cana-578	46	8	)	)	PUNCT
cana-578	46	9	𝑓𝑖𝑥(𝑓	𝑓𝑖𝑥(𝑓	PROPN
cana-578	46	10	)	)	PUNCT
cana-578	46	11	⊆	⊆	NUM
cana-578	46	12	𝑃𝑒𝑟(𝑓	𝑃𝑒𝑟(𝑓	NOUN
cana-578	46	13	)	)	PUNCT
cana-578	46	14	⊆	⊆	NUM
cana-578	46	15	ℛ(𝑓	ℛ(𝑓	PROPN
cana-578	46	16	)	)	PUNCT
cana-578	46	17	⊆	⊆	NUM
cana-578	46	18	ω(𝑓	ω(𝑓	NOUN
cana-578	46	19	)	)	PUNCT
cana-578	46	20	proof	proof	NOUN
cana-578	46	21	:	:	PUNCT
cana-578	47	1	1	1	X
cana-578	47	2	.	.	X
cana-578	47	3	the	the	DET
cana-578	47	4	first	first	ADJ
cana-578	47	5	two	two	NUM
cana-578	47	6	proofs	proof	NOUN
cana-578	47	7	are	be	AUX
cana-578	47	8	trivial	trivial	ADJ
cana-578	47	9	.	.	PUNCT
cana-578	48	1	for	for	ADP
cana-578	48	2	any	any	DET
cana-578	48	3	𝑥	𝑥	PRON
cana-578	48	4	∈	∈	PROPN
cana-578	48	5	𝑓𝑖𝑥(𝑓	𝑓𝑖𝑥(𝑓	NOUN
cana-578	48	6	)	)	PUNCT
cana-578	48	7	it	it	PRON
cana-578	48	8	is	be	AUX
cana-578	48	9	clear	clear	ADJ
cana-578	48	10	that	that	SCONJ
cana-578	48	11	fixed	fix	VERB
cana-578	48	12	point	point	NOUN
cana-578	48	13	are	be	AUX
cana-578	48	14	of	of	ADP
cana-578	48	15	period	period	NOUN
cana-578	48	16	one	one	NUM
cana-578	48	17	.	.	PUNCT
cana-578	49	1	2	2	NUM
cana-578	49	2	.	.	X
cana-578	49	3	for	for	ADP
cana-578	49	4	any	any	DET
cana-578	49	5	𝑥	𝑥	PRON
cana-578	49	6	∈	∈	PROPN
cana-578	49	7	𝑃𝑒𝑟(𝑓	𝑃𝑒𝑟(𝑓	NOUN
cana-578	49	8	)	)	PUNCT
cana-578	49	9	then	then	ADV
cana-578	49	10	for	for	ADP
cana-578	49	11	some	some	DET
cana-578	49	12	𝑛	𝑛	NOUN
cana-578	49	13	,	,	PUNCT
cana-578	49	14	𝑥𝑛	𝑥𝑛	PROPN
cana-578	49	15	=	=	NOUN
cana-578	49	16	𝑥.	𝑥.	VERB
cana-578	49	17	so	so	ADV
cana-578	49	18	for	for	ADP
cana-578	49	19	each	each	DET
cana-578	49	20	𝜖	𝜖	X
cana-578	49	21	>	>	X
cana-578	49	22	0	0	PUNCT
cana-578	49	23	for	for	SCONJ
cana-578	49	24	we	we	PRON
cana-578	49	25	get	get	VERB
cana-578	49	26	𝑑(𝑥𝑛	𝑑(𝑥𝑛	NOUN
cana-578	49	27	,	,	PUNCT
cana-578	49	28	𝑥	𝑥	NOUN
cana-578	49	29	)	)	PUNCT
cana-578	49	30	<	<	X
cana-578	49	31	𝜖	𝜖	PROPN
cana-578	50	1	that	that	PRON
cana-578	50	2	implies	imply	VERB
cana-578	50	3	𝑥	𝑥	X
cana-578	50	4	∈	∈	PROPN
cana-578	50	5	ℛ(𝑓	ℛ(𝑓	PROPN
cana-578	50	6	)	)	PUNCT
cana-578	50	7	.	.	PUNCT
cana-578	51	1	3	3	X
cana-578	51	2	.	.	X
cana-578	51	3	for	for	ADP
cana-578	51	4	any	any	DET
cana-578	51	5	𝑥	𝑥	PROPN
cana-578	51	6	∈	∈	PROPN
cana-578	51	7	ℛ(𝑓	ℛ(𝑓	PROPN
cana-578	51	8	)	)	PUNCT
cana-578	51	9	then	then	ADV
cana-578	51	10	from	from	ADP
cana-578	51	11	eq	eq	ADP
cana-578	51	12	.	.	PUNCT
cana-578	52	1	(	(	PUNCT
cana-578	52	2	1.6	1.6	NUM
cana-578	52	3	)	)	PUNCT
cana-578	52	4	,	,	PUNCT
cana-578	52	5	for	for	ADP
cana-578	52	6	any	any	DET
cana-578	52	7	𝜖	𝜖	PROPN
cana-578	52	8	>	>	X
cana-578	52	9	0	0	PUNCT
cana-578	52	10	there	there	PRON
cana-578	52	11	is	be	VERB
cana-578	52	12	𝑛	𝑛	DET
cana-578	52	13	∈	∈	PROPN
cana-578	52	14	ℕ	ℕ	PROPN
cana-578	52	15	such	such	ADJ
cana-578	52	16	that	that	SCONJ
cana-578	52	17	𝑑(𝑥𝑛	𝑑(𝑥𝑛	PROPN
cana-578	52	18	,	,	PUNCT
cana-578	52	19	𝑥	𝑥	NOUN
cana-578	52	20	)	)	PUNCT
cana-578	52	21	<	<	X
cana-578	52	22	𝜖.	𝜖.	NOUN
cana-578	52	23	let	let	VERB
cana-578	52	24	𝑧	𝑧	DET
cana-578	52	25	∈	∈	NOUN
cana-578	52	26	𝐵𝛿(𝑥	𝐵𝛿(𝑥	NOUN
cana-578	52	27	)	)	PUNCT
cana-578	52	28	for	for	ADP
cana-578	52	29	𝛿	𝛿	PROPN
cana-578	52	30	>	>	X
cana-578	52	31	0	0	NUM
cana-578	52	32	.	.	PUNCT
cana-578	53	1	then	then	ADV
cana-578	53	2	𝑑(𝑥	𝑑(𝑥	PROPN
cana-578	53	3	,	,	PUNCT
cana-578	53	4	𝑧	𝑧	NOUN
cana-578	53	5	)	)	PUNCT
cana-578	53	6	<	<	X
cana-578	53	7	𝛿	𝛿	X
cana-578	53	8	implies	imply	VERB
cana-578	53	9	𝑑(𝑥𝑛	𝑑(𝑥𝑛	PROPN
cana-578	53	10	,	,	PUNCT
cana-578	53	11	𝑧𝑛	𝑧𝑛	ADP
cana-578	53	12	)	)	PUNCT
cana-578	53	13	<	<	X
cana-578	53	14	𝜖	𝜖	PROPN
cana-578	53	15	as	as	ADP
cana-578	53	16	the	the	DET
cana-578	53	17	function	function	NOUN
cana-578	53	18	as	as	ADP
cana-578	53	19	the	the	DET
cana-578	53	20	function	function	NOUN
cana-578	53	21	𝑓	𝑓	PRON
cana-578	53	22	defined	define	VERB
cana-578	53	23	on	on	ADP
cana-578	53	24	the	the	DET
cana-578	53	25	compact	compact	ADJ
cana-578	53	26	metric	metric	ADJ
cana-578	53	27	space	space	NOUN
cana-578	53	28	.	.	PUNCT
cana-578	54	1	so	so	ADV
cana-578	54	2	,	,	PUNCT
cana-578	54	3	𝑑(𝑧𝑛	𝑑(𝑧𝑛	ADV
cana-578	54	4	,	,	PUNCT
cana-578	54	5	𝑥	𝑥	NOUN
cana-578	54	6	)	)	PUNCT
cana-578	54	7	≤	≤	NOUN
cana-578	54	8	𝑑(𝑧𝑛	𝑑(𝑧𝑛	ADV
cana-578	54	9	,	,	PUNCT
cana-578	54	10	𝑥𝑛	𝑥𝑛	NOUN
cana-578	54	11	)	)	PUNCT
cana-578	55	1	+	+	CCONJ
cana-578	55	2	𝑑(𝑥𝑛	𝑑(𝑥𝑛	PROPN
cana-578	55	3	,	,	PUNCT
cana-578	55	4	𝑥	𝑥	NOUN
cana-578	55	5	)	)	PUNCT
cana-578	55	6	<	<	X
cana-578	55	7	𝜖	𝜖	PROPN
cana-578	56	1	+	+	NUM
cana-578	56	2	𝜖	𝜖	X
cana-578	56	3	=	=	SYM
cana-578	56	4	2𝜖.	2𝜖.	NUM
cana-578	56	5	then	then	ADV
cana-578	56	6	𝑥	𝑥	X
cana-578	56	7	∈	∈	PROPN
cana-578	56	8	ω(𝑓	ω(𝑓	PROPN
cana-578	56	9	)	)	PUNCT
cana-578	56	10	.	.	PUNCT
cana-578	57	1	lemma	lemma	PROPN
cana-578	57	2	3.1	3.1	NUM
cana-578	57	3	.	.	PUNCT
cana-578	58	1	if	if	SCONJ
cana-578	58	2	an	an	DET
cana-578	58	3	orbit	orbit	NOUN
cana-578	58	4	of	of	ADP
cana-578	58	5	an	an	DET
cana-578	58	6	element	element	NOUN
cana-578	58	7	converges	converge	VERB
cana-578	58	8	then	then	ADV
cana-578	58	9	its	its	PRON
cana-578	58	10	converges	converge	NOUN
cana-578	58	11	to	to	ADP
cana-578	58	12	fixed	fix	VERB
cana-578	58	13	point	point	NOUN
cana-578	58	14	.	.	PUNCT
cana-578	59	1	proof	proof	NOUN
cana-578	59	2	:	:	PUNCT
cana-578	59	3	let	let	VERB
cana-578	59	4	𝑥	𝑥	X
cana-578	59	5	∈	∈	PROPN
cana-578	59	6	𝑋	𝑋	NOUN
cana-578	59	7	be	be	VERB
cana-578	59	8	any	any	DET
cana-578	59	9	point	point	NOUN
cana-578	59	10	as	as	ADP
cana-578	59	11	orbit	orbit	NOUN
cana-578	59	12	of	of	ADP
cana-578	59	13	𝑥	𝑥	PROPN
cana-578	59	14	and	and	CCONJ
cana-578	59	15	𝑂(𝑥	𝑂(𝑥	NUM
cana-578	59	16	)	)	PUNCT
cana-578	59	17	=	=	PRON
cana-578	59	18	{	{	PUNCT
cana-578	59	19	𝑥𝑛}0	𝑥𝑛}0	NOUN
cana-578	59	20	∞	∞	NUM
cana-578	59	21	converges	converge	VERB
cana-578	59	22	to	to	ADP
cana-578	59	23	for	for	ADP
cana-578	59	24	some	some	DET
cana-578	59	25	𝑝.	𝑝.	NOUN
cana-578	59	26	that	that	PRON
cana-578	59	27	is	be	AUX
cana-578	59	28	for	for	ADP
cana-578	59	29	any	any	DET
cana-578	59	30	𝜖	𝜖	PROPN
cana-578	59	31	>	>	X
cana-578	59	32	0	0	PUNCT
cana-578	60	1	there	there	PRON
cana-578	60	2	is	be	VERB
cana-578	60	3	𝑚	𝑚	PRON
cana-578	60	4	∈	∈	PROPN
cana-578	60	5	ℕ	ℕ	PROPN
cana-578	60	6	such	such	ADJ
cana-578	60	7	that	that	SCONJ
cana-578	60	8	𝑑(𝑥𝑛	𝑑(𝑥𝑛	PROPN
cana-578	60	9	,	,	PUNCT
cana-578	60	10	𝑝	𝑝	NOUN
cana-578	60	11	)	)	PUNCT
cana-578	60	12	<	<	X
cana-578	60	13	𝜖	𝜖	PROPN
cana-578	60	14	,	,	PUNCT
cana-578	60	15	𝑛	𝑛	DET
cana-578	60	16	≥	≥	NOUN
cana-578	60	17	𝑚.	𝑚.	ADJ
cana-578	60	18	by	by	ADP
cana-578	60	19	eq	eq	NOUN
cana-578	60	20	(	(	PUNCT
cana-578	60	21	1.1	1.1	NUM
cana-578	60	22	)	)	PUNCT
cana-578	60	23	,	,	PUNCT
cana-578	60	24	we	we	PRON
cana-578	60	25	can	can	AUX
cana-578	60	26	write	write	VERB
cana-578	60	27	𝑑(𝑓(𝑥𝑛−2	𝑑(𝑓(𝑥𝑛−2	PROPN
cana-578	60	28	,	,	PUNCT
cana-578	60	29	𝑥𝑛−1	𝑥𝑛−1	NOUN
cana-578	60	30	)	)	PUNCT
cana-578	60	31	,	,	PUNCT
cana-578	60	32	𝑝	𝑝	NOUN
cana-578	60	33	)	)	PUNCT
cana-578	60	34	<	<	X
cana-578	60	35	𝜖	𝜖	X
cana-578	60	36	⟹	⟹	X
cana-578	60	37	𝑑(𝑓(𝑝	𝑑(𝑓(𝑝	PROPN
cana-578	60	38	,	,	PUNCT
cana-578	60	39	𝑝	𝑝	NOUN
cana-578	60	40	)	)	PUNCT
cana-578	60	41	,	,	PUNCT
cana-578	60	42	𝑝	𝑝	NOUN
cana-578	60	43	)	)	PUNCT
cana-578	60	44	<	<	X
cana-578	61	1	𝜖	𝜖	X
cana-578	61	2	we	we	PRON
cana-578	61	3	can	can	AUX
cana-578	61	4	say	say	VERB
cana-578	61	5	𝑓(𝑝	𝑓(𝑝	PROPN
cana-578	61	6	,	,	PUNCT
cana-578	61	7	𝑝	𝑝	NOUN
cana-578	61	8	)	)	PUNCT
cana-578	61	9	=	=	PUNCT
cana-578	62	1	𝑝.	𝑝.	NOUN
cana-578	62	2	that	that	PRON
cana-578	62	3	is	be	AUX
cana-578	62	4	𝑝	𝑝	NOUN
cana-578	62	5	is	be	AUX
cana-578	62	6	a	a	DET
cana-578	62	7	fixed	fix	VERB
cana-578	62	8	point	point	NOUN
cana-578	62	9	.	.	PUNCT
cana-578	63	1	theorem	theorem	ADJ
cana-578	63	2	3.2	3.2	NUM
cana-578	63	3	.	.	PUNCT
cana-578	63	4	𝑓𝑖𝑥(𝑓	𝑓𝑖𝑥(𝑓	NOUN
cana-578	63	5	)	)	PUNCT
cana-578	63	6	closed	close	VERB
cana-578	63	7	in	in	ADP
cana-578	63	8	𝑋.	𝑋.	PROPN
cana-578	63	9	proof	proof	NOUN
cana-578	63	10	:	:	PUNCT
cana-578	63	11	let	let	VERB
cana-578	63	12	𝑝	𝑝	PROPN
cana-578	63	13	∈	∈	PROPN
cana-578	63	14	𝑓𝑖𝑥(𝑓)̅̅	𝑓𝑖𝑥(𝑓)̅̅	PROPN
cana-578	63	15	̅̅	̅̅	PROPN
cana-578	63	16	̅̅	̅̅	PROPN
cana-578	63	17	̅̅	̅̅	PROPN
cana-578	63	18	̅	̅	NOUN
cana-578	63	19	is	be	AUX
cana-578	63	20	the	the	DET
cana-578	63	21	closure	closure	NOUN
cana-578	63	22	of	of	ADP
cana-578	63	23	the	the	DET
cana-578	63	24	set	set	NOUN
cana-578	63	25	.	.	PUNCT
cana-578	64	1	there	there	PRON
cana-578	64	2	exist	exist	VERB
cana-578	64	3	an	an	DET
cana-578	64	4	orbit	orbit	NOUN
cana-578	64	5	𝑂(𝑥	𝑂(𝑥	NUM
cana-578	64	6	)	)	PUNCT
cana-578	64	7	=	=	PRON
cana-578	64	8	{	{	PUNCT
cana-578	64	9	𝑥𝑛}0	𝑥𝑛}0	NOUN
cana-578	64	10	∞	∞	NUM
cana-578	64	11	converges	converge	VERB
cana-578	64	12	to	to	PART
cana-578	64	13	𝑝.	𝑝.	VERB
cana-578	64	14	𝑝	𝑝	PROPN
cana-578	64	15	is	be	AUX
cana-578	64	16	a	a	DET
cana-578	64	17	fixed	fix	VERB
cana-578	64	18	point	point	NOUN
cana-578	64	19	by	by	ADP
cana-578	64	20	the	the	DET
cana-578	64	21	lemma	lemma	PROPN
cana-578	64	22	3.1	3.1	NUM
cana-578	64	23	.	.	PUNCT
cana-578	64	24	which	which	PRON
cana-578	64	25	means	mean	VERB
cana-578	64	26	set	set	NOUN
cana-578	64	27	of	of	ADP
cana-578	64	28	all	all	DET
cana-578	64	29	fixed	fix	VERB
cana-578	64	30	point	point	NOUN
cana-578	64	31	is	be	AUX
cana-578	64	32	a	a	DET
cana-578	64	33	closed	closed	ADJ
cana-578	64	34	set	set	NOUN
cana-578	64	35	as	as	SCONJ
cana-578	64	36	every	every	DET
cana-578	64	37	limit	limit	NOUN
cana-578	64	38	point	point	NOUN
cana-578	64	39	𝑝	𝑝	NOUN
cana-578	64	40	is	be	AUX
cana-578	64	41	in	in	ADP
cana-578	64	42	that	that	DET
cana-578	64	43	set	set	NOUN
cana-578	64	44	.	.	PUNCT
cana-578	65	1	theorem	theorem	VERB
cana-578	65	2	3.3	3.3	NUM
cana-578	65	3	.	.	PUNCT
cana-578	66	1	the	the	DET
cana-578	66	2	set	set	NOUN
cana-578	66	3	of	of	ADP
cana-578	66	4	strong	strong	ADJ
cana-578	66	5	non	non	ADJ
cana-578	66	6	-	-	ADJ
cana-578	66	7	wandering	wandering	ADJ
cana-578	66	8	points	point	NOUN
cana-578	66	9	is	be	AUX
cana-578	66	10	closed	close	VERB
cana-578	66	11	set	set	VERB
cana-578	66	12	in	in	ADP
cana-578	66	13	𝑋.	𝑋.	PROPN
cana-578	66	14	proof	proof	NOUN
cana-578	66	15	:	:	PUNCT
cana-578	66	16	as	as	SCONJ
cana-578	66	17	we	we	PRON
cana-578	66	18	defined	define	VERB
cana-578	66	19	𝑆	𝑆	PROPN
cana-578	66	20	is	be	AUX
cana-578	66	21	the	the	DET
cana-578	66	22	set	set	NOUN
cana-578	66	23	of	of	ADP
cana-578	66	24	strong	strong	ADJ
cana-578	66	25	non	non	ADJ
cana-578	66	26	-	-	ADJ
cana-578	66	27	wandering	wandering	ADJ
cana-578	66	28	points	point	NOUN
cana-578	66	29	in	in	ADP
cana-578	66	30	x.	x.	NOUN
cana-578	66	31	let	let	VERB
cana-578	66	32	𝑝	𝑝	NOUN
cana-578	66	33	is	be	AUX
cana-578	66	34	a	a	DET
cana-578	66	35	limit	limit	NOUN
cana-578	66	36	point	point	NOUN
cana-578	66	37	of	of	ADP
cana-578	66	38	𝑆.	𝑆.	PROPN
cana-578	66	39	then	then	ADV
cana-578	66	40	there	there	PRON
cana-578	66	41	exist	exist	VERB
cana-578	66	42	an	an	DET
cana-578	66	43	orbit	orbit	NOUN
cana-578	66	44	𝑂(𝑥	𝑂(𝑥	NUM
cana-578	66	45	)	)	PUNCT
cana-578	66	46	=	=	PRON
cana-578	66	47	{	{	PUNCT
cana-578	66	48	𝑥𝑛}0	𝑥𝑛}0	NOUN
cana-578	66	49	∞	∞	NUM
cana-578	66	50	converges	converge	VERB
cana-578	66	51	to	to	PART
cana-578	66	52	𝑝.	𝑝.	VERB
cana-578	66	53	so	so	ADV
cana-578	66	54	for	for	ADP
cana-578	66	55	every	every	DET
cana-578	66	56	𝛿	𝛿	PROPN
cana-578	66	57	>	>	X
cana-578	66	58	0	0	NUM
cana-578	66	59	we	we	PRON
cana-578	66	60	get	get	VERB
cana-578	66	61	𝐵𝛿(𝑝	𝐵𝛿(𝑝	NOUN
cana-578	66	62	)	)	PUNCT
cana-578	66	63	which	which	PRON
cana-578	66	64	contains	contain	VERB
cana-578	66	65	large	large	ADJ
cana-578	66	66	number	number	NOUN
cana-578	66	67	of	of	ADP
cana-578	66	68	non	non	ADJ
cana-578	66	69	wandering	wandering	ADJ
cana-578	66	70	points	point	NOUN
cana-578	66	71	𝑥𝑛.	𝑥𝑛.	NOUN
cana-578	66	72	for	for	ADP
cana-578	66	73	any	any	DET
cana-578	66	74	𝑦	𝑦	PROPN
cana-578	66	75	∈	∈	NOUN
cana-578	66	76	𝐵𝛿(𝑝	𝐵𝛿(𝑝	NOUN
cana-578	66	77	)	)	PUNCT
cana-578	66	78	which	which	PRON
cana-578	66	79	is	be	AUX
cana-578	66	80	non	non	ADJ
cana-578	66	81	-	-	ADJ
cana-578	66	82	wandering	wandering	ADJ
cana-578	66	83	then	then	ADV
cana-578	66	84	there	there	PRON
cana-578	66	85	exist	exist	VERB
cana-578	66	86	𝑧	𝑧	DET
cana-578	66	87	∈	∈	PROPN
cana-578	66	88	𝐵𝛿(𝑝	𝐵𝛿(𝑝	NOUN
cana-578	66	89	)	)	PUNCT
cana-578	66	90	and	and	CCONJ
cana-578	66	91	for	for	ADP
cana-578	66	92	some	some	DET
cana-578	66	93	natural	natural	ADJ
cana-578	66	94	number	number	NOUN
cana-578	66	95	𝑛,𝑑(𝑧𝑛	𝑛,𝑑(𝑧𝑛	PROPN
cana-578	66	96	,	,	PUNCT
cana-578	66	97	𝑦	𝑦	NOUN
cana-578	66	98	)	)	PUNCT
cana-578	66	99	<	<	X
cana-578	67	1	𝛿.	𝛿.	ADV
cana-578	68	1	this	this	PRON
cana-578	68	2	is	be	AUX
cana-578	68	3	true	true	ADJ
cana-578	68	4	for	for	ADP
cana-578	68	5	every	every	DET
cana-578	68	6	∈	∈	PROPN
cana-578	68	7	𝐵𝛿(𝑝	𝐵𝛿(𝑝	NOUN
cana-578	68	8	)	)	PUNCT
cana-578	68	9	.	.	PUNCT
cana-578	69	1	so	so	ADV
cana-578	69	2	𝑝	𝑝	PROPN
cana-578	69	3	is	be	AUX
cana-578	69	4	a	a	DET
cana-578	69	5	strong	strong	ADJ
cana-578	69	6	non	non	ADJ
cana-578	69	7	-	-	ADJ
cana-578	69	8	wandering	wandering	ADJ
cana-578	69	9	point	point	NOUN
cana-578	69	10	.	.	PUNCT
cana-578	70	1	communications	communication	NOUN
cana-578	70	2	on	on	ADP
cana-578	70	3	applied	apply	VERB
cana-578	70	4	nonlinear	nonlinear	ADJ
cana-578	70	5	analysis	analysis	NOUN
cana-578	70	6	issn	issn	NOUN
cana-578	70	7	:	:	PUNCT
cana-578	70	8	1074	1074	NUM
cana-578	70	9	-	-	PUNCT
cana-578	70	10	133x	133x	NUM
cana-578	70	11	vol	vol	NOUN
cana-578	70	12	31	31	NUM
cana-578	70	13	no	no	NOUN
cana-578	70	14	.	.	PUNCT
cana-578	71	1	1s	1s	NUM
cana-578	71	2	(	(	PUNCT
cana-578	71	3	2024	2024	NUM
cana-578	71	4	)	)	PUNCT
cana-578	71	5	190	190	NUM
cana-578	71	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-578	71	7	the	the	DET
cana-578	71	8	strong	strong	ADJ
cana-578	71	9	version	version	NOUN
cana-578	71	10	of	of	ADP
cana-578	71	11	the	the	DET
cana-578	71	12	theorem	theorem	NOUN
cana-578	71	13	is	be	AUX
cana-578	71	14	taken	take	VERB
cana-578	71	15	into	into	ADP
cana-578	71	16	consideration	consideration	NOUN
cana-578	71	17	because	because	SCONJ
cana-578	71	18	every	every	DET
cana-578	71	19	element	element	NOUN
cana-578	71	20	of	of	ADP
cana-578	71	21	the	the	DET
cana-578	71	22	nonwandering	nonwandering	ADJ
cana-578	71	23	orbit	orbit	NOUN
cana-578	71	24	is	be	AUX
cana-578	71	25	non	non	ADJ
cana-578	71	26	-	-	ADJ
cana-578	71	27	wandering	wandering	ADJ
cana-578	71	28	.	.	PUNCT
cana-578	72	1	lemma	lemma	PROPN
cana-578	72	2	3.2	3.2	NUM
cana-578	72	3	.	.	PUNCT
cana-578	73	1	set	set	VERB
cana-578	73	2	of	of	ADP
cana-578	73	3	fixed	fix	VERB
cana-578	73	4	points	point	NOUN
cana-578	73	5	,	,	PUNCT
cana-578	73	6	strong	strong	ADJ
cana-578	73	7	periodic	periodic	ADJ
cana-578	73	8	points	point	NOUN
cana-578	73	9	and	and	CCONJ
cana-578	73	10	strong	strong	ADJ
cana-578	73	11	non	non	NOUN
cana-578	73	12	-	-	ADJ
cana-578	73	13	wanderings	wandering	NOUN
cana-578	73	14	points	point	NOUN
cana-578	73	15	is	be	AUX
cana-578	73	16	invariant	invariant	ADJ
cana-578	73	17	in	in	ADP
cana-578	73	18	𝑋.	𝑋.	PROPN
cana-578	73	19	proof	proof	NOUN
cana-578	73	20	:	:	PUNCT
cana-578	73	21	the	the	DET
cana-578	73	22	proof	proof	NOUN
cana-578	73	23	is	be	AUX
cana-578	73	24	direct	direct	ADJ
cana-578	73	25	application	application	NOUN
cana-578	73	26	of	of	ADP
cana-578	73	27	the	the	DET
cana-578	73	28	definition	definition	NOUN
cana-578	73	29	of	of	ADP
cana-578	73	30	invariant	invariant	ADJ
cana-578	73	31	set	set	NOUN
cana-578	73	32	and	and	CCONJ
cana-578	73	33	consideration	consideration	NOUN
cana-578	73	34	strong	strong	ADJ
cana-578	73	35	form	form	NOUN
cana-578	73	36	.	.	PUNCT
cana-578	74	1	the	the	DET
cana-578	74	2	lemma	lemma	PROPN
cana-578	74	3	3.2	3.2	NUM
cana-578	74	4	is	be	AUX
cana-578	74	5	true	true	ADJ
cana-578	74	6	for	for	ADP
cana-578	74	7	set	set	NOUN
cana-578	74	8	of	of	ADP
cana-578	74	9	periodic	periodic	ADJ
cana-578	74	10	points	point	NOUN
cana-578	74	11	and	and	CCONJ
cana-578	74	12	non	non	ADJ
cana-578	74	13	-	-	ADJ
cana-578	74	14	wandering	wandering	ADJ
cana-578	74	15	points	point	NOUN
cana-578	74	16	in	in	ADP
cana-578	74	17	m	m	ADJ
cana-578	74	18	-	-	ADJ
cana-578	74	19	step	step	NOUN
cana-578	74	20	dynamical	dynamical	ADJ
cana-578	74	21	system	system	NOUN
cana-578	74	22	.	.	PUNCT
cana-578	75	1	4	4	X
cana-578	75	2	.	.	X
cana-578	75	3	transitivity	transitivity	NOUN
cana-578	75	4	function	function	VERB
cana-578	75	5	𝑓	𝑓	NOUN
cana-578	75	6	:	:	PUNCT
cana-578	75	7	𝑋	𝑋	PROPN
cana-578	75	8	×	×	NOUN
cana-578	75	9	𝑋	𝑋	PROPN
cana-578	75	10	→	→	SYM
cana-578	75	11	𝑋	𝑋	PROPN
cana-578	75	12	is	be	AUX
cana-578	75	13	topological	topological	ADJ
cana-578	75	14	transitive	transitive	ADJ
cana-578	75	15	if	if	SCONJ
cana-578	75	16	for	for	ADP
cana-578	75	17	every	every	DET
cana-578	75	18	pair	pair	NOUN
cana-578	75	19	of	of	ADP
cana-578	75	20	non	non	ADJ
cana-578	75	21	empty	empty	ADJ
cana-578	75	22	open	open	ADJ
cana-578	75	23	sets	set	NOUN
cana-578	75	24	𝑈	𝑈	PROPN
cana-578	75	25	,	,	PUNCT
cana-578	75	26	𝑉	𝑉	PROPN
cana-578	75	27	⊆	⊆	NUM
cana-578	75	28	𝑋	𝑋	NOUN
cana-578	75	29	then	then	ADV
cana-578	75	30	for	for	ADP
cana-578	75	31	some	some	DET
cana-578	75	32	𝑥	𝑥	DET
cana-578	75	33	∈	∈	PROPN
cana-578	75	34	𝑈	𝑈	PROPN
cana-578	75	35	then	then	ADV
cana-578	75	36	for	for	ADP
cana-578	75	37	some	some	DET
cana-578	75	38	∈	∈	PROPN
cana-578	75	39	ℕ	ℕ	PROPN
cana-578	75	40	,	,	PUNCT
cana-578	75	41	𝑥𝑛	𝑥𝑛	PROPN
cana-578	75	42	∈	∈	PROPN
cana-578	75	43	𝑉.	𝑉.	NOUN
cana-578	75	44	𝑥	𝑥	NOUN
cana-578	75	45	∈	∈	NOUN
cana-578	75	46	𝑋	𝑋	NOUN
cana-578	75	47	is	be	AUX
cana-578	75	48	a	a	DET
cana-578	75	49	transitive	transitive	ADJ
cana-578	75	50	point	point	NOUN
cana-578	75	51	if	if	SCONJ
cana-578	75	52	it	it	PRON
cana-578	75	53	has	have	VERB
cana-578	75	54	a	a	DET
cana-578	75	55	dense	dense	ADJ
cana-578	75	56	orbit	orbit	NOUN
cana-578	75	57	.	.	PUNCT
cana-578	76	1	every	every	DET
cana-578	76	2	element	element	NOUN
cana-578	76	3	of	of	ADP
cana-578	76	4	a	a	DET
cana-578	76	5	dense	dense	ADJ
cana-578	76	6	orbit	orbit	NOUN
cana-578	76	7	is	be	AUX
cana-578	76	8	also	also	ADV
cana-578	76	9	a	a	DET
cana-578	76	10	dense	dense	ADJ
cana-578	76	11	orbit	orbit	NOUN
cana-578	76	12	in	in	ADP
cana-578	76	13	a	a	DET
cana-578	76	14	strong	strong	ADJ
cana-578	76	15	dense	dense	ADJ
cana-578	76	16	orbit	orbit	NOUN
cana-578	76	17	.	.	PUNCT
cana-578	77	1	a	a	DET
cana-578	77	2	non	non	X
cana-578	77	3	empty	empty	ADJ
cana-578	77	4	,	,	PUNCT
cana-578	77	5	closed	closed	ADJ
cana-578	77	6	,	,	PUNCT
cana-578	77	7	invariant	invariant	ADJ
cana-578	77	8	subset	subset	NOUN
cana-578	77	9	𝑌	𝑌	PROPN
cana-578	77	10	of	of	ADP
cana-578	77	11	𝑋	𝑋	PROPN
cana-578	77	12	is	be	AUX
cana-578	77	13	minimal	minimal	ADJ
cana-578	77	14	set	set	VERB
cana-578	77	15	if	if	SCONJ
cana-578	77	16	it	it	PRON
cana-578	77	17	is	be	AUX
cana-578	77	18	not	not	PART
cana-578	77	19	contained	contain	VERB
cana-578	77	20	in	in	ADP
cana-578	77	21	any	any	DET
cana-578	77	22	proper	proper	ADJ
cana-578	77	23	invariant	invariant	ADJ
cana-578	77	24	closed	close	VERB
cana-578	77	25	set	set	NOUN
cana-578	77	26	,	,	PUNCT
cana-578	77	27	which	which	PRON
cana-578	77	28	is	be	AUX
cana-578	77	29	equivalent	equivalent	ADJ
cana-578	77	30	to	to	PART
cana-578	77	31	orbit	orbit	NOUN
cana-578	77	32	of	of	ADP
cana-578	77	33	every	every	DET
cana-578	77	34	element	element	NOUN
cana-578	77	35	of	of	ADP
cana-578	77	36	𝑌	𝑌	PROPN
cana-578	77	37	is	be	AUX
cana-578	77	38	dense	dense	ADJ
cana-578	77	39	.	.	PUNCT
cana-578	78	1	theorem	theorem	VERB
cana-578	78	2	4.1	4.1	NUM
cana-578	78	3	.	.	PUNCT
cana-578	79	1	the	the	DET
cana-578	79	2	following	follow	VERB
cana-578	79	3	are	be	AUX
cana-578	79	4	equivalent	equivalent	ADJ
cana-578	79	5	.	.	PUNCT
cana-578	80	1	(	(	PUNCT
cana-578	80	2	these	these	PRON
cana-578	80	3	also	also	ADV
cana-578	80	4	hold	hold	VERB
cana-578	80	5	true	true	ADJ
cana-578	80	6	for	for	ADP
cana-578	80	7	1	1	NUM
cana-578	80	8	-	-	PUNCT
cana-578	80	9	step	step	NOUN
cana-578	80	10	dynamical	dynamical	ADJ
cana-578	80	11	systems	system	NOUN
cana-578	80	12	,	,	PUNCT
cana-578	80	13	but	but	CCONJ
cana-578	80	14	additional	additional	ADJ
cana-578	80	15	proof	proof	NOUN
cana-578	80	16	is	be	AUX
cana-578	80	17	required	require	VERB
cana-578	80	18	in	in	ADP
cana-578	80	19	m	m	ADJ
cana-578	80	20	-	-	ADJ
cana-578	80	21	step	step	NOUN
cana-578	80	22	dynamical	dynamical	ADJ
cana-578	80	23	systems	system	NOUN
cana-578	80	24	case	case	NOUN
cana-578	80	25	.	.	PUNCT
cana-578	80	26	)	)	PUNCT
cana-578	81	1	1	1	X
cana-578	81	2	.	.	PUNCT
cana-578	81	3	topological	topological	ADJ
cana-578	81	4	transitivity	transitivity	NOUN
cana-578	81	5	.	.	PUNCT
cana-578	82	1	2	2	X
cana-578	82	2	.	.	X
cana-578	82	3	the	the	DET
cana-578	82	4	set	set	NOUN
cana-578	82	5	𝑉∗	𝑉∗	NOUN
cana-578	82	6	=	=	PRON
cana-578	82	7	{	{	PUNCT
cana-578	82	8	𝑥	𝑥	PRON
cana-578	82	9	∈	∈	PROPN
cana-578	82	10	𝑋	𝑋	NOUN
cana-578	82	11	:	:	PUNCT
cana-578	82	12	𝑥𝑛	𝑥𝑛	AUX
cana-578	82	13	∈	∈	PROPN
cana-578	82	14	𝑉𝑓𝑜𝑟	𝑉𝑓𝑜𝑟	PROPN
cana-578	82	15	𝑠𝑜𝑚𝑒	𝑠𝑜𝑚𝑒	PROPN
cana-578	82	16	𝑛	𝑛	DET
cana-578	82	17	∈	∈	PROPN
cana-578	82	18	ℕ	ℕ	PROPN
cana-578	82	19	}	}	PUNCT
cana-578	82	20	is	be	AUX
cana-578	82	21	dense	dense	ADJ
cana-578	82	22	for	for	ADP
cana-578	82	23	every	every	DET
cana-578	82	24	non	non	ADJ
cana-578	82	25	empty	empty	ADJ
cana-578	82	26	open	open	ADJ
cana-578	82	27	subset	subset	NOUN
cana-578	82	28	𝑉	𝑉	PROPN
cana-578	82	29	of	of	ADP
cana-578	82	30	𝑋.	𝑋.	PROPN
cana-578	82	31	3	3	NUM
cana-578	82	32	.	.	PUNCT
cana-578	83	1	if	if	SCONJ
cana-578	83	2	𝐾	𝐾	PROPN
cana-578	83	3	is	be	AUX
cana-578	83	4	any	any	DET
cana-578	83	5	invariant	invariant	ADJ
cana-578	83	6	subset	subset	NOUN
cana-578	83	7	of	of	ADP
cana-578	83	8	𝑋	𝑋	PROPN
cana-578	83	9	then	then	ADV
cana-578	83	10	either	either	CCONJ
cana-578	83	11	𝐾	𝐾	PROPN
cana-578	83	12	is	be	AUX
cana-578	83	13	dense	dense	ADJ
cana-578	83	14	or	or	CCONJ
cana-578	83	15	𝐾	𝐾	NOUN
cana-578	83	16	is	be	AUX
cana-578	83	17	nowhere	nowhere	ADV
cana-578	83	18	dense	dense	ADJ
cana-578	83	19	.	.	PUNCT
cana-578	84	1	proof	proof	NOUN
cana-578	84	2	:	:	PUNCT
cana-578	84	3	1⇒2	1⇒2	PROPN
cana-578	84	4	assume	assume	VERB
cana-578	84	5	𝑓	𝑓	PRON
cana-578	84	6	is	be	AUX
cana-578	84	7	a	a	DET
cana-578	84	8	topological	topological	ADJ
cana-578	84	9	transitive	transitive	NOUN
cana-578	84	10	.	.	PUNCT
cana-578	85	1	then	then	ADV
cana-578	85	2	for	for	ADP
cana-578	85	3	every	every	DET
cana-578	85	4	non	non	ADJ
cana-578	85	5	empty	empty	ADJ
cana-578	85	6	pair	pair	NOUN
cana-578	85	7	of	of	ADP
cana-578	85	8	open	open	ADJ
cana-578	85	9	subsets	subset	NOUN
cana-578	85	10	𝑊	𝑊	PROPN
cana-578	85	11	,	,	PUNCT
cana-578	85	12	𝑉	𝑉	PROPN
cana-578	85	13	⊆	⊆	NUM
cana-578	85	14	𝑋	𝑋	PROPN
cana-578	85	15	,	,	PUNCT
cana-578	85	16	there	there	PRON
cana-578	85	17	exist	exist	VERB
cana-578	85	18	𝑥	𝑥	DET
cana-578	85	19	∈	∈	PROPN
cana-578	85	20	𝑊	𝑊	NOUN
cana-578	85	21	and	and	CCONJ
cana-578	85	22	𝑥𝑛	𝑥𝑛	PROPN
cana-578	85	23	∈	∈	PROPN
cana-578	85	24	𝑉	𝑉	PROPN
cana-578	85	25	for	for	ADP
cana-578	85	26	some	some	DET
cana-578	85	27	𝑛.	𝑛.	NOUN
cana-578	85	28	so𝑉∗(≠	so𝑉∗(≠	NOUN
cana-578	85	29	∅	∅	NOUN
cana-578	85	30	)	)	PUNCT
cana-578	85	31	⊆	⊆	NUM
cana-578	85	32	𝑋.	𝑋.	PROPN
cana-578	85	33	and	and	CCONJ
cana-578	85	34	it	it	PRON
cana-578	85	35	is	be	AUX
cana-578	85	36	true	true	ADJ
cana-578	85	37	for	for	SCONJ
cana-578	85	38	every	every	DET
cana-578	85	39	pair	pair	NOUN
cana-578	85	40	of	of	ADP
cana-578	85	41	𝑊	𝑊	PROPN
cana-578	85	42	and	and	CCONJ
cana-578	85	43	fixed	fix	VERB
cana-578	85	44	𝑉.	𝑉.	PROPN
cana-578	85	45	then	then	ADV
cana-578	85	46	for	for	ADP
cana-578	85	47	every	every	DET
cana-578	85	48	𝑥	𝑥	PRON
cana-578	85	49	∈	∈	NOUN
cana-578	85	50	𝑋	𝑋	NOUN
cana-578	85	51	we	we	PRON
cana-578	85	52	get	get	VERB
cana-578	85	53	a	a	DET
cana-578	85	54	non	non	X
cana-578	85	55	empty	empty	ADJ
cana-578	85	56	open	open	ADJ
cana-578	85	57	set	set	NOUN
cana-578	85	58	which	which	PRON
cana-578	85	59	intersects	intersect	VERB
cana-578	85	60	the	the	DET
cana-578	85	61	given	give	VERB
cana-578	85	62	𝑉∗.	𝑉∗.	PROPN
cana-578	85	63	we	we	PRON
cana-578	85	64	can	can	AUX
cana-578	85	65	conclude	conclude	VERB
cana-578	85	66	𝑉∗	𝑉∗	NOUN
cana-578	85	67	is	be	AUX
cana-578	85	68	dense	dense	ADJ
cana-578	85	69	in	in	ADP
cana-578	85	70	𝑋.	𝑋.	PROPN
cana-578	85	71	2⇒	2⇒	PROPN
cana-578	86	1	3	3	NUM
cana-578	86	2	assume	assume	VERB
cana-578	86	3	𝐾	𝐾	PROPN
cana-578	86	4	be	be	VERB
cana-578	86	5	non	non	X
cana-578	86	6	empty	empty	ADJ
cana-578	86	7	invariant	invariant	ADJ
cana-578	86	8	subset	subset	NOUN
cana-578	86	9	of	of	ADP
cana-578	86	10	𝑋.we	𝑋.we	NOUN
cana-578	86	11	have	have	AUX
cana-578	86	12	𝐾	𝐾	PROPN
cana-578	86	13	is	be	AUX
cana-578	86	14	invariant	invariant	ADJ
cana-578	86	15	so	so	ADV
cana-578	86	16	compliment	compliment	NOUN
cana-578	86	17	of	of	ADP
cana-578	86	18	𝐾	𝐾	PROPN
cana-578	86	19	in	in	ADP
cana-578	86	20	𝑋	𝑋	PROPN
cana-578	86	21	does	do	AUX
cana-578	86	22	n’t	not	PART
cana-578	86	23	contain	contain	VERB
cana-578	86	24	any	any	DET
cana-578	86	25	element	element	NOUN
cana-578	86	26	in	in	ADP
cana-578	86	27	the	the	DET
cana-578	86	28	orbit	orbit	NOUN
cana-578	86	29	of	of	ADP
cana-578	86	30	element	element	NOUN
cana-578	86	31	of	of	ADP
cana-578	86	32	𝐾.	𝐾.	PROPN
cana-578	86	33	define	define	NOUN
cana-578	86	34	(	(	PUNCT
cana-578	86	35	𝐾𝑐)∗	𝐾𝑐)∗	X
cana-578	86	36	=	=	SYM
cana-578	86	37	{	{	PUNCT
cana-578	86	38	𝑥	𝑥	PRON
cana-578	86	39	∈	∈	PROPN
cana-578	86	40	𝑋	𝑋	NOUN
cana-578	86	41	:	:	PUNCT
cana-578	86	42	𝑥𝑛	𝑥𝑛	PROPN
cana-578	86	43	∈	∈	PROPN
cana-578	87	1	𝐾𝑐	𝐾𝑐	VERB
cana-578	87	2	𝑓𝑜𝑟	𝑓𝑜𝑟	ADJ
cana-578	87	3	𝑠𝑜𝑚𝑒	𝑠𝑜𝑚𝑒	PROPN
cana-578	87	4	𝑛	𝑛	VERB
cana-578	87	5	}	}	PUNCT
cana-578	87	6	is	be	AUX
cana-578	87	7	𝐾𝑐	𝐾𝑐	VERB
cana-578	87	8	only	only	ADV
cana-578	87	9	.	.	PUNCT
cana-578	88	1	if	if	SCONJ
cana-578	88	2	𝐾𝑐	𝐾𝑐	PROPN
cana-578	88	3	has	have	VERB
cana-578	88	4	an	an	DET
cana-578	88	5	interior	interior	ADJ
cana-578	88	6	point	point	NOUN
cana-578	88	7	then	then	ADV
cana-578	88	8	from	from	ADP
cana-578	88	9	consequence	consequence	NOUN
cana-578	88	10	of	of	ADP
cana-578	88	11	(	(	PUNCT
cana-578	88	12	2	2	X
cana-578	88	13	)	)	PUNCT
cana-578	88	14	it	it	PRON
cana-578	88	15	is	be	AUX
cana-578	88	16	dense	dense	ADJ
cana-578	88	17	and	and	CCONJ
cana-578	88	18	𝐾	𝐾	PROPN
cana-578	88	19	has	have	VERB
cana-578	88	20	empty	empty	ADJ
cana-578	88	21	interior	interior	NOUN
cana-578	88	22	.	.	PUNCT
cana-578	89	1	in	in	ADP
cana-578	89	2	other	other	ADJ
cana-578	89	3	case	case	NOUN
cana-578	89	4	𝐾𝑐	𝐾𝑐	NOUN
cana-578	89	5	has	have	VERB
cana-578	89	6	empty	empty	ADJ
cana-578	89	7	interior	interior	NOUN
cana-578	89	8	.	.	PUNCT
cana-578	90	1	in	in	ADP
cana-578	90	2	both	both	DET
cana-578	90	3	cases	case	NOUN
cana-578	90	4	𝐾	𝐾	PROPN
cana-578	90	5	or	or	CCONJ
cana-578	90	6	𝐾𝑐	𝐾𝑐	PROPN
cana-578	90	7	has	have	VERB
cana-578	90	8	empty	empty	ADJ
cana-578	90	9	interior	interior	NOUN
cana-578	90	10	.	.	PUNCT
cana-578	91	1	which	which	PRON
cana-578	91	2	means	mean	VERB
cana-578	91	3	𝐾	𝐾	PROPN
cana-578	91	4	or	or	CCONJ
cana-578	91	5	𝐾𝑐	𝐾𝑐	PROPN
cana-578	91	6	is	be	AUX
cana-578	91	7	dense	dense	ADJ
cana-578	91	8	.	.	PUNCT
cana-578	92	1	which	which	PRON
cana-578	92	2	is	be	AUX
cana-578	92	3	same	same	ADJ
cana-578	92	4	as	as	SCONJ
cana-578	92	5	either	either	CCONJ
cana-578	92	6	𝐾	𝐾	PROPN
cana-578	92	7	is	be	AUX
cana-578	92	8	dense	dense	ADJ
cana-578	92	9	or	or	CCONJ
cana-578	92	10	𝐾𝑐	𝐾𝑐	NOUN
cana-578	92	11	is	be	AUX
cana-578	92	12	dense	dense	ADJ
cana-578	92	13	.	.	PUNCT
cana-578	93	1	theorem	theorem	VERB
cana-578	93	2	4.2	4.2	NUM
cana-578	93	3	:	:	PUNCT
cana-578	93	4	𝑓	𝑓	PRON
cana-578	93	5	is	be	AUX
cana-578	93	6	topological	topological	ADJ
cana-578	93	7	transitive	transitive	NOUN
cana-578	93	8	then	then	ADV
cana-578	93	9	set	set	VERB
cana-578	93	10	of	of	ADP
cana-578	93	11	transitive	transitive	ADJ
cana-578	93	12	points	point	NOUN
cana-578	93	13	are	be	AUX
cana-578	93	14	residual	residual	ADJ
cana-578	93	15	.	.	PUNCT
cana-578	94	1	communications	communication	NOUN
cana-578	94	2	on	on	ADP
cana-578	94	3	applied	apply	VERB
cana-578	94	4	nonlinear	nonlinear	ADJ
cana-578	94	5	analysis	analysis	NOUN
cana-578	94	6	issn	issn	NOUN
cana-578	94	7	:	:	PUNCT
cana-578	94	8	1074	1074	NUM
cana-578	94	9	-	-	PUNCT
cana-578	94	10	133x	133x	NUM
cana-578	94	11	vol	vol	NOUN
cana-578	94	12	31	31	NUM
cana-578	94	13	no	no	NOUN
cana-578	94	14	.	.	PUNCT
cana-578	95	1	1s	1s	NUM
cana-578	95	2	(	(	PUNCT
cana-578	95	3	2024	2024	NUM
cana-578	95	4	)	)	PUNCT
cana-578	95	5	191	191	NUM
cana-578	96	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-578	96	2	proof	proof	NOUN
cana-578	96	3	:	:	PUNCT
cana-578	96	4	𝑥	𝑥	PRON
cana-578	96	5	is	be	AUX
cana-578	96	6	a	a	DET
cana-578	96	7	transitive	transitive	ADJ
cana-578	96	8	point	point	NOUN
cana-578	96	9	then	then	ADV
cana-578	96	10	𝑂(𝑥	𝑂(𝑥	NUM
cana-578	96	11	)	)	PUNCT
cana-578	96	12	is	be	AUX
cana-578	96	13	dense	dense	ADJ
cana-578	96	14	in	in	ADP
cana-578	96	15	𝑋.	𝑋.	PROPN
cana-578	96	16	let	let	VERB
cana-578	96	17	{	{	PUNCT
cana-578	96	18	𝑈𝑖	𝑈𝑖	ADV
cana-578	96	19	}	}	PUNCT
cana-578	96	20	be	be	AUX
cana-578	96	21	the	the	DET
cana-578	96	22	arbitary	arbitary	ADJ
cana-578	96	23	collection	collection	NOUN
cana-578	96	24	of	of	ADP
cana-578	96	25	open	open	ADJ
cana-578	96	26	sets	set	NOUN
cana-578	96	27	whose	whose	DET
cana-578	96	28	union	union	NOUN
cana-578	96	29	is	be	AUX
cana-578	96	30	𝑋.	𝑋.	PROPN
cana-578	96	31	define	define	VERB
cana-578	96	32	𝐷(𝑈𝑖	𝐷(𝑈𝑖	ADV
cana-578	96	33	)	)	PUNCT
cana-578	97	1	=	=	PRON
cana-578	97	2	{	{	PUNCT
cana-578	97	3	𝑥	𝑥	PUNCT
cana-578	97	4	∈	∈	PROPN
cana-578	97	5	𝑋	𝑋	NOUN
cana-578	97	6	:	:	PUNCT
cana-578	97	7	𝑂(𝑥	𝑂(𝑥	NUM
cana-578	97	8	)	)	PUNCT
cana-578	97	9	∩	∩	NOUN
cana-578	97	10	𝑈𝑖	𝑈𝑖	PROPN
cana-578	97	11	≠	≠	PROPN
cana-578	97	12	∅	∅	NOUN
cana-578	97	13	}	}	PUNCT
cana-578	97	14	which	which	PRON
cana-578	97	15	is	be	AUX
cana-578	97	16	non	non	X
cana-578	97	17	empty	empty	ADJ
cana-578	97	18	as	as	SCONJ
cana-578	97	19	𝑓	𝑓	PRON
cana-578	97	20	is	be	AUX
cana-578	97	21	topological	topological	ADJ
cana-578	97	22	transitive	transitive	NOUN
cana-578	97	23	and	and	CCONJ
cana-578	97	24	is	be	AUX
cana-578	97	25	open	open	ADJ
cana-578	97	26	subset	subset	NOUN
cana-578	97	27	of	of	ADP
cana-578	97	28	𝑋	𝑋	PROPN
cana-578	97	29	as	as	SCONJ
cana-578	97	30	𝑓	𝑓	PRON
cana-578	97	31	is	be	AUX
cana-578	97	32	continuous	continuous	ADJ
cana-578	97	33	[	[	X
cana-578	97	34	6	6	NUM
cana-578	97	35	]	]	PUNCT
cana-578	97	36	.	.	PUNCT
cana-578	98	1	𝐷(𝑈𝑖	𝐷(𝑈𝑖	ADJ
cana-578	98	2	)	)	PUNCT
cana-578	98	3	is	be	AUX
cana-578	98	4	open	open	ADJ
cana-578	98	5	dense	dense	ADJ
cana-578	98	6	set	set	NOUN
cana-578	98	7	for	for	ADP
cana-578	98	8	each	each	DET
cana-578	98	9	𝑖.	𝑖.	NOUN
cana-578	98	10	as	as	SCONJ
cana-578	98	11	𝑋	𝑋	NOUN
cana-578	98	12	is	be	AUX
cana-578	98	13	compact	compact	ADJ
cana-578	98	14	so	so	ADV
cana-578	98	15	complete	complete	ADJ
cana-578	98	16	and	and	CCONJ
cana-578	98	17	we	we	PRON
cana-578	98	18	have	have	VERB
cana-578	98	19	∩	∩	NOUN
cana-578	98	20	𝑈𝑖	𝑈𝑖	PROPN
cana-578	98	21	≠	≠	PROPN
cana-578	98	22	∅	∅	NOUN
cana-578	98	23	for	for	ADP
cana-578	98	24	each	each	DET
cana-578	98	25	𝑖	𝑖	NOUN
cana-578	98	26	and	and	CCONJ
cana-578	98	27	𝑥	𝑥	DET
cana-578	98	28	∈	∈	PROPN
cana-578	98	29	(	(	PUNCT
cana-578	98	30	∩	∩	ADJ
cana-578	98	31	𝑈𝑖	𝑈𝑖	NOUN
cana-578	98	32	)	)	PUNCT
cana-578	98	33	≠	≠	PROPN
cana-578	98	34	∅.	∅.	NOUN
cana-578	98	35	which	which	PRON
cana-578	98	36	means	mean	VERB
cana-578	98	37	𝑥	𝑥	PROPN
cana-578	98	38	is	be	AUX
cana-578	98	39	the	the	DET
cana-578	98	40	residual	residual	ADJ
cana-578	98	41	point	point	NOUN
cana-578	98	42	.	.	PUNCT
cana-578	99	1	theorem	theorem	VERB
cana-578	99	2	4.3	4.3	NUM
cana-578	99	3	:	:	PUNCT
cana-578	99	4	(	(	PUNCT
cana-578	99	5	𝑋	𝑋	NOUN
cana-578	99	6	,	,	PUNCT
cana-578	99	7	𝑑	𝑑	NOUN
cana-578	99	8	)	)	PUNCT
cana-578	99	9	is	be	AUX
cana-578	99	10	a	a	DET
cana-578	99	11	compact	compact	ADJ
cana-578	99	12	metric	metric	ADJ
cana-578	99	13	space	space	NOUN
cana-578	99	14	and	and	CCONJ
cana-578	99	15	𝑓	𝑓	PRON
cana-578	99	16	:	:	PUNCT
cana-578	99	17	𝑋𝑚	𝑋𝑚	PROPN
cana-578	99	18	⟶	⟶	NOUN
cana-578	99	19	𝑋	𝑋	NOUN
cana-578	99	20	is	be	AUX
cana-578	99	21	continuous	continuous	ADJ
cana-578	99	22	.	.	PUNCT
cana-578	100	1	if	if	SCONJ
cana-578	100	2	𝑂(𝑥)̅̅	𝑂(𝑥)̅̅	ADJ
cana-578	100	3	̅̅	̅̅	NOUN
cana-578	100	4	̅̅	̅̅	NOUN
cana-578	100	5	is	be	AUX
cana-578	100	6	minimal	minimal	ADJ
cana-578	100	7	for	for	ADP
cana-578	100	8	some	some	DET
cana-578	100	9	𝑥	𝑥	DET
cana-578	100	10	∈	∈	NOUN
cana-578	100	11	𝑋	𝑋	NOUN
cana-578	100	12	then	then	ADV
cana-578	100	13	for	for	ADP
cana-578	100	14	any	any	DET
cana-578	100	15	non	non	ADJ
cana-578	100	16	empty	empty	ADJ
cana-578	100	17	open	open	ADJ
cana-578	100	18	subset	subset	ADJ
cana-578	100	19	𝑈	𝑈	PROPN
cana-578	100	20	of	of	ADP
cana-578	100	21	𝑋	𝑋	PROPN
cana-578	100	22	the	the	DET
cana-578	100	23	cardinality	cardinality	NOUN
cana-578	100	24	of	of	ADP
cana-578	100	25	the	the	DET
cana-578	100	26	set	set	NOUN
cana-578	100	27	𝑍	𝑍	NOUN
cana-578	100	28	=	=	SYM
cana-578	100	29	{	{	PUNCT
cana-578	100	30	𝑛	𝑛	NOUN
cana-578	100	31	:	:	PUNCT
cana-578	100	32	(	(	PUNCT
cana-578	100	33	𝑥𝑛)𝑝	𝑥𝑛)𝑝	NOUN
cana-578	100	34	∈	∈	PROPN
cana-578	100	35	𝑈	𝑈	PROPN
cana-578	100	36	𝑓𝑜𝑟	𝑓𝑜𝑟	NOUN
cana-578	100	37	𝑒𝑣𝑒𝑟𝑦	𝑒𝑣𝑒𝑟𝑦	VERB
cana-578	100	38	𝑛	𝑛	DET
cana-578	100	39	∈	∈	PROPN
cana-578	100	40	ℕ	ℕ	PROPN
cana-578	100	41	,	,	PUNCT
cana-578	100	42	𝑝	𝑝	PROPN
cana-578	100	43	∈	∈	PROPN
cana-578	100	44	ℕ	ℕ	PROPN
cana-578	100	45	}	}	PUNCT
cana-578	100	46	is	be	AUX
cana-578	100	47	finite	finite	ADJ
cana-578	100	48	.	.	PUNCT
cana-578	101	1	proof	proof	NOUN
cana-578	101	2	:	:	PUNCT
cana-578	101	3	assume	assume	VERB
cana-578	101	4	𝑌	𝑌	PROPN
cana-578	101	5	=	=	PUNCT
cana-578	101	6	𝑂(𝑥)̅̅	𝑂(𝑥)̅̅	PROPN
cana-578	101	7	̅̅	̅̅	NOUN
cana-578	101	8	̅̅	̅̅	NOUN
cana-578	101	9	is	be	AUX
cana-578	101	10	minimal	minimal	ADJ
cana-578	101	11	.	.	PUNCT
cana-578	102	1	assume	assume	VERB
cana-578	102	2	𝑈	𝑈	PROPN
cana-578	102	3	be	be	AUX
cana-578	102	4	open	open	ADJ
cana-578	102	5	subset	subset	NOUN
cana-578	102	6	of	of	ADP
cana-578	102	7	𝑌.	𝑌.	PROPN
cana-578	102	8	given	give	VERB
cana-578	102	9	𝑌	𝑌	PROPN
cana-578	102	10	is	be	AUX
cana-578	102	11	minimal	minimal	ADJ
cana-578	102	12	then	then	ADV
cana-578	102	13	orbit	orbit	NOUN
cana-578	102	14	of	of	ADP
cana-578	102	15	every	every	DET
cana-578	102	16	element	element	NOUN
cana-578	102	17	is	be	AUX
cana-578	102	18	dense	dense	ADJ
cana-578	102	19	in	in	ADP
cana-578	102	20	𝑌.	𝑌.	PROPN
cana-578	102	21	so	so	ADV
cana-578	102	22	for	for	ADP
cana-578	102	23	every	every	DET
cana-578	102	24	𝑦	𝑦	NOUN
cana-578	102	25	∈	∈	NOUN
cana-578	102	26	𝑌	𝑌	PROPN
cana-578	102	27	there	there	PRON
cana-578	102	28	is	be	VERB
cana-578	102	29	some	some	PRON
cana-578	102	30	𝑛	𝑛	DET
cana-578	102	31	such	such	ADJ
cana-578	102	32	that	that	DET
cana-578	102	33	𝑦𝑛	𝑦𝑛	PROPN
cana-578	102	34	∈	∈	PROPN
cana-578	102	35	𝑈.	𝑈.	PROPN
cana-578	102	36	we	we	PRON
cana-578	102	37	define	define	VERB
cana-578	102	38	𝑂−𝑛(𝑈	𝑂−𝑛(𝑈	PROPN
cana-578	102	39	)	)	PUNCT
cana-578	103	1	=	=	PRON
cana-578	103	2	{	{	PUNCT
cana-578	103	3	𝑦	𝑦	NOUN
cana-578	103	4	∈	∈	PROPN
cana-578	103	5	𝑋	𝑋	NOUN
cana-578	103	6	:	:	PUNCT
cana-578	103	7	𝑦𝑛	𝑦𝑛	ADP
cana-578	103	8	∈	∈	PROPN
cana-578	103	9	𝑈	𝑈	PROPN
cana-578	103	10	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-578	103	11	𝑠𝑜𝑚𝑒	𝑠𝑜𝑚𝑒	PROPN
cana-578	103	12	𝑛	𝑛	PROPN
cana-578	103	13	}	}	PUNCT
cana-578	103	14	.	.	PUNCT
cana-578	104	1	which	which	PRON
cana-578	104	2	is	be	AUX
cana-578	104	3	open	open	ADJ
cana-578	104	4	as	as	SCONJ
cana-578	104	5	𝑓	𝑓	PRON
cana-578	104	6	is	be	AUX
cana-578	104	7	continuous	continuous	ADJ
cana-578	104	8	.	.	PUNCT
cana-578	105	1	so	so	ADV
cana-578	105	2	ℳ	ℳ	PROPN
cana-578	105	3	=	=	SYM
cana-578	105	4	{	{	PUNCT
cana-578	105	5	𝑂−𝑛(𝑈	𝑂−𝑛(𝑈	PROPN
cana-578	105	6	):	):	PUNCT
cana-578	105	7	𝑓𝑜𝑟	𝑓𝑜𝑟	X
cana-578	105	8	𝑒𝑣𝑒𝑟𝑦	𝑒𝑣𝑒𝑟𝑦	VERB
cana-578	105	9	𝑛	𝑛	PROPN
cana-578	105	10	}	}	PUNCT
cana-578	105	11	is	be	AUX
cana-578	105	12	open	open	ADJ
cana-578	105	13	cover	cover	NOUN
cana-578	105	14	of	of	ADP
cana-578	105	15	𝑋.	𝑋.	PROPN
cana-578	105	16	being	be	AUX
cana-578	105	17	𝑋	𝑋	PROPN
cana-578	105	18	is	be	AUX
cana-578	105	19	compact	compact	ADJ
cana-578	105	20	we	we	PRON
cana-578	105	21	have	have	VERB
cana-578	105	22	a	a	DET
cana-578	105	23	finite	finite	ADJ
cana-578	105	24	subcover	subcover	NOUN
cana-578	105	25	for	for	ADP
cana-578	105	26	ℳ.then	ℳ.then	ADV
cana-578	105	27	the	the	DET
cana-578	105	28	set	set	NOUN
cana-578	105	29	{	{	PUNCT
cana-578	105	30	𝑈	𝑈	PROPN
cana-578	105	31	,	,	PUNCT
cana-578	105	32	𝑂−1(𝑈	𝑂−1(𝑈	NOUN
cana-578	105	33	)	)	PUNCT
cana-578	105	34	,	,	PUNCT
cana-578	105	35	…	…	PUNCT
cana-578	105	36	𝑂−(𝑝−1)(𝑈	𝑂−(𝑝−1)(𝑈	NUM
cana-578	105	37	)	)	PUNCT
cana-578	105	38	}	}	PUNCT
cana-578	105	39	is	be	AUX
cana-578	105	40	a	a	DET
cana-578	105	41	finite	finite	ADJ
cana-578	105	42	sub	sub	NOUN
cana-578	105	43	cover	cover	NOUN
cana-578	105	44	for	for	ADP
cana-578	105	45	ℳ.	ℳ.	PROPN
cana-578	105	46	then	then	ADV
cana-578	105	47	for	for	ADP
cana-578	105	48	every	every	DET
cana-578	105	49	𝑛	𝑛	PRON
cana-578	105	50	∈	∈	PROPN
cana-578	105	51	ℕ	ℕ	PROPN
cana-578	105	52	there	there	ADV
cana-578	105	53	exist	exist	VERB
cana-578	105	54	𝑘	𝑘	DET
cana-578	105	55	<	<	X
cana-578	105	56	𝑝	𝑝	PROPN
cana-578	105	57	and	and	CCONJ
cana-578	105	58	𝑥𝑛	𝑥𝑛	PROPN
cana-578	105	59	∈	∈	PROPN
cana-578	105	60	𝑂−𝑘(𝑈	𝑂−𝑘(𝑈	NOUN
cana-578	105	61	)	)	PUNCT
cana-578	105	62	.	.	PUNCT
cana-578	106	1	which	which	PRON
cana-578	106	2	is	be	AUX
cana-578	106	3	equivalent	equivalent	ADJ
cana-578	106	4	to	to	ADP
cana-578	106	5	(	(	PUNCT
cana-578	106	6	𝑥𝑛)𝑘	𝑥𝑛)𝑘	PROPN
cana-578	106	7	∈	∈	PROPN
cana-578	106	8	𝑈	𝑈	PROPN
cana-578	106	9	,	,	PUNCT
cana-578	106	10	0	0	NUM
cana-578	106	11	≤	≤	NOUN
cana-578	107	1	𝑘	𝑘	DET
cana-578	107	2	<	<	X
cana-578	107	3	𝑝	𝑝	PROPN
cana-578	107	4	that	that	PRON
cana-578	107	5	is	be	AUX
cana-578	107	6	the	the	DET
cana-578	107	7	set	set	ADJ
cana-578	107	8	𝑍	𝑍	NOUN
cana-578	107	9	=	=	SYM
cana-578	107	10	{	{	PUNCT
cana-578	107	11	𝑛	𝑛	NOUN
cana-578	107	12	:	:	PUNCT
cana-578	107	13	(	(	PUNCT
cana-578	107	14	𝑥𝑛)𝑝	𝑥𝑛)𝑝	NOUN
cana-578	107	15	∈	∈	PROPN
cana-578	107	16	𝑈	𝑈	PROPN
cana-578	107	17	𝑓𝑜𝑟	𝑓𝑜𝑟	NOUN
cana-578	107	18	𝑒𝑣𝑒𝑟𝑦	𝑒𝑣𝑒𝑟𝑦	VERB
cana-578	107	19	𝑛	𝑛	DET
cana-578	107	20	∈	∈	PROPN
cana-578	107	21	ℕ	ℕ	PROPN
cana-578	107	22	,	,	PUNCT
cana-578	107	23	𝑝	𝑝	PROPN
cana-578	107	24	∈	∈	PROPN
cana-578	107	25	ℕ	ℕ	PROPN
cana-578	107	26	}	}	PUNCT
cana-578	107	27	is	be	AUX
cana-578	107	28	finite	finite	ADJ
cana-578	107	29	.	.	PUNCT
cana-578	108	1	5	5	NUM
cana-578	108	2	.	.	NOUN
cana-578	108	3	periods	period	NOUN
cana-578	108	4	and	and	CCONJ
cana-578	108	5	periodic	periodic	ADJ
cana-578	108	6	points	point	NOUN
cana-578	108	7	in	in	ADP
cana-578	108	8	this	this	DET
cana-578	108	9	section	section	NOUN
cana-578	108	10	we	we	PRON
cana-578	108	11	find	find	VERB
cana-578	108	12	period	period	NOUN
cana-578	108	13	and	and	CCONJ
cana-578	108	14	periodic	periodic	ADJ
cana-578	108	15	points	point	NOUN
cana-578	108	16	for	for	ADP
cana-578	108	17	the	the	DET
cana-578	108	18	general	general	ADJ
cana-578	108	19	system	system	NOUN
cana-578	108	20	in	in	ADP
cana-578	108	21	the	the	DET
cana-578	108	22	form	form	NOUN
cana-578	108	23	of	of	ADP
cana-578	108	24	,	,	PUNCT
cana-578	108	25	𝑓	𝑓	X
cana-578	108	26	:	:	PUNCT
cana-578	108	27	ℝ	ℝ	PROPN
cana-578	108	28	×	×	NOUN
cana-578	108	29	ℝ	ℝ	PROPN
cana-578	108	30	→	→	SYM
cana-578	108	31	ℝ	ℝ	PROPN
cana-578	108	32	defined	define	VERB
cana-578	108	33	as	as	ADP
cana-578	108	34	,	,	PUNCT
cana-578	108	35	𝑓(𝑥	𝑓(𝑥	NOUN
cana-578	108	36	,	,	PUNCT
cana-578	108	37	𝑦	𝑦	NOUN
cana-578	108	38	)	)	PUNCT
cana-578	108	39	=	=	VERB
cana-578	108	40	𝑠𝑥	𝑠𝑥	ADP
cana-578	109	1	+	+	CCONJ
cana-578	109	2	𝑡𝑦	𝑡𝑦	NOUN
cana-578	110	1	+	+	NUM
cana-578	110	2	𝑐	𝑐	PROPN
cana-578	110	3	(	(	PUNCT
cana-578	110	4	eq	eq	NOUN
cana-578	110	5	5.1	5.1	NUM
cana-578	110	6	)	)	PUNCT
cana-578	110	7	where	where	SCONJ
cana-578	110	8	𝑠	𝑠	X
cana-578	110	9	,	,	PUNCT
cana-578	110	10	𝑡	𝑡	PROPN
cana-578	110	11	𝑎𝑛𝑑	𝑎𝑛𝑑	PROPN
cana-578	110	12	𝑐	𝑐	PROPN
cana-578	110	13	are	be	AUX
cana-578	110	14	constants	constant	NOUN
cana-578	110	15	.	.	PUNCT
cana-578	111	1	case	case	NOUN
cana-578	112	1	i	i	PRON
cana-578	112	2	(	(	PUNCT
cana-578	112	3	if	if	SCONJ
cana-578	112	4	𝒄	𝒄	PROPN
cana-578	112	5	≠	≠	PROPN
cana-578	112	6	𝟎	𝟎	NUM
cana-578	112	7	)	)	PUNCT
cana-578	112	8	orbit	orbit	NOUN
cana-578	112	9	of	of	ADP
cana-578	112	10	any	any	DET
cana-578	112	11	point	point	NOUN
cana-578	112	12	𝑂(𝑥	𝑂(𝑥	NUM
cana-578	112	13	)	)	PUNCT
cana-578	112	14	=	=	PRON
cana-578	112	15	{	{	PUNCT
cana-578	112	16	𝑥0	𝑥0	NOUN
cana-578	112	17	=	=	SYM
cana-578	112	18	𝑥	𝑥	PROPN
cana-578	112	19	,	,	PUNCT
cana-578	112	20	𝑥1	𝑥1	NOUN
cana-578	112	21	=	=	SYM
cana-578	112	22	𝑥	𝑥	PROPN
cana-578	112	23	,	,	PUNCT
cana-578	112	24	𝑥2	𝑥2	NOUN
cana-578	112	25	=	=	SYM
cana-578	112	26	𝑓(𝑥	𝑓(𝑥	NOUN
cana-578	112	27	,	,	PUNCT
cana-578	112	28	𝑥	𝑥	NOUN
cana-578	112	29	)	)	PUNCT
cana-578	112	30	,	,	PUNCT
cana-578	112	31	…	…	PUNCT
cana-578	112	32	}	}	PUNCT
cana-578	112	33	we	we	PRON
cana-578	112	34	examine	examine	VERB
cana-578	112	35	fixed	fix	VERB
cana-578	112	36	points	point	NOUN
cana-578	112	37	for	for	ADP
cana-578	112	38	this	this	DET
cana-578	112	39	map	map	NOUN
cana-578	112	40	by	by	ADP
cana-578	112	41	𝑥2	𝑥2	NOUN
cana-578	112	42	=	=	SYM
cana-578	112	43	𝑥	𝑥	PROPN
cana-578	112	44	⇒	⇒	NOUN
cana-578	112	45	𝑓(𝑥	𝑓(𝑥	NOUN
cana-578	112	46	,	,	PUNCT
cana-578	112	47	𝑥	𝑥	NOUN
cana-578	112	48	)	)	PUNCT
cana-578	112	49	=	=	SYM
cana-578	113	1	𝑥	𝑥	NOUN
cana-578	113	2	⟹	⟹	PUNCT
cana-578	113	3	𝑠𝑥	𝑠𝑥	VERB
cana-578	113	4	+	+	CCONJ
cana-578	113	5	𝑡𝑥	𝑡𝑥	PROPN
cana-578	113	6	+	+	NOUN
cana-578	113	7	𝑐	𝑐	NOUN
cana-578	113	8	=	=	SYM
cana-578	113	9	𝑥	𝑥	PROPN
cana-578	113	10	⇒	⇒	NOUN
cana-578	113	11	(	(	PUNCT
cana-578	113	12	𝑠	𝑠	PROPN
cana-578	113	13	+	+	X
cana-578	113	14	𝑡)𝑥	𝑡)𝑥	X
cana-578	113	15	−	−	ADP
cana-578	113	16	𝑥	𝑥	NOUN
cana-578	113	17	=	=	PUNCT
cana-578	113	18	−𝑐	−𝑐	NOUN
cana-578	113	19	⇒	⇒	NOUN
cana-578	113	20	𝑥(𝑠	𝑥(𝑠	PROPN
cana-578	113	21	+	+	CCONJ
cana-578	113	22	𝑡	𝑡	VERB
cana-578	113	23	−	−	NOUN
cana-578	113	24	1	1	NUM
cana-578	113	25	)	)	PUNCT
cana-578	113	26	=	=	NOUN
cana-578	113	27	−𝑐	−𝑐	PUNCT
cana-578	113	28	⇒	⇒	VERB
cana-578	113	29	𝑥	𝑥	X
cana-578	113	30	=	=	SYM
cana-578	113	31	𝑐	𝑐	PROPN
cana-578	113	32	1−𝑠−𝑡	1−𝑠−𝑡	INTJ
cana-578	113	33	(	(	PUNCT
cana-578	113	34	eq	eq	ADP
cana-578	113	35	5.2	5.2	NUM
cana-578	113	36	)	)	PUNCT
cana-578	113	37	if	if	SCONJ
cana-578	113	38	any	any	DET
cana-578	113	39	map	map	NOUN
cana-578	113	40	defined	define	VERB
cana-578	113	41	as	as	ADP
cana-578	113	42	in	in	ADP
cana-578	113	43	(	(	PUNCT
cana-578	113	44	eq	eq	NOUN
cana-578	113	45	5.1	5.1	NUM
cana-578	113	46	)	)	PUNCT
cana-578	113	47	then	then	ADV
cana-578	113	48	𝑥	𝑥	X
cana-578	113	49	=	=	SYM
cana-578	113	50	𝑐	𝑐	PROPN
cana-578	113	51	1−𝑠−𝑡	1−𝑠−𝑡	INTJ
cana-578	113	52	,	,	PUNCT
cana-578	113	53	1	1	NUM
cana-578	113	54	−	−	NOUN
cana-578	113	55	𝑠	𝑠	INTJ
cana-578	113	56	−	−	PROPN
cana-578	113	57	𝑡	𝑡	PROPN
cana-578	113	58	≠	≠	PROPN
cana-578	113	59	0	0	NUM
cana-578	113	60	is	be	AUX
cana-578	113	61	a	a	DET
cana-578	113	62	fixed	fix	VERB
cana-578	113	63	point	point	NOUN
cana-578	113	64	.	.	PUNCT
cana-578	114	1	now	now	ADV
cana-578	114	2	we	we	PRON
cana-578	114	3	find	find	VERB
cana-578	114	4	the	the	DET
cana-578	114	5	periodic	periodic	ADJ
cana-578	114	6	points	point	NOUN
cana-578	114	7	of	of	ADP
cana-578	114	8	period	period	NOUN
cana-578	114	9	2	2	NUM
cana-578	114	10	.	.	PUNCT
cana-578	114	11	𝑂(𝑥	𝑂(𝑥	NUM
cana-578	114	12	)	)	PUNCT
cana-578	114	13	=	=	PRON
cana-578	114	14	{	{	PUNCT
cana-578	114	15	𝑥0	𝑥0	NOUN
cana-578	114	16	=	=	SYM
cana-578	114	17	𝑥	𝑥	PROPN
cana-578	114	18	,	,	PUNCT
cana-578	114	19	𝑥1	𝑥1	NOUN
cana-578	114	20	=	=	SYM
cana-578	114	21	𝑥	𝑥	PROPN
cana-578	114	22	,	,	PUNCT
cana-578	114	23	𝑥2	𝑥2	NOUN
cana-578	114	24	=	=	SYM
cana-578	114	25	𝑠𝑥	𝑠𝑥	ADP
cana-578	115	1	+	+	CCONJ
cana-578	115	2	𝑡𝑥	𝑡𝑥	PROPN
cana-578	116	1	+	+	CCONJ
cana-578	116	2	𝑐	𝑐	NOUN
cana-578	116	3	,	,	PUNCT
cana-578	116	4	𝑥3	𝑥3	NOUN
cana-578	116	5	=	=	PUNCT
cana-578	116	6	𝑠𝑥1	𝑠𝑥1	X
cana-578	116	7	+	+	CCONJ
cana-578	116	8	𝑡𝑥2	𝑡𝑥2	NOUN
cana-578	116	9	+	+	CCONJ
cana-578	116	10	𝑐	𝑐	NOUN
cana-578	116	11	,	,	PUNCT
cana-578	116	12	…	…	PUNCT
cana-578	116	13	}	}	PUNCT
cana-578	116	14	communications	communication	NOUN
cana-578	116	15	on	on	ADP
cana-578	116	16	applied	apply	VERB
cana-578	116	17	nonlinear	nonlinear	ADJ
cana-578	116	18	analysis	analysis	NOUN
cana-578	116	19	issn	issn	NOUN
cana-578	116	20	:	:	PUNCT
cana-578	116	21	1074	1074	NUM
cana-578	116	22	-	-	PUNCT
cana-578	116	23	133x	133x	NUM
cana-578	116	24	vol	vol	NOUN
cana-578	116	25	31	31	NUM
cana-578	116	26	no	no	NOUN
cana-578	116	27	.	.	PUNCT
cana-578	117	1	1s	1s	NUM
cana-578	117	2	(	(	PUNCT
cana-578	117	3	2024	2024	NUM
cana-578	117	4	)	)	PUNCT
cana-578	117	5	192	192	NUM
cana-578	117	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-578	117	7	for	for	ADP
cana-578	117	8	period	period	NOUN
cana-578	117	9	2	2	NUM
cana-578	117	10	points	point	NOUN
cana-578	117	11	𝑥3	𝑥3	NOUN
cana-578	117	12	=	=	SYM
cana-578	117	13	𝑥	𝑥	NOUN
cana-578	117	14	,	,	PUNCT
cana-578	117	15	⇒	⇒	VERB
cana-578	117	16	𝑠𝑥	𝑠𝑥	VERB
cana-578	118	1	+	+	CCONJ
cana-578	118	2	𝑡(𝑠𝑥	𝑡(𝑠𝑥	PROPN
cana-578	118	3	+	+	CCONJ
cana-578	118	4	𝑡𝑥	𝑡𝑥	PROPN
cana-578	118	5	+	+	NOUN
cana-578	118	6	𝑐	𝑐	X
cana-578	118	7	)	)	PUNCT
cana-578	118	8	+	+	NOUN
cana-578	118	9	𝑐	𝑐	NOUN
cana-578	118	10	=	=	SYM
cana-578	118	11	𝑥	𝑥	NOUN
cana-578	118	12	⇒	⇒	NOUN
cana-578	118	13	𝑥(𝑠	𝑥(𝑠	PROPN
cana-578	118	14	+	+	CCONJ
cana-578	118	15	𝑠𝑡	𝑠𝑡	PROPN
cana-578	118	16	+	+	PROPN
cana-578	118	17	𝑡2	𝑡2	PROPN
cana-578	118	18	−	−	PROPN
cana-578	118	19	1	1	NUM
cana-578	118	20	)	)	PUNCT
cana-578	118	21	=	=	PUNCT
cana-578	118	22	−𝑐(1	−𝑐(1	PROPN
cana-578	119	1	+	+	PUNCT
cana-578	119	2	𝑡	𝑡	X
cana-578	119	3	)	)	PUNCT
cana-578	119	4	⇒	⇒	NOUN
cana-578	119	5	𝑥(𝑠(1	𝑥(𝑠(1	ADP
cana-578	119	6	+	+	SYM
cana-578	119	7	𝑡	𝑡	X
cana-578	119	8	)	)	PUNCT
cana-578	119	9	+	+	CCONJ
cana-578	119	10	(	(	PUNCT
cana-578	119	11	1	1	NUM
cana-578	119	12	+	+	CCONJ
cana-578	119	13	𝑡)(𝑡	𝑡)(𝑡	SYM
cana-578	119	14	−	−	PROPN
cana-578	119	15	1	1	NUM
cana-578	119	16	)	)	PUNCT
cana-578	119	17	)	)	PUNCT
cana-578	120	1	=	=	PUNCT
cana-578	120	2	−𝑐(1	−𝑐(1	PROPN
cana-578	121	1	+	+	PUNCT
cana-578	121	2	𝑡	𝑡	X
cana-578	121	3	)	)	PUNCT
cana-578	121	4	⇒	⇒	NOUN
cana-578	121	5	𝑖𝑓	𝑖𝑓	X
cana-578	122	1	(	(	PUNCT
cana-578	122	2	1	1	NUM
cana-578	122	3	+	+	NUM
cana-578	122	4	𝑡	𝑡	NOUN
cana-578	122	5	)	)	PUNCT
cana-578	122	6	≠	≠	PROPN
cana-578	122	7	0	0	NUM
cana-578	122	8	,	,	PUNCT
cana-578	122	9	𝑥(𝑠	𝑥(𝑠	PROPN
cana-578	122	10	+	+	CCONJ
cana-578	122	11	𝑡	𝑡	NUM
cana-578	122	12	−	−	NOUN
cana-578	122	13	1	1	NUM
cana-578	122	14	)	)	PUNCT
cana-578	122	15	=	=	NOUN
cana-578	122	16	−𝑐	−𝑐	PUNCT
cana-578	122	17	⇒	⇒	VERB
cana-578	122	18	𝑥	𝑥	X
cana-578	122	19	=	=	SYM
cana-578	122	20	𝑐	𝑐	PROPN
cana-578	122	21	1	1	NUM
cana-578	122	22	−	−	NOUN
cana-578	122	23	𝑠	𝑠	INTJ
cana-578	122	24	−	−	PROPN
cana-578	122	25	𝑡	𝑡	NOUN
cana-578	122	26	which	which	PRON
cana-578	122	27	is	be	AUX
cana-578	122	28	same	same	ADJ
cana-578	122	29	as	as	ADP
cana-578	122	30	fixed	fix	VERB
cana-578	122	31	point	point	NOUN
cana-578	122	32	so	so	ADV
cana-578	122	33	for	for	ADP
cana-578	122	34	(	(	PUNCT
cana-578	122	35	1	1	NUM
cana-578	122	36	+	+	NUM
cana-578	122	37	𝑡	𝑡	NOUN
cana-578	122	38	)	)	PUNCT
cana-578	122	39	≠	≠	PROPN
cana-578	122	40	0	0	NUM
cana-578	122	41	,	,	PUNCT
cana-578	122	42	then	then	ADV
cana-578	122	43	,	,	PUNCT
cana-578	122	44	the	the	DET
cana-578	122	45	map	map	NOUN
cana-578	122	46	in	in	ADP
cana-578	122	47	(	(	PUNCT
cana-578	122	48	eq	eq	NOUN
cana-578	122	49	5.1	5.1	NUM
cana-578	122	50	)	)	PUNCT
cana-578	122	51	does	do	AUX
cana-578	122	52	n’t	not	PART
cana-578	122	53	have	have	VERB
cana-578	122	54	any	any	DET
cana-578	122	55	periodic	periodic	ADJ
cana-578	122	56	point	point	NOUN
cana-578	122	57	with	with	ADP
cana-578	122	58	period	period	NOUN
cana-578	122	59	2	2	NUM
cana-578	122	60	.	.	PUNCT
cana-578	123	1	if	if	SCONJ
cana-578	123	2	𝑡	𝑡	PROPN
cana-578	123	3	+	+	VERB
cana-578	123	4	1	1	NUM
cana-578	123	5	=	=	SYM
cana-578	123	6	0	0	NUM
cana-578	123	7	⇒	⇒	NOUN
cana-578	123	8	𝑡	𝑡	PROPN
cana-578	123	9	=	=	NOUN
cana-578	123	10	−1	−1	NOUN
cana-578	123	11	then	then	ADV
cana-578	123	12	the	the	DET
cana-578	123	13	orbit	orbit	NOUN
cana-578	123	14	will	will	AUX
cana-578	123	15	be	be	AUX
cana-578	123	16	,	,	PUNCT
cana-578	123	17	𝑂(𝑥	𝑂(𝑥	NOUN
cana-578	123	18	)	)	PUNCT
cana-578	123	19	=	=	PRON
cana-578	123	20	{	{	PUNCT
cana-578	123	21	𝑥0	𝑥0	NOUN
cana-578	123	22	=	=	SYM
cana-578	123	23	𝑥	𝑥	PROPN
cana-578	123	24	,	,	PUNCT
cana-578	123	25	𝑥1	𝑥1	NOUN
cana-578	123	26	=	=	SYM
cana-578	123	27	𝑥	𝑥	PROPN
cana-578	123	28	,	,	PUNCT
cana-578	123	29	𝑥2	𝑥2	NOUN
cana-578	123	30	=	=	SYM
cana-578	123	31	𝑠𝑥	𝑠𝑥	ADP
cana-578	123	32	−	−	PROPN
cana-578	124	1	𝑥	𝑥	PROPN
cana-578	125	1	+	+	SYM
cana-578	125	2	𝑐	𝑐	NOUN
cana-578	125	3	,	,	PUNCT
cana-578	125	4	𝑥3	𝑥3	ADJ
cana-578	125	5	=	=	PUNCT
cana-578	125	6	𝑠𝑥1	𝑠𝑥1	NOUN
cana-578	125	7	−	−	PROPN
cana-578	125	8	𝑥2	𝑥2	NOUN
cana-578	125	9	+	+	CCONJ
cana-578	125	10	𝑐	𝑐	NOUN
cana-578	125	11	,	,	PUNCT
cana-578	125	12	…	…	PUNCT
cana-578	125	13	}	}	PUNCT
cana-578	125	14	⇒	⇒	VERB
cana-578	125	15	𝑥3	𝑥3	NOUN
cana-578	125	16	=	=	SYM
cana-578	125	17	𝑠𝑥	𝑠𝑥	ADP
cana-578	125	18	−	−	PROPN
cana-578	125	19	(	(	PUNCT
cana-578	125	20	𝑠𝑥	𝑠𝑥	ADP
cana-578	125	21	−	−	PROPN
cana-578	125	22	𝑥	𝑥	PROPN
cana-578	125	23	+	+	NUM
cana-578	125	24	𝑐	𝑐	X
cana-578	125	25	)	)	PUNCT
cana-578	125	26	+	+	NOUN
cana-578	125	27	𝑐	𝑐	PROPN
cana-578	125	28	⇒	⇒	X
cana-578	125	29	𝑥3	𝑥3	NOUN
cana-578	125	30	=	=	PUNCT
cana-578	125	31	𝑠𝑥	𝑠𝑥	ADP
cana-578	125	32	−	−	PROPN
cana-578	125	33	𝑠𝑥	𝑠𝑥	INTJ
cana-578	126	1	+	+	CCONJ
cana-578	127	1	𝑥	𝑥	PROPN
cana-578	127	2	−	−	PROPN
cana-578	127	3	𝑐	𝑐	NOUN
cana-578	127	4	−	−	PROPN
cana-578	127	5	𝑐	𝑐	PROPN
cana-578	127	6	⇒	⇒	VERB
cana-578	127	7	𝑥3	𝑥3	NOUN
cana-578	127	8	=	=	PUNCT
cana-578	127	9	𝑥	𝑥	PROPN
cana-578	127	10	(	(	PUNCT
cana-578	127	11	eq	eq	NOUN
cana-578	127	12	5.3	5.3	NUM
cana-578	127	13	)	)	PUNCT
cana-578	127	14	we	we	PRON
cana-578	127	15	can	can	AUX
cana-578	127	16	conclude	conclude	VERB
cana-578	127	17	if	if	SCONJ
cana-578	127	18	(	(	PUNCT
cana-578	127	19	1	1	NUM
cana-578	127	20	+	+	NUM
cana-578	127	21	𝑡	𝑡	NOUN
cana-578	127	22	)	)	PUNCT
cana-578	127	23	=	=	SYM
cana-578	127	24	0	0	PUNCT
cana-578	128	1	then	then	ADV
cana-578	128	2	every	every	DET
cana-578	128	3	point	point	NOUN
cana-578	128	4	is	be	AUX
cana-578	128	5	of	of	ADP
cana-578	128	6	period	period	NOUN
cana-578	128	7	2	2	NUM
cana-578	128	8	.	.	PUNCT
cana-578	128	9	case	case	NOUN
cana-578	128	10	ii	ii	PROPN
cana-578	128	11	(	(	PUNCT
cana-578	128	12	if	if	SCONJ
cana-578	128	13	𝒄	𝒄	PROPN
cana-578	128	14	=	=	SYM
cana-578	128	15	𝟎	𝟎	PROPN
cana-578	128	16	)	)	PUNCT
cana-578	128	17	then	then	ADV
cana-578	128	18	(	(	PUNCT
cana-578	128	19	eq	eq	NOUN
cana-578	128	20	5.1	5.1	NUM
cana-578	128	21	)	)	PUNCT
cana-578	128	22	becomes	become	VERB
cana-578	128	23	,	,	PUNCT
cana-578	128	24	𝑓(𝑥	𝑓(𝑥	NOUN
cana-578	128	25	,	,	PUNCT
cana-578	128	26	𝑦	𝑦	NOUN
cana-578	128	27	)	)	PUNCT
cana-578	128	28	=	=	VERB
cana-578	129	1	𝑠𝑥	𝑠𝑥	ADP
cana-578	129	2	+	+	CCONJ
cana-578	129	3	𝑡𝑦	𝑡𝑦	NOUN
cana-578	129	4	(	(	PUNCT
cana-578	129	5	eq	eq	NOUN
cana-578	129	6	5.4	5.4	NUM
cana-578	129	7	)	)	PUNCT
cana-578	129	8	orbit	orbit	NOUN
cana-578	129	9	of	of	ADP
cana-578	129	10	any	any	DET
cana-578	129	11	point	point	NOUN
cana-578	129	12	𝑂(𝑥	𝑂(𝑥	NUM
cana-578	129	13	)	)	PUNCT
cana-578	129	14	=	=	PRON
cana-578	129	15	{	{	PUNCT
cana-578	129	16	𝑥0	𝑥0	NOUN
cana-578	129	17	=	=	SYM
cana-578	129	18	𝑥	𝑥	PROPN
cana-578	129	19	,	,	PUNCT
cana-578	129	20	𝑥1	𝑥1	NOUN
cana-578	129	21	=	=	SYM
cana-578	129	22	𝑥	𝑥	PROPN
cana-578	129	23	,	,	PUNCT
cana-578	129	24	𝑥2	𝑥2	NOUN
cana-578	129	25	=	=	SYM
cana-578	129	26	𝑓(𝑥	𝑓(𝑥	NOUN
cana-578	129	27	,	,	PUNCT
cana-578	129	28	𝑥	𝑥	NOUN
cana-578	129	29	)	)	PUNCT
cana-578	129	30	,	,	PUNCT
cana-578	129	31	𝑥3	𝑥3	NOUN
cana-578	129	32	=	=	SYM
cana-578	129	33	𝑓(𝑥1	𝑓(𝑥1	ADJ
cana-578	129	34	,	,	PUNCT
cana-578	129	35	𝑥2	𝑥2	NOUN
cana-578	129	36	)	)	PUNCT
cana-578	129	37	…	…	PUNCT
cana-578	129	38	}	}	PUNCT
cana-578	129	39	fixed	fix	VERB
cana-578	129	40	points	point	NOUN
cana-578	129	41	are	be	AUX
cana-578	129	42	𝑥2	𝑥2	NOUN
cana-578	129	43	=	=	SYM
cana-578	129	44	𝑥	𝑥	PROPN
cana-578	129	45	⇒	⇒	NOUN
cana-578	129	46	𝑓(𝑥	𝑓(𝑥	NOUN
cana-578	129	47	,	,	PUNCT
cana-578	129	48	𝑥	𝑥	NOUN
cana-578	129	49	)	)	PUNCT
cana-578	129	50	=	=	SYM
cana-578	130	1	𝑥	𝑥	NOUN
cana-578	130	2	⟹	⟹	PUNCT
cana-578	130	3	𝑠𝑥	𝑠𝑥	VERB
cana-578	130	4	+	+	CCONJ
cana-578	130	5	𝑡𝑥	𝑡𝑥	NOUN
cana-578	130	6	=	=	SYM
cana-578	130	7	𝑥	𝑥	PROPN
cana-578	130	8	⇒	⇒	NOUN
cana-578	130	9	(	(	PUNCT
cana-578	130	10	𝑠	𝑠	PROPN
cana-578	130	11	+	+	X
cana-578	130	12	𝑡)𝑥	𝑡)𝑥	X
cana-578	130	13	=	=	SYM
cana-578	130	14	𝑥	𝑥	PROPN
cana-578	130	15	⇒	⇒	NOUN
cana-578	130	16	(	(	PUNCT
cana-578	130	17	𝑠	𝑠	PROPN
cana-578	130	18	+	+	SYM
cana-578	130	19	𝑡	𝑡	X
cana-578	130	20	)	)	PUNCT
cana-578	130	21	=	=	SYM
cana-578	130	22	1	1	NUM
cana-578	130	23	if	if	SCONJ
cana-578	130	24	𝑠	𝑠	PROPN
cana-578	130	25	+	+	NUM
cana-578	130	26	𝑡	𝑡	X
cana-578	130	27	=	=	NOUN
cana-578	130	28	1	1	NUM
cana-578	130	29	then	then	ADV
cana-578	130	30	every	every	DET
cana-578	130	31	point	point	NOUN
cana-578	130	32	is	be	AUX
cana-578	130	33	fixed	fix	VERB
cana-578	130	34	point	point	NOUN
cana-578	130	35	.	.	PUNCT
cana-578	131	1	(	(	PUNCT
cana-578	131	2	eq	eq	ADP
cana-578	131	3	5.5	5.5	NUM
cana-578	131	4	)	)	PUNCT
cana-578	131	5	for	for	ADP
cana-578	131	6	period	period	NOUN
cana-578	131	7	2	2	NUM
cana-578	131	8	points	point	NOUN
cana-578	131	9	we	we	PRON
cana-578	131	10	have	have	VERB
cana-578	131	11	𝑥3	𝑥3	NOUN
cana-578	131	12	=	=	SYM
cana-578	131	13	𝑥	𝑥	PROPN
cana-578	131	14	⇒	⇒	NOUN
cana-578	131	15	𝑓(𝑥	𝑓(𝑥	NOUN
cana-578	131	16	,	,	PUNCT
cana-578	131	17	𝑠𝑥	𝑠𝑥	ADP
cana-578	131	18	+	+	CCONJ
cana-578	131	19	𝑡𝑥	𝑡𝑥	NOUN
cana-578	131	20	)	)	PUNCT
cana-578	132	1	=	=	SYM
cana-578	132	2	𝑥	𝑥	PROPN
cana-578	132	3	⇒	⇒	NOUN
cana-578	132	4	𝑠𝑥	𝑠𝑥	VERB
cana-578	132	5	+	+	CCONJ
cana-578	132	6	𝑡(𝑠𝑥	𝑡(𝑠𝑥	PROPN
cana-578	132	7	+	+	CCONJ
cana-578	132	8	𝑡𝑥	𝑡𝑥	NOUN
cana-578	132	9	)	)	PUNCT
cana-578	132	10	=	=	SYM
cana-578	133	1	𝑥	𝑥	PROPN
cana-578	133	2	⇒	⇒	NOUN
cana-578	133	3	𝑥	𝑥	PROPN
cana-578	133	4	≠	≠	PROPN
cana-578	133	5	0	0	NUM
cana-578	133	6	,	,	PUNCT
cana-578	133	7	𝑠	𝑠	PROPN
cana-578	133	8	+	+	NUM
cana-578	133	9	𝑡	𝑡	PROPN
cana-578	133	10	≠	≠	PROPN
cana-578	133	11	1	1	NUM
cana-578	133	12	,	,	PUNCT
cana-578	133	13	𝑠	𝑠	PROPN
cana-578	133	14	+	+	CCONJ
cana-578	133	15	𝑠𝑡	𝑠𝑡	PROPN
cana-578	133	16	+	+	CCONJ
cana-578	133	17	𝑡2	𝑡2	ADJ
cana-578	133	18	=	=	SYM
cana-578	133	19	1	1	NUM
cana-578	133	20	⇒	⇒	X
cana-578	133	21	𝑠(1	𝑠(1	PROPN
cana-578	133	22	+	+	CCONJ
cana-578	133	23	𝑡	𝑡	NOUN
cana-578	133	24	)	)	PUNCT
cana-578	133	25	+	+	CCONJ
cana-578	133	26	(	(	PUNCT
cana-578	133	27	1	1	NUM
cana-578	133	28	+	+	CCONJ
cana-578	133	29	𝑡)(𝑡	𝑡)(𝑡	SYM
cana-578	133	30	−	−	PROPN
cana-578	133	31	1	1	NUM
cana-578	133	32	)	)	PUNCT
cana-578	133	33	=	=	SYM
cana-578	133	34	0	0	NUM
cana-578	133	35	⇒	⇒	NOUN
cana-578	133	36	(	(	PUNCT
cana-578	133	37	1	1	NUM
cana-578	133	38	+	+	NUM
cana-578	133	39	𝑡)(𝑠	𝑡)(𝑠	PROPN
cana-578	133	40	+	+	NUM
cana-578	133	41	𝑡	𝑡	NUM
cana-578	133	42	−	−	NOUN
cana-578	133	43	1	1	NUM
cana-578	133	44	)	)	PUNCT
cana-578	133	45	=	=	SYM
cana-578	133	46	0	0	PUNCT
cana-578	134	1	as	as	SCONJ
cana-578	134	2	we	we	PRON
cana-578	134	3	have	have	AUX
cana-578	134	4	assumed	assume	VERB
cana-578	134	5	𝑠	𝑠	PROPN
cana-578	134	6	+	+	PROPN
cana-578	134	7	𝑡	𝑡	PROPN
cana-578	134	8	−	−	PROPN
cana-578	134	9	1	1	NUM
cana-578	134	10	≠	≠	PROPN
cana-578	134	11	0	0	NUM
cana-578	134	12	⇒	⇒	NOUN
cana-578	134	13	1	1	NUM
cana-578	135	1	+	+	CCONJ
cana-578	135	2	𝑡	𝑡	PROPN
cana-578	135	3	=	=	SYM
cana-578	135	4	0	0	NUM
cana-578	135	5	⇒	⇒	PROPN
cana-578	135	6	𝑡	𝑡	PROPN
cana-578	135	7	=	=	NOUN
cana-578	135	8	−1	−1	NOUN
cana-578	135	9	for	for	ADP
cana-578	135	10	𝑠	𝑠	PROPN
cana-578	135	11	+	+	CCONJ
cana-578	135	12	𝑡	𝑡	PROPN
cana-578	135	13	−	−	PROPN
cana-578	135	14	1	1	NUM
cana-578	135	15	≠	≠	PROPN
cana-578	135	16	0	0	NUM
cana-578	135	17	and	and	CCONJ
cana-578	135	18	𝑡	𝑡	PROPN
cana-578	135	19	=	=	NOUN
cana-578	135	20	−1	−1	NOUN
cana-578	135	21	every	every	DET
cana-578	135	22	point	point	NOUN
cana-578	135	23	is	be	AUX
cana-578	135	24	of	of	ADP
cana-578	135	25	period	period	NOUN
cana-578	135	26	2	2	NUM
cana-578	135	27	.	.	PUNCT
cana-578	136	1	(	(	PUNCT
cana-578	136	2	eq	eq	NOUN
cana-578	136	3	5.6	5.6	NUM
cana-578	136	4	)	)	PUNCT
cana-578	136	5	.	.	PUNCT
cana-578	137	1	6	6	X
cana-578	137	2	.	.	X
cana-578	137	3	problems	problem	NOUN
cana-578	137	4	problem	problem	VERB
cana-578	137	5	6.1	6.1	NUM
cana-578	137	6	for	for	ADP
cana-578	137	7	the	the	DET
cana-578	137	8	map	map	NOUN
cana-578	137	9	𝑓	𝑓	X
cana-578	137	10	:	:	PUNCT
cana-578	137	11	ℝ	ℝ	PROPN
cana-578	137	12	×	×	NOUN
cana-578	137	13	ℝ	ℝ	PROPN
cana-578	137	14	→	→	SYM
cana-578	137	15	ℝ	ℝ	PROPN
cana-578	137	16	defined	define	VERB
cana-578	137	17	as	as	ADP
cana-578	137	18	𝑓(𝑥	𝑓(𝑥	NOUN
cana-578	137	19	,	,	PUNCT
cana-578	137	20	𝑦	𝑦	NOUN
cana-578	137	21	)	)	PUNCT
cana-578	137	22	=	=	SYM
cana-578	137	23	2𝑥	2𝑥	NOUN
cana-578	137	24	−	−	NOUN
cana-578	138	1	3𝑦	3𝑦	NOUN
cana-578	139	1	+	+	CCONJ
cana-578	139	2	4	4	NUM
cana-578	139	3	which	which	PRON
cana-578	139	4	is	be	AUX
cana-578	139	5	continuous	continuous	ADJ
cana-578	139	6	so	so	SCONJ
cana-578	139	7	it	it	PRON
cana-578	139	8	is	be	AUX
cana-578	139	9	a	a	DET
cana-578	139	10	general	general	ADJ
cana-578	139	11	system	system	NOUN
cana-578	139	12	.	.	PUNCT
cana-578	140	1	communications	communication	NOUN
cana-578	140	2	on	on	ADP
cana-578	140	3	applied	apply	VERB
cana-578	140	4	nonlinear	nonlinear	ADJ
cana-578	140	5	analysis	analysis	NOUN
cana-578	140	6	issn	issn	NOUN
cana-578	140	7	:	:	PUNCT
cana-578	140	8	1074	1074	NUM
cana-578	140	9	-	-	PUNCT
cana-578	140	10	133x	133x	NUM
cana-578	140	11	vol	vol	NOUN
cana-578	140	12	31	31	NUM
cana-578	140	13	no	no	NOUN
cana-578	140	14	.	.	PUNCT
cana-578	141	1	1s	1s	NUM
cana-578	141	2	(	(	PUNCT
cana-578	141	3	2024	2024	NUM
cana-578	141	4	)	)	PUNCT
cana-578	141	5	193	193	NUM
cana-578	141	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-578	141	7	by	by	ADP
cana-578	141	8	(	(	PUNCT
cana-578	141	9	eq	eq	ADP
cana-578	141	10	5.2	5.2	NUM
cana-578	141	11	)	)	PUNCT
cana-578	141	12	fixed	fix	VERB
cana-578	141	13	points	point	NOUN
cana-578	141	14	for	for	ADP
cana-578	141	15	the	the	DET
cana-578	141	16	map	map	NOUN
cana-578	141	17	is	be	AUX
cana-578	141	18	𝑥	𝑥	NOUN
cana-578	141	19	=	=	SYM
cana-578	141	20	4	4	NUM
cana-578	141	21	1−2−(−3	1−2−(−3	NUM
cana-578	141	22	)	)	PUNCT
cana-578	141	23	=	=	SYM
cana-578	142	1	2	2	X
cana-578	142	2	.	.	PUNCT
cana-578	142	3	by	by	ADP
cana-578	142	4	(	(	PUNCT
cana-578	142	5	eq	eq	NOUN
cana-578	142	6	5.3	5.3	NUM
cana-578	142	7	)	)	PUNCT
cana-578	142	8	it	it	PRON
cana-578	142	9	does	do	AUX
cana-578	142	10	n’t	not	PART
cana-578	142	11	have	have	VERB
cana-578	142	12	any	any	DET
cana-578	142	13	periodic	periodic	ADJ
cana-578	142	14	points	point	NOUN
cana-578	142	15	of	of	ADP
cana-578	142	16	period	period	NOUN
cana-578	142	17	2	2	NUM
cana-578	142	18	as	as	ADP
cana-578	142	19	1	1	NUM
cana-578	142	20	+	+	NUM
cana-578	142	21	𝑡	𝑡	NOUN
cana-578	142	22	≠	≠	PROPN
cana-578	142	23	0	0	NUM
cana-578	142	24	.	.	PUNCT
cana-578	142	25	problem	problem	NOUN
cana-578	142	26	6.2	6.2	NUM
cana-578	142	27	for	for	ADP
cana-578	142	28	the	the	DET
cana-578	142	29	map	map	NOUN
cana-578	142	30	𝑓	𝑓	X
cana-578	143	1	:	:	PUNCT
cana-578	143	2	ℝ	ℝ	PROPN
cana-578	143	3	×	×	NOUN
cana-578	143	4	ℝ	ℝ	PROPN
cana-578	143	5	→	→	SYM
cana-578	143	6	ℝ	ℝ	PROPN
cana-578	143	7	defined	define	VERB
cana-578	143	8	as	as	ADP
cana-578	143	9	𝑓(𝑥	𝑓(𝑥	NOUN
cana-578	143	10	,	,	PUNCT
cana-578	143	11	𝑦	𝑦	NOUN
cana-578	143	12	)	)	PUNCT
cana-578	143	13	=	=	SYM
cana-578	144	1	𝑥	𝑥	DET
cana-578	144	2	−	−	NUM
cana-578	144	3	𝑦	𝑦	NOUN
cana-578	144	4	+	+	CCONJ
cana-578	144	5	9	9	NUM
cana-578	144	6	which	which	PRON
cana-578	144	7	is	be	AUX
cana-578	144	8	continuous	continuous	ADJ
cana-578	144	9	so	so	SCONJ
cana-578	144	10	it	it	PRON
cana-578	144	11	is	be	AUX
cana-578	144	12	a	a	DET
cana-578	144	13	general	general	ADJ
cana-578	144	14	system	system	NOUN
cana-578	144	15	.	.	PUNCT
cana-578	145	1	by	by	ADP
cana-578	145	2	(	(	PUNCT
cana-578	145	3	eq	eq	NOUN
cana-578	145	4	5.2	5.2	NUM
cana-578	145	5	)	)	PUNCT
cana-578	145	6	fixed	fix	VERB
cana-578	145	7	points	point	NOUN
cana-578	145	8	for	for	ADP
cana-578	145	9	the	the	DET
cana-578	145	10	map	map	NOUN
cana-578	145	11	is	be	AUX
cana-578	145	12	𝑥	𝑥	NOUN
cana-578	145	13	=	=	SYM
cana-578	145	14	9	9	NUM
cana-578	145	15	1−1−(−1	1−1−(−1	NUM
cana-578	145	16	)	)	PUNCT
cana-578	146	1	=	=	SYM
cana-578	146	2	9	9	X
cana-578	146	3	.	.	PUNCT
cana-578	147	1	by	by	ADP
cana-578	147	2	(	(	PUNCT
cana-578	147	3	eq	eq	NOUN
cana-578	147	4	5.3	5.3	NUM
cana-578	147	5	)	)	PUNCT
cana-578	147	6	1	1	NUM
cana-578	147	7	+	+	CCONJ
cana-578	147	8	𝑡	𝑡	X
cana-578	147	9	=	=	SYM
cana-578	147	10	0	0	PUNCT
cana-578	148	1	so	so	ADV
cana-578	148	2	every	every	DET
cana-578	148	3	point	point	NOUN
cana-578	148	4	is	be	AUX
cana-578	148	5	of	of	ADP
cana-578	148	6	period	period	NOUN
cana-578	148	7	2	2	NUM
cana-578	148	8	except	except	SCONJ
cana-578	148	9	fixed	fix	VERB
cana-578	148	10	points	point	NOUN
cana-578	148	11	.	.	PUNCT
cana-578	149	1	problem	problem	NOUN
cana-578	149	2	6.3	6.3	NUM
cana-578	149	3	for	for	ADP
cana-578	149	4	the	the	DET
cana-578	149	5	map	map	NOUN
cana-578	149	6	𝑓	𝑓	X
cana-578	149	7	:	:	PUNCT
cana-578	149	8	ℝ	ℝ	PROPN
cana-578	149	9	×	×	NOUN
cana-578	149	10	ℝ	ℝ	PROPN
cana-578	149	11	→	→	SYM
cana-578	149	12	ℝ	ℝ	PROPN
cana-578	149	13	defined	define	VERB
cana-578	149	14	as	as	ADP
cana-578	149	15	𝑓(𝑥	𝑓(𝑥	NOUN
cana-578	149	16	,	,	PUNCT
cana-578	149	17	𝑦	𝑦	NOUN
cana-578	149	18	)	)	PUNCT
cana-578	149	19	=	=	SYM
cana-578	149	20	2𝑥	2𝑥	NUM
cana-578	150	1	−	−	PROPN
cana-578	150	2	𝑦	𝑦	NOUN
cana-578	150	3	which	which	PRON
cana-578	150	4	is	be	AUX
cana-578	150	5	continuous	continuous	ADJ
cana-578	150	6	so	so	SCONJ
cana-578	150	7	it	it	PRON
cana-578	150	8	is	be	AUX
cana-578	150	9	a	a	DET
cana-578	150	10	general	general	ADJ
cana-578	150	11	system	system	NOUN
cana-578	150	12	.	.	PUNCT
cana-578	151	1	by	by	ADP
cana-578	151	2	(	(	PUNCT
cana-578	151	3	eq	eq	ADP
cana-578	151	4	5.5	5.5	NUM
cana-578	151	5	)	)	PUNCT
cana-578	151	6	every	every	DET
cana-578	151	7	point	point	NOUN
cana-578	151	8	is	be	AUX
cana-578	151	9	fixed	fix	VERB
cana-578	151	10	point	point	NOUN
cana-578	151	11	as	as	ADP
cana-578	151	12	𝑠	𝑠	PROPN
cana-578	151	13	+	+	CCONJ
cana-578	151	14	𝑡	𝑡	X
cana-578	151	15	=	=	NOUN
cana-578	151	16	1	1	X
cana-578	151	17	.	.	PUNCT
cana-578	151	18	by	by	ADP
cana-578	151	19	(	(	PUNCT
cana-578	151	20	eq	eq	NOUN
cana-578	151	21	5.3	5.3	NUM
cana-578	151	22	)	)	PUNCT
cana-578	151	23	every	every	DET
cana-578	151	24	point	point	NOUN
cana-578	151	25	is	be	AUX
cana-578	151	26	of	of	ADP
cana-578	151	27	period	period	NOUN
cana-578	151	28	2	2	NUM
cana-578	151	29	except	except	SCONJ
cana-578	151	30	fixed	fix	VERB
cana-578	151	31	points	point	NOUN
cana-578	151	32	as	as	ADP
cana-578	151	33	𝑡	𝑡	X
cana-578	151	34	=	=	NOUN
cana-578	151	35	−1	−1	NOUN
cana-578	151	36	.	.	PUNCT
cana-578	152	1	7	7	X
cana-578	152	2	.	.	X
cana-578	152	3	conclusion	conclusion	NOUN
cana-578	152	4	the	the	DET
cana-578	152	5	major	major	ADJ
cana-578	152	6	part	part	NOUN
cana-578	152	7	of	of	ADP
cana-578	152	8	the	the	DET
cana-578	152	9	fundamental	fundamental	ADJ
cana-578	152	10	dynamical	dynamical	ADJ
cana-578	152	11	system	system	NOUN
cana-578	152	12	is	be	AUX
cana-578	152	13	still	still	ADV
cana-578	152	14	unknown	unknown	ADJ
cana-578	152	15	in	in	ADP
cana-578	152	16	the	the	DET
cana-578	152	17	generalized	generalized	ADJ
cana-578	152	18	system	system	NOUN
cana-578	152	19	and	and	CCONJ
cana-578	152	20	m	m	NOUN
cana-578	152	21	-	-	PUNCT
cana-578	152	22	step	step	NOUN
cana-578	152	23	systems	system	NOUN
cana-578	152	24	,	,	PUNCT
cana-578	152	25	which	which	PRON
cana-578	152	26	are	be	AUX
cana-578	152	27	novel	novel	ADJ
cana-578	152	28	concepts	concept	NOUN
cana-578	152	29	.	.	PUNCT
cana-578	153	1	we	we	PRON
cana-578	153	2	developed	develop	VERB
cana-578	153	3	relationships	relationship	NOUN
cana-578	153	4	between	between	ADP
cana-578	153	5	periodic	periodic	ADJ
cana-578	153	6	concepts	concept	NOUN
cana-578	153	7	such	such	ADJ
cana-578	153	8	as	as	ADP
cana-578	153	9	fixed	fix	VERB
cana-578	153	10	points	point	NOUN
cana-578	153	11	,	,	PUNCT
cana-578	153	12	recurrent	recurrent	ADJ
cana-578	153	13	points	point	NOUN
cana-578	153	14	,	,	PUNCT
cana-578	153	15	and	and	CCONJ
cana-578	153	16	non	non	ADJ
cana-578	153	17	-	-	ADJ
cana-578	153	18	wandering	wandering	ADJ
cana-578	153	19	points	point	NOUN
cana-578	153	20	in	in	ADP
cana-578	153	21	this	this	DET
cana-578	153	22	study	study	NOUN
cana-578	153	23	.	.	PUNCT
cana-578	154	1	additionally	additionally	ADV
cana-578	154	2	,	,	PUNCT
cana-578	154	3	we	we	PRON
cana-578	154	4	looked	look	VERB
cana-578	154	5	at	at	ADP
cana-578	154	6	the	the	DET
cana-578	154	7	theorems	theorem	NOUN
cana-578	154	8	connected	connect	VERB
cana-578	154	9	to	to	ADP
cana-578	154	10	transitivity	transitivity	NOUN
cana-578	154	11	and	and	CCONJ
cana-578	154	12	,	,	PUNCT
cana-578	154	13	finally	finally	ADV
cana-578	154	14	,	,	PUNCT
cana-578	154	15	were	be	AUX
cana-578	154	16	provided	provide	VERB
cana-578	154	17	with	with	ADP
cana-578	154	18	examples	example	NOUN
cana-578	154	19	.	.	PUNCT
cana-578	155	1	references	reference	NOUN
cana-578	155	2	[	[	X
cana-578	155	3	1	1	NUM
cana-578	155	4	]	]	X
cana-578	155	5	akbar	akbar	NOUN
cana-578	155	6	,	,	PUNCT
cana-578	155	7	k.a	k.a	PROPN
cana-578	155	8	.	.	PROPN
cana-578	155	9	,	,	PUNCT
cana-578	155	10	kannan	kannan	PROPN
cana-578	155	11	,	,	PUNCT
cana-578	155	12	v.	v.	PROPN
cana-578	155	13	,	,	PUNCT
cana-578	155	14	gopal	gopal	PROPN
cana-578	155	15	,	,	PUNCT
cana-578	155	16	s.	s.	PROPN
cana-578	155	17	,	,	PUNCT
cana-578	155	18	&	&	CCONJ
cana-578	155	19	chiranjeevi	chiranjeevi	PROPN
cana-578	155	20	,	,	PUNCT
cana-578	155	21	p.	p.	NOUN
cana-578	155	22	(	(	PUNCT
cana-578	155	23	2009	2009	NUM
cana-578	155	24	)	)	PUNCT
cana-578	155	25	.	.	PUNCT
cana-578	156	1	the	the	DET
cana-578	156	2	set	set	NOUN
cana-578	156	3	of	of	ADP
cana-578	156	4	periods	period	NOUN
cana-578	156	5	of	of	ADP
cana-578	156	6	periodic	periodic	ADJ
cana-578	156	7	points	point	NOUN
cana-578	156	8	of	of	ADP
cana-578	156	9	a	a	DET
cana-578	156	10	linear	linear	ADJ
cana-578	156	11	operator	operator	NOUN
cana-578	156	12	.	.	PUNCT
cana-578	157	1	linear	linear	PROPN
cana-578	157	2	algebra	algebra	NOUN
cana-578	157	3	and	and	CCONJ
cana-578	157	4	its	its	PRON
cana-578	157	5	applications	application	NOUN
cana-578	157	6	,	,	PUNCT
cana-578	157	7	431(1	431(1	NUM
cana-578	157	8	-	-	SYM
cana-578	157	9	2	2	NUM
cana-578	157	10	)	)	PUNCT
cana-578	157	11	,	,	PUNCT
cana-578	157	12	241	241	PROPN
cana-578	157	13	-	-	SYM
cana-578	157	14	246	246	NUM
cana-578	157	15	.	.	PUNCT
cana-578	158	1	https://doi.org/10.1016/j.laa.2009.02.027	https://doi.org/10.1016/j.laa.2009.02.027	PROPN
cana-578	158	2	[	[	X
cana-578	158	3	2	2	NUM
cana-578	158	4	]	]	PUNCT
cana-578	158	5	chiranjeevi	chiranjeevi	NOUN
cana-578	158	6	,	,	PUNCT
cana-578	158	7	p.	p.	PROPN
cana-578	158	8	,	,	PUNCT
cana-578	158	9	kannan	kannan	PROPN
cana-578	158	10	,	,	PUNCT
cana-578	158	11	v.	v.	PROPN
cana-578	158	12	,	,	PUNCT
cana-578	158	13	&	&	CCONJ
cana-578	158	14	sharan	sharan	PROPN
cana-578	158	15	,	,	PUNCT
cana-578	158	16	g.	g.	PROPN
cana-578	158	17	(	(	PUNCT
cana-578	158	18	2013	2013	NUM
cana-578	158	19	)	)	PUNCT
cana-578	158	20	.	.	PUNCT
cana-578	159	1	periodic	periodic	ADJ
cana-578	159	2	points	point	NOUN
cana-578	159	3	and	and	CCONJ
cana-578	159	4	periods	period	NOUN
cana-578	159	5	for	for	ADP
cana-578	159	6	operators	operator	NOUN
cana-578	159	7	on	on	ADP
cana-578	159	8	hilbert	hilbert	NOUN
cana-578	159	9	space	space	NOUN
cana-578	159	10	.	.	PUNCT
cana-578	160	1	discrete	discrete	ADJ
cana-578	160	2	and	and	CCONJ
cana-578	160	3	continuous	continuous	ADJ
cana-578	160	4	dynamical	dynamical	ADJ
cana-578	160	5	systems	system	NOUN
cana-578	160	6	,	,	PUNCT
cana-578	160	7	33(9	33(9	NUM
cana-578	160	8	)	)	PUNCT
cana-578	160	9	,	,	PUNCT
cana-578	160	10	4233	4233	NUM
cana-578	160	11	-	-	SYM
cana-578	160	12	4237	4237	NUM
cana-578	160	13	.	.	PUNCT
cana-578	161	1	[	[	X
cana-578	161	2	3	3	X
cana-578	161	3	]	]	X
cana-578	161	4	kumar	kumar	PROPN
cana-578	161	5	,	,	PUNCT
cana-578	161	6	l.p	l.p	PROPN
cana-578	161	7	.	.	PROPN
cana-578	161	8	,	,	PUNCT
cana-578	161	9	&	&	CCONJ
cana-578	161	10	kumar	kumar	PROPN
cana-578	161	11	,	,	PUNCT
cana-578	161	12	v.s.	v.s.	X
cana-578	161	13	(	(	PUNCT
cana-578	161	14	2023	2023	NUM
cana-578	161	15	)	)	PUNCT
cana-578	161	16	.	.	PUNCT
cana-578	161	17	periods	period	NOUN
cana-578	161	18	and	and	CCONJ
cana-578	161	19	periodic	periodic	ADJ
cana-578	161	20	points	point	NOUN
cana-578	161	21	of	of	ADP
cana-578	161	22	linear	linear	ADJ
cana-578	161	23	cellular	cellular	ADJ
cana-578	161	24	automata	automata	NOUN
cana-578	161	25	.	.	PUNCT
cana-578	162	1	the	the	DET
cana-578	162	2	scientific	scientific	ADJ
cana-578	162	3	temper	temper	NOUN
cana-578	162	4	,	,	PUNCT
cana-578	162	5	14(03	14(03	PROPN
cana-578	162	6	)	)	PUNCT
cana-578	162	7	,	,	PUNCT
cana-578	162	8	1020–1023	1020–1023	NUM
cana-578	162	9	.	.	PUNCT
cana-578	162	10	https://doi.org/10.58414/scientifictemper.2023.14.3.71	https://doi.org/10.58414/scientifictemper.2023.14.3.71	NOUN
cana-578	163	1	[	[	X
cana-578	163	2	4	4	NUM
cana-578	163	3	]	]	X
cana-578	163	4	mihail	mihail	NOUN
cana-578	163	5	,	,	PUNCT
cana-578	163	6	a.	a.	NOUN
cana-578	163	7	(	(	PUNCT
cana-578	163	8	2008	2008	NUM
cana-578	163	9	)	)	PUNCT
cana-578	163	10	.	.	PUNCT
cana-578	164	1	recurrent	recurrent	ADJ
cana-578	164	2	iterated	iterate	VERB
cana-578	164	3	function	function	NOUN
cana-578	164	4	systems	system	NOUN
cana-578	164	5	.	.	PUNCT
cana-578	165	1	revue	revue	PROPN
cana-578	165	2	roumaine	roumaine	NOUN
cana-578	165	3	de	de	X
cana-578	165	4	mathematiques	mathematique	NOUN
cana-578	165	5	pures	pure	NOUN
cana-578	165	6	et	et	PROPN
cana-578	165	7	appliquees	appliquee	NOUN
cana-578	165	8	,	,	PUNCT
cana-578	165	9	53(1	53(1	NOUN
cana-578	165	10	)	)	PUNCT
cana-578	165	11	,	,	PUNCT
cana-578	165	12	43	43	NUM
cana-578	165	13	-	-	SYM
cana-578	165	14	54	54	NUM
cana-578	165	15	.	.	PUNCT
cana-578	166	1	[	[	X
cana-578	166	2	5	5	NUM
cana-578	166	3	]	]	X
cana-578	166	4	dumitru	dumitru	X
cana-578	166	5	,	,	PUNCT
cana-578	166	6	d.	d.	PROPN
cana-578	166	7	,	,	PUNCT
cana-578	166	8	ioana	ioana	PROPN
cana-578	166	9	,	,	PUNCT
cana-578	166	10	l.	l.	PROPN
cana-578	166	11	,	,	PUNCT
cana-578	166	12	sfetcu	sfetcu	ADJ
cana-578	166	13	,	,	PUNCT
cana-578	166	14	r.c	r.c	PROPN
cana-578	166	15	.	.	PROPN
cana-578	166	16	,	,	PUNCT
cana-578	166	17	&	&	CCONJ
cana-578	166	18	strobin	strobin	PROPN
cana-578	166	19	,	,	PUNCT
cana-578	166	20	f.	f.	PROPN
cana-578	166	21	(	(	PUNCT
cana-578	166	22	2015	2015	NUM
cana-578	166	23	)	)	PUNCT
cana-578	166	24	.	.	PUNCT
cana-578	167	1	topological	topological	ADJ
cana-578	167	2	version	version	NOUN
cana-578	167	3	of	of	ADP
cana-578	167	4	generalized	generalized	ADJ
cana-578	167	5	(	(	PUNCT
cana-578	167	6	infinite	infinite	NOUN
cana-578	167	7	)	)	PUNCT
cana-578	167	8	iterated	iterate	VERB
cana-578	167	9	function	function	NOUN
cana-578	167	10	systems	system	NOUN
cana-578	167	11	.	.	PUNCT
cana-578	168	1	chaos	chaos	NOUN
cana-578	168	2	,	,	PUNCT
cana-578	168	3	solitons	soliton	NOUN
cana-578	168	4	&	&	CCONJ
cana-578	168	5	fractals	fractal	NOUN
cana-578	168	6	,	,	PUNCT
cana-578	168	7	71	71	NUM
cana-578	168	8	,	,	PUNCT
cana-578	168	9	78	78	NUM
cana-578	168	10	-	-	SYM
cana-578	168	11	90	90	NUM
cana-578	168	12	.	.	PUNCT
cana-578	169	1	https://doi.org/10.1016/j.chaos.2014.12.005	https://doi.org/10.1016/j.chaos.2014.12.005	PROPN
cana-578	169	2	.	.	PUNCT
cana-578	170	1	[	[	X
cana-578	170	2	6	6	NUM
cana-578	170	3	]	]	SYM
cana-578	170	4	nia	nia	PROPN
cana-578	170	5	,	,	PUNCT
cana-578	170	6	m.f	m.f	PROPN
cana-578	170	7	.	.	PROPN
cana-578	170	8	,	,	PUNCT
cana-578	170	9	&	&	CCONJ
cana-578	170	10	bahabadi	bahabadi	PROPN
cana-578	170	11	,	,	PUNCT
cana-578	170	12	a.z	a.z	PROPN
cana-578	170	13	.	.	PROPN
cana-578	170	14	(	(	PUNCT
cana-578	170	15	2022	2022	NUM
cana-578	170	16	)	)	PUNCT
cana-578	170	17	.	.	PUNCT
cana-578	171	1	chaos	chaos	NOUN
cana-578	171	2	and	and	CCONJ
cana-578	171	3	shadowing	shadow	VERB
cana-578	171	4	in	in	ADP
cana-578	171	5	general	general	ADJ
cana-578	171	6	systems	system	NOUN
cana-578	171	7	.	.	PUNCT
cana-578	172	1	kragujevac	kragujevac	PROPN
cana-578	172	2	journal	journal	PROPN
cana-578	172	3	of	of	ADP
cana-578	172	4	mathematics	mathematics	PROPN
cana-578	172	5	,	,	PUNCT
cana-578	172	6	46(3	46(3	NOUN
cana-578	172	7	)	)	PUNCT
cana-578	172	8	,	,	PUNCT
cana-578	172	9	383	383	NUM
cana-578	172	10	-	-	SYM
cana-578	172	11	394	394	NUM
cana-578	172	12	.	.	PUNCT
cana-578	173	1	[	[	X
cana-578	173	2	7	7	NUM
cana-578	173	3	]	]	X
cana-578	173	4	secelean	secelean	ADJ
cana-578	173	5	,	,	PUNCT
cana-578	173	6	n.a	n.a	PROPN
cana-578	173	7	.	.	PROPN
cana-578	173	8	(	(	PUNCT
cana-578	173	9	2015	2015	NUM
cana-578	173	10	)	)	PUNCT
cana-578	173	11	.	.	PUNCT
cana-578	174	1	generalized	generalize	VERB
cana-578	174	2	f	f	X
cana-578	174	3	-	-	PUNCT
cana-578	174	4	iterated	iterate	VERB
cana-578	174	5	function	function	NOUN
cana-578	174	6	systems	system	NOUN
cana-578	174	7	on	on	ADP
cana-578	174	8	product	product	NOUN
cana-578	174	9	of	of	ADP
cana-578	174	10	metric	metric	ADJ
cana-578	174	11	spaces	space	NOUN
cana-578	174	12	.	.	PUNCT
cana-578	175	1	journal	journal	NOUN
cana-578	175	2	of	of	ADP
cana-578	175	3	fixed	fix	VERB
cana-578	175	4	point	point	NOUN
cana-578	175	5	theory	theory	NOUN
cana-578	175	6	and	and	CCONJ
cana-578	175	7	applications	application	NOUN
cana-578	175	8	,	,	PUNCT
cana-578	175	9	17	17	NUM
cana-578	175	10	,	,	PUNCT
cana-578	175	11	575	575	NUM
cana-578	175	12	-	-	SYM
cana-578	175	13	595	595	NUM
cana-578	175	14	.	.	PUNCT
cana-578	176	1	https://doi.org/10.1007/s11784-015-0235-2	https://doi.org/10.1007/s11784-015-0235-2	NUM
cana-578	176	2	https://doi.org/10.1016/j.chaos.2014.12.005	https://doi.org/10.1016/j.chaos.2014.12.005	NUM