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Chapter 7Input–Output Mechanism of the DiscreteChaos Extension

Marat Akhmet and Mehmet Onur Fen

Abstract In this chapter the extension of chaos in difference equations is discussed.The theoretical results are based on chaos in the sense of Devaney and period-doubling cascades. The existence of homoclinic and heteroclinic orbits is rigorouslyproved, and a theoretical control technique for the extended chaos is proposed.The results are supported with the aid of simulations. Arbitrarily high-dimensionalchaotic discrete-time dynamical systems can be designed by means of the presentedtechnique. A discrete gonorrhea model is utilized to generate chaotic behavior inpopulation dynamics.

7.1 Introduction

Chaotic dynamics in discrete-time systems have been widely investigated in theliterature [22, 23, 25, 36, 38, 40, 42, 54]. As a mathematical notion, the termchaos was first used by Li and Yorke [38] for one-dimensional (1D) differenceequations. Another definition of chaos was introduced by Devaney [23]. The period-doubling cascade, which was first observed in quadratic maps by Myrberg [41], isthe most prominent among the discovered routes to chaos [28]. It is known thatperiod-doubling onset of chaos exhibits universal behavior [26]. An example of adiffeomorphism that is structurally stable and possesses a chaotic invariant set is theSmale horseshoe map [23, 35, 39, 51, 55]. The horseshoe map is prominent becauseof its use in the recognition of chaotic dynamics and can arise both in discrete andcontinuous cases [19, 24, 29, 31].

Consider the discrete equation unC1 D LŒun� C hn, where L is a linear operatorwith spectra inside the unit circle in the complex plane. If the sequence fhng isconsidered as an input with a certain property such as boundedness, periodicity, or

M. Akhmet (�)Department of Mathematics, Middle East Technical University, 06800 Ankara, Turkeye-mail: [email protected]

M.O. FenDepartment of Mathematics, Middle East Technical University, 06800 Ankara, Turkeye-mail: [email protected]

© Springer International Publishing Switzerland 2016V. Afraimovich et al. (eds.), Complex Motions and Chaos in Nonlinear Systems,Nonlinear Systems and Complexity 15, DOI 10.1007/978-3-319-28764-5_7

203

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204 M. Akhmet and M.O. Fen

almost periodicity, then the discrete equation produces a solution, output, with asimilar feature, boundedness/periodicity/almost periodicity [21, 37]. Motivated bythis fact, in this study we take into account the problem of whether chaotic inputsgenerate chaotic outputs.

Throughout the chapter, Z, N, and R will denote the sets of integers, naturalnumbers, and real numbers, respectively. Moreover, we will make use of the usualEuclidean norm for vectors and the norm induced by the Euclidean norm formatrices [33].

In this chapter, we consider the discrete equations

xnC1 D F.xn/ (7.1)

and

ynC1 D Ayn C g.xn; yn/; (7.2)

where n 2 Z; A is a nonsingular, constant q�q real-valued matrix, and the functionsF W � ! � and g W � � R

q ! Rq are continuous in all their arguments, and �

is a compact subset of Rp. We will rigorously prove that if Eq. (7.1) is chaotic, thenEq. (7.2) is also chaotic. Our results are based on chaos in the sense of Devaney [23]and the one through a period-doubling cascade [26, 41, 47, 48].

Chaos extension problems for continuous-time dynamics were considered in[1–13]. In particular, the paper [11] deals with the extension of specific types ofchaos, such as Devaney and Li–Yorke chaos and the one obtained through a period-doubling cascade. In [11], a system of ordinary differential equations in the form

x0 D K.t; x/ (7.3)

is utilized as the source of chaos, where K W R�Rm ! R

m is a continuous functionin all its arguments, in order to create chaotic motions in the system

y0 D Cy C M.x.t/; y/; (7.4)

where x.t/ is a solution of (7.3), M W Rm � R

n ! Rn is a continuous function

in all its arguments, and the constant n � n real-valued matrix C has real parts ofeigenvalues all negative. The rigorous results of [11] specify that the chaos of (7.3)influences (7.4) such that the latter also possesses chaos, and the type of chaos ispreserved in the process. On the other hand, the entrainment of limit cycles by chaos,which results as the appearance of cyclic irregular behavior, was discussed in thepaper [13]. The extension of discrete Li–Yorke chaos [38] and intermittency [44],as well as chaos generation around periodic orbits were studied in [14]. However, inthe present chapter, we consider the extension of Devaney chaos, period-doublingcascades, and homoclinic and heteroclinic orbits.

In the literature, control of chaos is understood as the stabilization of unstableperiodic orbits embedded in a chaotic attractor. The studies on the control of chaos

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7 Input–Output Mechanism of the Discrete Chaos Extension 205

originated with Ott, Grebogi and Yorke [45]. The Ott-Grebogi-Yorke (OGY) controlmethod depends on the usage of small time-dependent perturbations in an accessiblesystem parameter to stabilize an already existing periodic orbit, which is initiallyunstable [27, 45, 50]. In the present chapter, a technique to control the extendedchaos is also proposed.

The chapter is organized as follows. In Sect. 7.2 we discuss bounded solutionsof Eq. (7.2) and present the collections of bounded sequences associated withEqs. (7.1) and (7.2). Section 7.3 is devoted to the input–output mechanism for chaosin the sense of Devaney [23]. In Sect. 7.3 we introduce the ingredients of Devaneychaos for collections of sequences and prove their extension in equations of theform (7.1) C (7.2). In Sect. 7.4 we study the extension of period-doubling cascades.Thereafter, in Sect. 7.5 we consider the existence of homoclinic and heteroclinicorbits in a chaotic attractor. The control of the extended chaos is discussed inSect. 7.6, and Sect. 7.7 is concerned with an application of our results to a discretegonorrhea model. Finally, some concluding remarks are made in Sect. 7.8.

7.2 Preliminaries

The following conditions are required throughout the chapter:

(A1) There exists a positive number L1 such that kg.x; y/ � g.x; y/k � L1 kx � xkfor all x; x 2 �; y 2 R

q;(A2) There exists a positive number L2 such that kg.x; y/ � g.x; y/k � L2 kx � xk

for all x; x 2 �; y 2 Rq;

(A3) There exists a positive number L3 such that kg.x; y/ � g.x; y/k � L3 ky � ykfor all x 2 �; y; y 2 R

q;(A4) There exists a positive number Mg such that sup

x2�;y2Rqkg.x; y/k � Mg;

(A5) kAk C L3 < 1.

For a given solution x D fxng of (7.1), using the standard technique for maps[37], one can confirm under conditions A3–A5 that there exists a unique boundedsolution of (7.2), which will be denoted by �x D ˚

�xn

�. In the notation �x, the

superscript x is utilized to indicate the dependence on the chosen solution x D fxngof (7.1). The bounded solution �x satisfies the following relation:

�xn D

nX

jD�1An�jg.xj�1; �x

j�1/; n 2 Z: (7.5)

Let us denote by Ax the set of all bounded solutions of (7.1), with initial datafrom �, and define the sets

Ay D f�x W x 2 Axg

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and

A D f.x; �x/ W x 2 Axg :

Relation (7.5) implies that supn2Z

kynk � Mg

1 � kAk for each fyng 2 Ay. It can be

verified that the set Ay is an attractor, that is, for each solution fyng of (7.2), thereexists a solution fQyng 2 Ay such that kyn � Qynk ! 0 as n ! 1.

7.3 Extension of Devaney Chaos

Our purpose in this section is to introduce ingredients of Devaney chaos [23] forcollections of sequences and rigorously prove the extension of this type of chaos inequations of the form (7.1) C (7.2).

Let us consider a set of uniformly bounded sequences

B D

f�ngn2Z W supn2Z

k�nk � MB

�; (7.6)

where MB is a positive number.We start with a definition of sensitivity [23], which is one of the ingredients of

Devaney chaos. Collection B is called sensitive if there exists a number � > 0 suchthat for any sequence f�ng 2 B and an arbitrary number ı > 0 there exist a sequencef�ng 2 B and a number k 2 N such that k�0 � �0k < ı and k�k � �kk > �.

In the definition of sensitivity for maps, generally, one considers the solutions fornonnegative integers and, consequently, uses the closeness of two sequences only atthe initial moment. However, in our case, we make use of solutions for all integers,and to provide the extension of chaos in equations of the form (7.1) C (7.2), weneed to extend the classical definition of sensitivity to the following one, which wecall strong sensitivity.

We say that collection B is strongly sensitive if there exists a number � > 0

such that for any sequence f�ng 2 B and an arbitrary number ı > 0 there exist asequence f�ng 2 B and a number k 2 N such that

��j � �j

�� < ı for each j � 0 andk�k � �kk > �.

Note that in the case of strong sensitivity, the closeness of solutions is considerednot only at the initial moment j D 0, but also for each negative value of the integer j.It is clear that strong sensitivity implies sensitivity. The converse is also true for theset Ax if, for example, there exists a positive number �0 such that the function Fsatisfies the property kF.z1/ � F.z2/k � �0 kz1 � z2k for all z1; z2 2 �.

Now, we will emphasize how it is possible to obtain a strongly sensitivecollection of sequences by means of symbolic dynamics [16, 17, 25, 43, 46, 55].To this end, let us continue with a brief description of symbolic dynamics. Considerthe sequence space [23, 25]

†2 D ˚s D .s0s1s2 : : :/ W sj D 0 or 1

�;

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with the metric

dŒs; Ns� D1X

iD0

jsi � Nsij2i

;

where s D .s0s1s2 : : :/ and Ns D .Ns0Ns1Ns2 : : :/. On †2 we define the shift map � W†2 ! †2 by letting �.s/ D .s1s2s3 : : :/. The semidynamics .†2; �/ is the symbolicdynamics [23, 25, 55].

Suppose that the map F on � is topologically conjugate to the symbolic dynamics.†2; �/. In this case, there exists a homeomorphism S W � ! †2 such thatS ı F D � ı S.

For each x0 2 � one can construct an arbitrary solution fxng ; n 2 Z, of Eq. (7.1)in the following way. If n � 0, then we let xnC1 D F.xn/. Let us describe how onecan define the sequence for negative values of integer n. Denote the itinerary S.x0/

of x0 by s0 D �s00s0

1s02 : : :

. Consider the elements s D �

0s00s0

1 : : :

and s D �1s0

0s01 : : :

such that � .s/ D � .s/ D s0 and x D S�1 .s/ ; x D S�1 .s/. In this case, by means oftopological conjugacy, we have F .x/ D F .x/ D x0. The set F�1.x0/ consists of theelements x and x, and each of these values can be assigned as x�1. One can continuein this way to expand the sequence to �1 by choosing the value of x�.nC1/ 2 �

from the set F�1 .xn/. This procedure completes the construction of the sequencefxng 2 Ax.

In the following discussion, we indicate the strong sensitivity of the set Ax in thecase where the map F on � is topologically conjugate to the symbolic dynamics.†2; �/.

Fix an arbitrary positive number ı and a sequence fxng 2 Ax. Suppose that thesequence fxng admits the value x0 at the initial moment in such a way that S.x0/ Ds0 D �

s00s0

1s02 : : :

. By the continuity of the function S�1 W †2 ! �, one can find

a number ı D ı.ı; s0/ > 0 such that for any s1 2 †2 with dŒs0; s1� < ı we have��S�1�s0� S�1

�s1�� < ı. Fix a sufficiently large natural number k satisfying the

inequality 2�k � ı. Take a point x0 2 � such that S .x0/ D s0 D �s00s0

1s02 : : :

,

where s0k ¤ s0

k , and s0i D s0

i for all i ¤ k. Moreover, assume that the sequencesfxng 2 Ax and fxng 2 Ax have extensions in the negative direction in such a waythat for each positive integer m; S.x�m/ D �

s0�ms0�mC1s0�mC2 : : :

and S.x�m/ D�s0�ms0�mC1s0�mC2 : : :

, where s0

k ¤ s0k and s0

i D s0i for all i ¤ k. In the present case,

it is obvious that kxm � xmk < ı for all m � 0. Since d��k.s0/; �k.s1/

D 1, and Sis uniformly continuous on �, one can find a positive number � < 1, independentof the sequences fxng and fxng, such that kxk � xkk > �. This discussion reveals thatif the map F on � is topologically conjugate to the symbolic dynamics, then Ax isstrongly sensitive.

Another ingredient of Devaney chaos is the transitivity property. The collectionB possesses a dense sequence

˚��

n

� 2 B if for every sequence f�ng 2 B, arbitrarysmall positive number �, and arbitrary large natural number E there exist an integerm0 and a natural number k such that

��n � ��nCk

�� < � for m0 � n � m0 C E. Wesay that the collection B is transitive if it possesses a dense sequence.

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A sequence f�ng 2 B is called k0-periodic for some natural number k0 if for eachn 2 Z we have �n D �nCk0 . The next definition concerns the density of periodicsequences inside B.

B possesses a dense collection P B of periodic sequences if for everysequence f�ng 2 B, arbitrary small positive number �, and arbitrary large naturalnumber E there exist f Q�ng 2 P and an integer m0 such that k�n � Q�nk < � form0 � n � m0 C E.

Collection B is called Devaney chaotic if: (i) B is sensitive, (ii) B is transitive,and (iii) B possesses a dense collection of periodic sequences.

The following lemma is concerned with the sensitivity feature in Eq. (7.2). Theextension of the transitivity and density of periodic solutions will be provided inLemmas 2 and 3, respectively.

Lemma 1. Suppose that conditions A1–A5 are valid. If Ax is strongly sensitive,then Ay is sensitive.

Proof. Fix an arbitrary positive number ı and a solution �x D ˚�x

n

� 2 Ay ofEq. (7.2), where x D fxng 2 Ax. Take any positive number ı1 satisfying the

inequality ı1 ��

1 � kAk � L3

1 C L2 � kAk � L3

�ı and fix a negative integer r, sufficiently

large in absolute value, such that2Mg

1 � kAk .kAk C L3/�r < ı1.

Since Ax is strongly sensitive, there exist a positive number �0 and a sequencex D fxng 2 Ax such that

��xj � xj

�� < ı1 for each j � 0 and kxk � xkk > �0 for somek 2 N. For each integer n, let us use the notation yn D �x

n and yn D �xn.

First we shall verify that ky0 � y0k < ı. Using the relations

yn DnX

jD�1An�jg.xj�1; yj�1/

and

yn DnX

jD�1An�jg.xj�1; yj�1/;

we obtain that

yn � yn DnX

jD�1An�j

�g.xj�1; yj�1/ � g.xj�1; yj�1/

DrX

jD�1An�j

�g.xj�1; yj�1/ � g.xj�1; yj�1/

CnX

jDrC1

An�j�g.xj�1; yj�1/ � g.xj�1; yj�1/

CnX

jDrC1

An�j�g.xj�1; yj�1/ � g.xj�1; yj�1/

:

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Therefore, the inequality

kyn � ynk �rX

jD�12Mg kAkn�j C

nX

jDrC1

L2ı1 kAkn�j

CnX

jDrC1

kAkn�j L3

��yj�1 � yj�1

��

D 2Mg

1 � kAk kAkn�r C L2ı1

1 � kAk .1 � kAkn�r/ Cn�1X

jDr

L3

kAk kAkn�j��yj � yj

��

holds for each n 2 Z satisfying r C 1 � n � 0. Multiplication of both sides of thelast inequality by kAk�n gives us

kAk�n kyn � ynk � 2Mg

1 � kAk kAk�r C L2ı1

1 � kAk .kAk�n � kAk�r/

Cn�1X

jDr

L3


��

D�

2Mg � L2ı1

1 � kAk�

kAk�r C L2ı1

1 � kAk kAk�n

Cn�1X

jDr

L3


�� :

Using Gronwall’s lemma, one can confirm that

kAk�n kyn � ynk ��

2Mg � L2ı1

1 � kAk�

kAk�r C L2ı1

1 � kAk kAk�n

Cn�1X

jDr

L3

kAk��

2Mg � L2ı1

1 � kAk�

kAk�r C L2ı1

1 � kAk kAk�j��

1 C L3

kAk�n�1�j

D L2ı1

1 � kAk kAk�n C�

2Mg � L2ı1

1 � kAk�

kAk�r� kAk

kAk C L3

�r�n

C L2L3ı1

.1 � kAk/ .1 � kAk � L3/kAk�n .1 � .kAk C L3/

n�r/ :

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Thus, we have that

kyn � ynk � L2ı1

1 � kAk C�

2Mg � L2ı1

1 � kAk�

.kAk C L3/n�r

C L2L3ı1

.1 � kAk/ .1 � kAk � L3/.1 � .kAk C L3/

n�r/

D 2Mg

1 � kAk .kAk C L3/n�r C L2ı1

1 � kAk � L3

.1 � .kAk C L3/n�r/ ;

and hence,

ky0 � y0k � 2Mg

1 � kAk .kAk C L3/�r C L2ı1

1 � kAk � L3

<

�1 C L2

1 � kAk � L3

�ı1

� ı:

In the remaining part of the proof, we will show that the sequences fyng and fyngdiverge at some moment.

By means of the relations ykC1 D Ayk C g.xk; yk/ and ykC1 D Ayk C g .xk; yk/,one can obtain that

��ykC1 � ykC1

�� kg.xk; yk/ � g .xk; yk/k� kg.xk; yk/ � g .xk; yk/k � kAk kyk � ykk

� L1 kxk � xkk � L3 kyk � ykk � kAk kyk � ykk> L1�0 � .kAk C L3/ kyk � ykk :

The last inequality implies that

.1 C kAk C L3/ max˚kyk � ykk ;

��ykC1 � ykC1

�� > L1�0:

Therefore, we have either kyk � ykk > � or��ykC1 � ykC1

�� > �, where

� D L1�0

1 C kAk C L3

:

Consequently, Ay is sensitive. �

In Lemma 1, the extension of the sensitivity feature from the collection ofsequences Ax to Ay is mentioned. It is also reasonable to determine the unpre-dictability of the solutions of the coupled system (7.1) C (7.2). In other words, weshall analyze the sensitivity of the collection A , which consists of the bounded

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solutions of (7.1) C (7.2). We say that A is sensitive provided that Ay is sensitive.This description of sensitivity for the collection A is a natural one, since otherwisethe inequality kxk � xkk > �0 implies that

��xk; �xk

� �xk; �x

k

�� > �0, which alreadysignifies the sensitivity of A . In the input–output mechanism of chaos extension, thecrucial idea is the extension of sensitivity not only through the collectionA , but alsothrough Ay. For this reason, the sensitivity of the collection A is a property that isequivalent to the sensitivity of the collection Ay. According to this explanation, itis worth noting that if Ax is strongly sensitive, then Lemma 1 implies that A issensitive.

We discuss the extension of transitivity in the next lemma.

Lemma 2. Assume that conditions A2–A5 are fulfilled. If Ax is transitive, then thesame is valid for Ay.

Proof. Fix an arbitrary small positive number � and a natural number E such that

�E

2

�� ln.��/

ln .kAk C L3/� 1;

where the notation

�E

2

�denotes the greatest integer that is not larger than

E

2.

Consider an arbitrary solution fyng 2 Ay of (7.2). There exists a sequence x Dfxng 2 Ax such that for each integer n we have yn D �x

n. Let � be a positive numbersuch that

� ��

L2

1 � kAk � L3

C 2Mg

1 � kAk��1

:

Since Ax is transitive, there exists a sequence x� D ˚x�

n

� 2 Ax such that theinequality

��xn � x�nCk

�� < ��; m0 � n � m0 C E, is valid for some m0 2 Z andk 2 N. For each integer n, let us use the notation y�

n D �x�

n .For n � m0 C 1, the solutions fyng and

˚y�

n

�satisfy the relation

yn � y�nCk D

nX

jD�1An�j

�g.xj�1; yj�1/ � g.x�

jCk�1; y�jCk�1/

Dm0X

jD�1An�j

�g.xj�1; yj�1/ � g.x�

jCk�1; y�jCk�1/

CnX

jDm0C1

An�j�g.xj�1; yj�1/ � g.x�

jCk�1; yj�1/

CnX

jDm0C1

An�j�g.x�

jCk�1; yj�1/ � g.x�jCk�1; y�

jCk�1/ :

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Therefore, we have for m0 C 1 � n � m0 C E C 1 that

��yn � y�nCk

�� 2Mg

1 � kAk kAkn�m0 C L2��

1 � kAk .1 � kAkn�m0 /

Cn�1X

jDm0

L3 kAkn�1�j��yj � y�

jCk

�� :

Multiplying both sides of the last inequality by the term kAk�n, we attain thefollowing inequality:

kAk�n��yn � y�

nCk

��

2Mg � L2��

1 � kAk�

kAk�m0

C L2��

1 � kAk kAk�n Cn�1X

jDm0

L3

kAk kAk�j��yj � y�

jCk

�� :

Application of Gronwall’s lemma to the last inequality leads to

kAk�n��yn � y�

nCk

�� L2��


2Mg � L2��

1 � kAk�

kAk�n .kAk C L3/n�m0

C L2L3��

.1 � kAk/ .1 � kAk � L3/kAk�n .1 � .kAk C L3/n�m0 / :

Thus, one obtains that


�� <2Mg

1 � kAk .kAk C L3/n�m0 C L2��

1 � kAk � L3

:

Because the number E is sufficiently large such that the inequality

�E

2

��

ln.��/

ln .kAk C L3/� 1 holds, we have for m0 C

�E

2

�C 1 � n � m0 C E C 1 that

.kAk C L3/n�m0 � ��:

Hence,


�� <

�L2

1 � kAk � L3

C 2Mg

1 � kAk�

�� :

Consequently, Ay is transitive. �Suppose that there exists a nonempty set Px Ax of periodic solutions of

Eq. (7.1). By means of relation (7.5), it can be verified under conditions A3–A5that if x D fxng 2 Px is a p0-periodic solution of (7.1) for some p0 2 N, then

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the bounded solution �x of (7.2) is p0-periodic, too. If, additionally, condition A1holds, then the converse is also true. Thus, there is a one-to-one correspondencebetween the sets Px andPy, wherePy D f�x W x 2 Pxg Ay is the set of periodicsolutions of Eq. (7.2).

The following assertion can be proved in a similar way to Lemma 2.

Lemma 3. Assume that conditions A2–A5 are valid. If Ax admits a dense collectionof periodic solutions, then the same is true for Ay.

The main result of the present section is mentioned in the next theorem, whichcan be proved using Lemmas 1, 2, and 3.

Theorem 1. Suppose that conditions A1–A5 are fulfilled. If Ax is strongly sensitive,transitive and possesses a dense collection of periodic sequences, then Ay isDevaney chaotic.

A corollary of Theorem 1 is the following one.

Corollary 1. Under the assumptions of Theorem 1, collection A is a Devaneychaotic set.

In the following example, we will make use of the logistic map

xnC1 D f�.xn/; (7.7)

where f�.s/ D �s.1 � s/ and � is a parameter.

7.3.1 An Example

According to the results of [23, 46], for � > 4 the logistic map (7.7) is topologicallyconjugate to the symbolic dynamics .†2; �/, possesses an invariant Cantor set � Œ0; 1�, and is chaotic in the sense of Devaney.

Let us take into account the system of discrete equations

ynC1 D 1

6yn � 1

10zn C 2

5sin.yn C zn/;

znC1 D 1

12yn C 1

9zn C xn � 1

1 C xn; (7.8)

where fxng ; n 2 Z, is a solution of (7.7), with x0 2 �.Equation (7.8) is in the form of (7.2), where

A D

0

B@1

6� 1

101

12

1

9

1

CA

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and

g.xn; yn; zn/ D

0

B@

2

5sin.yn C zn/

xn � 1

1 C xn

1

CA :

One can confirm that conditions A1–A5 hold for (7.8), with L1 D 5=4; L2 D 2,and L3 D 2

p2=5. Therefore, according to Theorem 1, system (7.8) is chaotic in the

sense of Devaney provided that � > 4.In the next section, we will consider the extension of a period-doubling cascade.

7.4 Extension of a Period-Doubling Cascade

In this section, we will discuss the extension of a period-doubling cascade in coupledequations of the form (7.1) C (7.2). We will first indicate the existence of chaosthrough a period-doubling cascade for Eq. (7.1) and then continue with its extensionthrough (7.2).

We start by considering the discrete-time equation

xnC1 D G.xn; �/; (7.9)

where n 2 Z, � is a parameter, and the function G W � � R ! � satisfies, for alls 2 �, the property that F.s/ D G.s; �1/ for some finite value �1 of the parameter�, which will be explained subsequently.

Suppose that there exists a sequence of period-doubling bifurcation valuesf�mg ; m 2 N, of the parameter � such that for each m, Eq. (7.9) undergoesa period-doubling bifurcation as the parameter � increases or decreases through�m, and the sequence f�mg converges to a finite value �1 as m ! 1. As aresult, for � D �1 there exist infinitely many unstable periodic orbits of (7.9)and, consequently, of (7.1), all lying in a bounded region. In this case, we say thatEq. (7.1) admits a period-doubling cascade. For details see [26, 39, 47, 48].

Because there is a one-to-one correspondence between the sets Px and Py,and the corresponding orbits inside Px and Py have the same periods, one canconclude that if Eq. (7.1) possesses infinitely many periodic orbits, then the sameis true for (7.2), with no additional periodic orbits of any other period. Moreover,

all the periodic orbits of (7.2) lie inside a tube with radiusMg

1 � kAk . The instability

of the existing infinitely many periodic orbits can be proved in a similar way toLemma 1. We combine and emphasize these results in the next theorem.

Theorem 2. Suppose that conditions A1–A5 are fulfilled. If Eq. (7.1) admits aperiod-doubling cascade, then the same is true for (7.2).

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A corollary of Theorem 2 is the following one.

Corollary 2. Under the assumptions of Theorem 2, the coupled system (7.1) C(7.2) admits a period-doubling cascade.

Our theoretical discussions reveal that Eq. (7.2), like Eq. (7.9), undergoes period-doubling bifurcations as the parameter � increases or decreases through the values�m; m 2 N. In other words, the sequence f�mg of bifurcation parameters isexactly the same for both equations. Therefore, if Eq. (7.9) obeys the Feigenbaumuniversality [26], then one can confirm that the same is also true for Eq. (7.2).

That is, when limm!1

�m � �mC1

�mC1 � �mC2

is evaluated, the universal constant known as the

Feigenbaum number, 4:6692016 : : :, is obtained, and this universal number is thesame for both equations, and hence for the coupled Eqs. (7.9) C (7.2).

7.4.1 An Example

One of the well-known maps that is chaotic through a period-doubling cascade andsuitable for simulations is the logistic map (7.7). The set � D Œ0; 1� is invariantunder the iterations of (7.7) for 0 < � � 4 [23, 25, 46].

The map (7.7) undergoes period-doubling bifurcations at the values �1 D3; �2 D 3:449489; �3 D 3:544090; �4 D 3:564407; : : :, of the parameter � suchthat the sequence f�mg ; m 2 N, of period-doubling bifurcation values accumulatesat �1 � 3:570, after which chaotic behavior arises [25].

To illustrate the extension of a period-doubling cascade, we set up the followingsystem of discrete equations:

ynC1 D �1

8yn C 1

12zn C 2xn;

znC1 D 1

4yn C 1

8zn C tan

�xn C yn

5

�; (7.10)

where fxng ; n 2 Z, with x0 2 Œ0; 1�, is a solution of (7.7).Note that for each parameter value 0 < � � 4, the bounded solutions of the

coupled system (7.7) C (7.10) satisfy the inequalities jxnj � 1 and jynj � 2.Therefore, conditions A1–A5 are valid for Eq. (7.10), and, according to Theorem 2,system (7.10) admits a period-doubling cascade. To attain chaotic behavior in thedynamics of (7.10), we will need those values of the parameter � that are between�1 and 4 such that the period-doubling cascade accumulates there to provide thechaotic structure [23, 46]. According to [34], the measure of such � is positive.

The bifurcation diagram for the y-coordinate of the coupled equations (7.7)C(7.10) is depicted in Fig. 7.1. The picture in Fig. 7.1a represents the successiveintervals of chaos and stable periodic solutions for values of the parameter �

between 2:2 and 4. It is observable in the figure that, as in the case of map (7.7),

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2.5 3 3.5 4

0

0.5

1

1.5

2

μ

ya

3.8 3.82 3.84 3.86 3.88 3.90

0.5

1

1.5

2

μ

y

b

Fig. 7.1 Bifurcation diagram of discrete equation (7.10). (a) For 2:2 � � � 4, (b) for 3:8 � � �3:9. Successive intervals of chaos and stable periodic motions are observable in panel (a). Themagnification of the period-3 window appearing in the whole bifurcation diagram is presented inpanel (b). The diagrams in (a) and (b) demonstrate the extension of the period-doubling cascadein Eq. (7.10)

the period-doubling bifurcation values accumulate at �1 � 3:570, after whichchaos arises. In the range of parameter values greater than �1, correlatively to thebehavior of the logistic map, periodic windows appear in the diagram. In Fig. 7.1b,the parameter � is adjusted to take values between 3:8 and 3:9 for the illustrationof the period-3 window. The remarkable similarities of the presented bifurcationdiagram with that of the logistic map are a manifestation of the chaos extension andstrongly support our theoretical results. Moreover, Fig. 7.1 indicates that the period-doubling bifurcation values for Eq. (7.10) coincide with those for the logistic mapin such a way that Feigenbaum’s universality is valid for Eq. (7.10). This featurerepresents an additional evidence for the chaos extension.

7.5 Homoclinic and Heteroclinic Orbits

This section is devoted to the existence of homoclinic and heteroclinic orbits in thedynamics of Eq. (7.2) as well as the coupled system (7.1) C (7.2).

Consider once again the collection B of uniformly bounded sequences definedin (7.6). The stable set of a sequence � D f�ng 2 B is defined as

Ws.�/ D ff�ng 2 B W k�n � �nk ! 0 as n ! 1g ;

and the unstable set of � is

Wu.�/ D ff�ng 2 B W k�n � �nk ! 0 as n ! �1g :

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The set B is called hyperbolic if for each � D f�ng 2 B the stable and unstablesets of � contain at least one element different from �. A sequence � D f�ng 2 B ishomoclinic to another sequence � D f�ng 2 B if � 2 Ws.�/ \ Wu.�/. Moreover, �

is heteroclinic to the sequences �1 D ˚�1

n

� 2 B; �2 D ˚�2

n

� 2 B; � ¤ �1; � ¤ �2

if � 2 Ws.�1/ \ Wu.�2/.

Lemma 4. Assume that conditions A2–A5 are valid and the sequences x D fxngand x D fxng are elements of Ax. If x 2 Ws.x/, then �x 2 Ws.�x/.

Proof. Fix an arbitrary positive number �, and let � be a positive number such that

� ��

2Mg

1 � kAk C L2

1 � kAk � L3

��1

:

Since x 2 Ax belongs to the stable set Ws.x/ of x 2 Ax, one can find an integer n0

such that kxn � xnk < �� whenever n � n0. For each n 2 Z, let us denote yn D �xn

and yn D �xn.

For n � n0 C 1, using the relation

yn � yn Dn0X

jD�1An�j

�g.xj�1; yj�1/ � g.xj�1; yj�1/

CnX

jDn0C1

An�j�g.xj�1; yj�1/ � g.xj�1; yj�1/

CnX

jDn0C1

An�j�g.xj�1; yj�1/ � g.xj�1; yj�1/

it can be verified that

kyn � ynk �n0X

jD�12Mg kAkn�j C

nX

jDn0C1

L2 kAkn�j��xj�1 � xj�1

��

CnX

jDn0C1

L3 kAkn�j��yj�1 � yj�1

��

� 2Mg

1 � kAk kAkn�n0 C L2��

1 � kAk .1 � kAkn�n0 /

Cn�1X

jDn0

L3


�� :

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Therefore, one obtains that


2Mg � L2��

1 � kAk�

kAk�n0

C L2��


jDn0

L3

kAk kAk�j��yj � yj

�� :

Applying Gronwall’s lemma to the last inequality we deduce that


2Mg � L2��

1 � kAk�

kAk�n0 C L2��

1 � kAk kAk�n

Cn�1X

jDn0

L3

kAk��

2Mg � L2��

1 � kAk�

kAk�n0

C L2��

1 � kAk kAk�j��

1 C L3

kAk�n�1�j

D L2��


2Mg � L2��

1 � kAk�

kAk�n .kAk C L3/n�n0

C L2L3��

.1 � kAk/ .1 � kAk � L3/kAk�n .1 � .kAk C L3/n�n0 / :

Hence, for n � n0 C 1, the inequality

kyn � ynk � L2��

1 � kAk C�

2Mg � L2��

1 � kAk�

.kAk C L3/n�n0

C L2L3��

.1 � kAk/ .1 � kAk � L3/.1 � .kAk C L3/

n�n0 /

D 2Mg

1 � kAk .kAk C L3/n�n0 C L2��

1 � kAk � L3

.1 � .kAk C L3/n�n0 /

<2Mg

1 � kAk .kAk C L3/n�n0 C L2��

1 � kAk � L3

is valid.Now, let n0 be an integer that is greater than n0 such that .kAk C L3/

n0�n0 � ��.If n � n0, then we have that

kyn � ynk <

�2Mg

1 � kAk C L2

1 � kAk � L3

�� :

Consequently, �x belongs to Ws.�x/. �

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Lemma 5. Assume that conditions A2–A5 are valid and the sequences x D fxngand x D fxng are elements of Ax. If x 2 Wu.x/, then �x 2 Wu.�x/.

Proof. Fix an arbitrary positive number �, and let � be a positive number such that

� <1 � kAk � L3

L2

:

Because the sequence x D fxng 2 Ax belongs to the unstable set of x D fxng 2 Ax,there exists an integer n0 such that if n � n0, then kxn � xnk < ��. Let us use thenotations yn D �x

n and yn D �xn for each n 2 Z.

By means of the relation

yn � yn DnX

jD�1An�jŒg.xj�1; yj�1/ � g.xj�1; yj�1/�

CnX

jD�1An�jŒg.xj�1; yj�1/ � g.xj�1; yj�1/�

one can confirm that

kyn � ynk <

nX

jD�1L2�� kAkn�j C

nX

jD�1L3 kAkn�j

��yj�1 � yj�1

��

� L2��

1 � kAk C L3

1 � kAk supn�n0

kyn � ynk :

Therefore, we have that

supn�n0C1

kyn � ynk � L2��

1 � kAk � L3

< �:

The last inequality implies that kyn � ynk ! 0 as n ! �1. Consequently, �x 2 Ay

is an element of the unstable set Wu.�x/ of �x 2 Ay. �

The following theorem can be proved using Lemmas 4 and 5.

Theorem 3. Under conditions A2–A5, the following assertions are valid:

(i) If x 2 Ax is homoclinic to x 2 Ax, then �x 2 Ay is homoclinic to �x 2 Ay;(ii) If x 2 Ax is heteroclinic to x1; x2 2 Ax, then �x 2 Ay is heteroclinic to

�x1; �x2 2 Ay;

(iii) If Ax is hyperbolic, then the same is true for Ay.

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A corollary of Theorem 3 is as follows.

Corollary 3. Under conditions A2–A5, the following assertions are valid:

(i) If x 2 Ax is homoclinic to x 2 Ax, then .x; �x/ 2 A is homoclinic to .x; �x/ 2A ;

(ii) If x 2 Ax is heteroclinic to x1; x2 2 Ax, then .x; �x/ 2 A is heteroclinic to.x1; �x1

/; .x2; �x2/ 2 A ;

(iii) If Ax is hyperbolic, then the same is true for A .

7.5.1 An Example

Consider the coupled discrete equations (7.7) C (7.8), introduced in Sect. 7.3.1. Theinverses of the function f�.s/ defined in (7.7) on the intervals Œ0; 1=2� and Œ1=2; 1�

are h1.s/ D 1

2

1 �

s

1 � 4s

�

!and h2.s/ D 1

2

1 C

s

1 � 4s

�

!; respectively.

Let us take � D 3:9 in map (7.7). It was demonstrated in [15] that the orbit

x Dn: : : ; h3

2.x0/; h22.x0/; h2.x0/; x0; f�.x0/; f 2

�.x0/; f 3�.x0/; : : :

o;

where x0 D 1=3:9, is homoclinic to the fixed point x� D 2:9=3:9 of (7.7). Denoteby �x and �x�

the bounded solutions of (7.8) corresponding to the sequences x andx�, respectively. Theorem 3 implies that �x is homoclinic to �x�

. Figure 7.2 showsthe y-coordinates of the orbits �x and �x�

. In the figure, the orbit �x is representedin red, while �x�

is represented in blue. The figure supports the result of Theorem 3such that �x is homoclinic to �x�

.Now we set � D 4 in Eq. (7.7). According to [15], the orbit

x Dn: : : ; h3

1.x0/; h21.x0/; h1.x0/; x0; f�.x0/; f 2

�.x0/; f 3�.x0/; : : :

o;

−10 −5 0 5 10 15

−0.05

0

0.05

0.1

0.15

n

y n

Fig. 7.2 A homoclinic orbit of Eq. (7.8), with � D 3:9. The bounded orbits �x and �x�

are shownin red and blue, respectively. One can observe that �x is homoclinic to �x�

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−5 0 5 10 15

−0.3

−0.2

−0.1

0

0.1

n

y n

Fig. 7.3 A heteroclinic orbit of Eq. (7.8), with � D 4. The orbits �x; �x1, and �x2

are in red, blue,and green, respectively. It is seen that �x is heteroclinic to the orbits �x1

and �x2

where x0 D 1=4, is heteroclinic to the fixed points x1 D 3=4 and x2 D 0 of thelogistic map (7.7). Suppose that �x, �x1

, and �x2are the bounded solutions of (7.8)

corresponding to the orbits x; x1, and x2, respectively. One can conclude by usingTheorem 3 that �x is heteroclinic to �x1

and �x2. In Fig. 7.3, we depict the orbits

�x; �x1, and �x2

in red, blue, and green, respectively. One can observe from the

figure that��x

n � �x1

n

�� ! 0 as n ! 1 and��x

n � �x2

n

�� ! 0 as n ! �1, i.e., �x is

heteroclinic to �x1and �x2

.

7.6 Control of Extended Chaos

In this section, we will focus on controlling the chaos of the coupled system (7.1)C (7.2). The next theorem emphasizes that to control extended chaos, it is sufficientto stabilize periodic solutions of the prior Eq. (7.1), which is used as the source ofchaos.

Theorem 4. Assume that conditions A1–A5 hold and a periodic solution � D f�ngof Eq. (7.1) is stabilized such that for arbitrary � > 0 and any solution fxng of (7.1)there exist integers k and N > 0 such that the inequality kxn � �nk < � is valid fork � n � k C N.

In this case, the periodic solution˚�

�n�

of Eq. (7.2) is stabilized such that for anysolution fyng of Eq. (7.2) there exists an integer l � k C 1 such that the inequality��yn � ��

n

�� <

�1 C L2

1 � kAk � L3

�� holds for l � n � k C N C 1, provided that

the number N is sufficiently large.

Proof. Let us take an arbitrary solution fyng of (7.2) for some solution fxng of (7.1).According to our assumption, one can find integers k and N > 0 such that theinequality kxn � �nk < � holds for k � n � k C N. For each n 2 Z, let us use thenotation yn D �

�n .

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By means of the relations

yn D An�kyk CnX

jDkC1

An�jg.xj�1; yj�1/

and

yn D An�kyk CnX

jDkC1

An�jg.�j�1; yj�1/

one obtains the equation

yn � yn D An�k.yk � yk/ CnX

jDkC1

An�jŒg.xj�1; yj�1/ � g.�j�1; yj�1/�

CnX

jDkC1

An�jŒg.�j�1; yj�1/ � g.�j�1; yj�1/�:

Therefore, we have for k C 1 � n � k C N C 1 that

kyn � ynk � kAkn�k kyk � ykk CnX

jDkC1

kAkn�j L2�

CnX

jDkC1

kAkn�j L3

��yj�1 � yj�1

��

D kAkn�k�

kyk � ykk � L2�

1 � kAk�

C L2�

1 � kAk

Cn�1X

jDk

kAkn�j�1 L3

��yj � yj

�� :

The last inequality implies that

kAk�n kyn � ynk � kAk�k�


1 � kAk�

C L2�


jDk

L3

kAk kAk�j��yj � yj

�� :

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If Gronwall’s lemma is applied, then the inequality

kAk�n kyn � ynk � kAk�k�


1 � kAk�

C kAk�k�


1 � kAk�

L3

kAkn�1X

jDk

�1 C L3

kAk�n�j�1

C L2L3�

kAk .1 � kAk/

n�1X

jDk

kAk�j�

1 C L3

kAk�n�j�1

D L2�

1 � kAk kAk�n C kAk�k�


1 � kAk�� kAk

kAk C L3

�k�n

C L2L3�

.1 � kAk/.1 � kAk � L3/kAk�n

h1 � .kAk C L3/

n�ki

is obtained. Now, multiplying both sides by the term kAkn, we get

kyn � ynk � kAk�k�


1 � kAk�� kAk

kAk C L3

�k

.kAk C L3/n

C L2�

1 � kAk C L2L3�

.1 � kAk/.1 � kAk � L3/

h1 � .kAk C L3/

n�ki

D kyk � ykk .kAk C L3/n�k C L2�

1 � kAk � L3

�1 � .kAk C L3/

n�k

< kyk � ykk .kAk C L3/n�k C L2�

1 � kAk � L3

for all n satisfying k C 1 � n � k C N.In the case where yk D yk, the result of the theorem is obviously true. Suppose

that yk ¤ yk. If n � k Cln

��

kyk�ykk�

ln.kAk C L3/, then the inequality kyk � ykk .kAk C

L3/n�k � � holds. Assume that the integer N is sufficiently large such that N >666664

ln

��

kyk�ykk�

ln.kAk C L3/

777775, where bsc denotes the greatest integer that is not larger than s.

Let

l D max

8ˆ<

ˆ:k C 1; k C 1 C

666664

ln

��

kyk�ykk�

ln.kAk C L3/

777775

9>>=

>>;

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and

QN D min

8ˆ<

ˆ:N; N �

666664ln

��

kyk�ykk�

ln.kAk C L3/

777775

9>>=

>>;:

Note that the positive integer QN is the duration of control for Eq. (7.2) and l C QN Dk C N C 1. Consequently, the inequality kyn � ynk <

�1 C L2

1 � kAk � L3

�� holds

for l � n � k C N C 1. �

The following corollary of Theorem 4 mentions that if the control of Eq. (7.1) isperformed, then the chaos of the coupled system (7.1) C (7.2) is also controlled.

Corollary 4. Under the conditions of Theorem 4, the periodic solution f�ng of thecoupled system (7.1) C (7.2), where �n D �

�n; ��n

, is stabilized such that for any

solution fzng of (7.1) C (7.2) there exists an integer l � kC1 such that the inequality

kzn � �nk <

�2 C L2

1 � kAk � L3

�� holds for l � n � k C N, provided that the

number N is sufficiently large.

7.6.1 An Example

In this part, we will control the chaos of the coupled discrete equations (7.7) C(7.10) with � D 3:8. It was demonstrated in Sect. 7.4.1 that the dynamics of (7.7)C (7.10) is chaotic for the aforementioned value of the parameter. The OGY controlmethod [45] is convenient for controlling the chaos of (7.7) [50]. Therefore, wecontinue with a brief explanation of the OGY control method applied to a logisticmap [50].

Suppose that the parameter � in the logistic map (7.7) is allowed to vary in therange Œ3:8 � ı; 3:8 C ı�, where ı is a given small number. Consider an arbitrarysolution fxng ; x0 2 Œ0; 1�, of the logistic map and denote by x.j/; j D 1; 2; : : : ; p0,the target unstable p0-periodic orbit to be stabilized. In the procedure of the OGYcontrol method, at each iteration step n, we consider the logistic map with theparameter value � D N�n, where

N�n D 3:8

�1 C Œ2x.j/ � 1�Œxn � x.j/�

x.j/Œ1 � x.j/�

�; (7.11)

provided that the number on the right-hand side of formula (7.11) belongs to theinterval Œ3:8 � ı; 3:8 C ı�. In other words, formula (7.11) is valid if the trajectoryfxng is sufficiently close to the target periodic orbit. Otherwise, we take N�n D 3:8,so that the system evolves at its nominal parameter value, and wait until the

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trajectory fxng enters in a sufficiently small neighborhood of the periodic orbitx.j/; j D 1; 2; : : : ; p0, such that the inequality

� ı � 3:8Œ2x.j/ � 1�Œxn � x.j/�

x.j/Œ1 � x.j/�� ı (7.12)

holds [50]. It is worth noting that the stabilization of the desired unstable periodicorbit is not achieved immediately after the control procedure is initiated; rather, thereis a transient time before the stabilization takes place. The transient time increasesif the number ı decreases [27, 50].

According to Theorem 4, the unstable 2-periodic solution of Eq. (7.10), where� D 3:8, can be stabilized by controlling the corresponding 2-periodic orbit x.1/ �0:3737; x.2/ � 0:8894 of the logistic map. We consider a solution of (7.10) withinitial data x0 D 0:573; y0 D 1:212, and z0 D 0:526 and apply the OGY controlmethod around the 2-periodic orbit of the logistic map. We use the value ı D 0:035.The simulation results are represented in Fig. 7.4, which supports the result ofTheorem 4. The x, y, and z coordinates of the solution are shown in Fig. 7.4a–c,respectively. The control mechanism is switched on at the iteration number n D 40

and switched off at n D 120. It is seen in Fig. 7.4 that there is a transient time in thecontrol mechanism such that the stabilization of the 2-periodic solution is achievedapproximately 30 iterations after the control is switched on. During this transitoryevolution, the equation remains chaotic until the chaotic orbit passes close enoughto the 2-periodic orbit such that condition (7.12) is satisfied. Although the control isswitched off at n D 120, the control lasts approximately until n D 180, after whichthe instability becomes dominant and irregular behavior develops again. Comparedto the first coordinate, the control duration in the second and third coordinates startsand ends with a delay of one iteration. This is a prospective behavior in accordancewith Theorem 4.

In the next section, we will apply our theoretically approved results to the discretegonorrhea model, which was proposed in [25].

7.7 Application to Gonorrhea Model

Gonorrhea is a sexually transmitted bacterial infection that tends to attack themucous membranes of the body, and this disease is caused by the growth andproliferation of the gram-negative bacteria Neisseria gonorrhoeae [49]. The firstmathematical formulation for gonorrhea was introduced in the paper [20], whichaddresses the problem through continuous-time dynamics. In this section wewill apply our theoretical results to the discrete model for gonorrhea [25] byusing so-called toy perturbations through the Hénon, Ikeda, and logistic maps.To demonstrate the effects of perturbations, we provide the simulations shown inFigs. 7.6, 7.8, and 7.9. The pictures presented in Figs. 7.5 and 7.7 are constructedto emphasize the input–output concept of the chaos extension mechanism in sucha way that the applied chaotic perturbations (inputs) generate chaotic solutions

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0 50 100 150 200 250

0.2

0.4

0.6

0.8

1

n

x n

a

0 50 100 150 200 2500

1

2

n

y n

b

0 50 100 150 200 2500.2

0.4

0.6

0.8

1

n

z n

c

Fig. 7.4 OGY control method applied to discrete equation (7.10). (a) Graph of x-coordinate.(b) Graph of y-coordinate. (c) Graph of z-coordinate. The OGY control technique is applied tothe logistic map (7.7), with � D 3:8, around the 2-periodic orbit x.1/ � 0:3737; x.2/ � 0:8894.In the control mechanism, the value ı D 0:035 is used. Control is switched on at n D 40 andswitched off at n D 120. Although the control is switched off at n D 120, it lasts approximatelyuntil n D 180, after which chaos is observable in all coordinates again. The pictures presented in(a), (b), and (c) support Theorem 4 and Corollary 4 such that stabilizing the periodic solutions ofthe logistic map, which is used as the source of chaos in (7.10), is sufficient to control the chaosof (7.10) as well as the coupled system (7.7) C (7.10)

(outputs). All these pictures affirm our mathematically approved results such thatthe chaotic behavior is extended through the model for gonorrhea. The establishedmodels in the present section may not coincide with reality in population dynamics.Nevertheless, the novelty in our examples is that one can achieve chaos in thediscrete gonorrhea model through different forms of perturbations.

Suppose that P1 and P2 are two distinct heterosexual populations infected byNeisseria gonorrhoeae. If we denote by un and vn the infected fraction of thepopulations P1 and P2 at time period n, respectively, then 1 � un and 1 � vn arethe susceptible population fractions. We assume that the infected members of apopulation can transmit the disease to a susceptible person in the other population.Under the assumption that populations P1 and P2 are constant, the discrete model

unC1 D a1vn.1 � un/ C .1 � b1/un;

vnC1 D a2un.1 � vn/ C .1 � b2/vn; (7.13)

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was proposed in [25], where a1; a2; b1, and b2 are constants such that 0 < ai < 1;

0 < bi < 1; i D 1; 2. In model (7.13), the susceptible fraction 1 � un of populationP1 is assumed to become ill through sexual encounters with the other population.Therefore, the term a1vn.1 � un/ is added to the model, and the reason for the terma2un.1 � vn/ is similar. Furthermore, the terms b1un and b2vn indicate the fractionsof populations P1 and P2, respectively, that have been cured.

Note that if the condition a1a2 �b1b2 � 0 holds, then the origin is asymptoticallystable for system (7.13). On the other hand, in the case where a1a2 � b1b2 > 0, the

fixed point

�a1a2 � b1b2

a1a2 C a2b1

;a1a2 � b1b2

a1a2 C a1b2

�is asymptotically stable, while the origin

is unstable.One should understand the importance of the perturbations applied to the

gonorrhea model in the sense that the source of the perturbation terms can be in therole of the dynamics of outer effects, such as interactions with other populations,which comprise the chaotic behavior of epidemics, or a treatment process. Here, weapply perturbations to indicate that when there is an exterior chaotic influence onthe dynamics of populations P1 and P2, the dynamics should also display chaoticbehavior.

In our first illustration, we will utilize perturbations displaying chaotic behaviorin connection with the Hénon map [32],

xnC1 D 1 � ˛1x2n C yn;

ynC1 D ˛2xn; (7.14)

where ˛1 and ˛2 are constants. Map (7.14), with ˛1 D 1:4 and ˛2 D 0:3, is chaotic[52, 53], and, depending on the initial point .x0; y0/, the solution of the map eitherdiverges to infinity or tends to a strange attractor, which appears to be the productof a 1D manifold by a Cantor set [32].

We perturb model (7.13) with the solutions of (7.14) and set up the followingsystem of discrete equations:

unC1 D a1vn.1 � un/ C .1 � b1/un C 0:4yn � 0:2 cos yn C 0:6;

vnC1 D a2un.1 � vn/ C .1 � b2/vn C 0:2xn C 0:1 arctan.2xn/ C 0:5; (7.15)

where a1 D 0:1; a2 D 0:07; b1 D 0:9; b2 D 0:92. It is worth noting that with thechosen values of a1; a2; b1, and b2, the origin is asymptotically stable for (7.13).

Equation (7.15) is in the form of (7.2), where A D�

0:1 0:1

0:07 0:08

�: One can

verify that kAk < 0:18 and conditions A1–A5 are valid for Eq. (7.15), withL1 D p

2=10; L2 D 3p

2=5, and L3 D 1=4.In Fig. 7.5, we represent the solution of (7.14) with the initial data x0 D

�0:3239; y0 D 0:2731. It is seen in Fig. 7.5 that the solution of the Hénon mapis chaotic.

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0 10 20 30 40 50 60 70 80 90 100

−1

0

1

n

x n

0 10 20 30 40 50 60 70 80 90 100−0.4

−0.2

0

0.2

0.4

n

y n

Fig. 7.5 Chaotic behavior in Hénon map (7.14). The solution with x0 D �0:3239 and y0 D0:2731 is shown in the figure

0 10 20 30 40 50 60 70 80 90 1000.3

0.4

0.5

0.6

0.7

n

u n

0 10 20 30 40 50 60 70 80 90 1000.20.40.60.8

1

n

v n

Fig. 7.6 The chaotic output of (7.15). It is seen that the chaos of the Hénon map (7.14) makesEq. (7.15) also behave chaotically, even if a stable equilibrium takes place in the dynamics of (7.13)

According to our theoretical results, Eq. (7.15), as well as the coupled Eqs. (7.14)C (7.15), displays chaotic behavior. Using the solution shown in Fig. 7.5 as theinput for (7.15), we obtain the output shown in Fig. 7.6. The initial data u0 D0:4149; v0 D 0:8471 are used. Figure 7.6 assures that the chaotic behaviorproduced by the Hénon map is extended to (7.15). From an input–output point ofview, Figs. 7.5 and 7.6 confirm that when a chaotic perturbation (input) is appliedto Eq. (7.13), a solution (output) appears in a chaotic region, even though thenonperturbed equation (7.13) possesses an asymptotically stable equilibrium.

Now, let us consider the chaotic Ikeda map [52]

xnC1 D 1 C 0:9xn cos.0:4 � 6=.1 C x2n C y2

n//

�0:9yn sin.0:4 � 6=.1 C x2n C y2

n//

ynC1 D 0:9xn sin.0:4 � 6=.1 C x2n C y2

n//

C0:9yn cos.0:4 � 6=.1 C x2n C y2

n//: (7.16)

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Map (7.16) is combined with the gonorrhea model (7.13) to build the followingdiscrete-time system:

unC1 D a1vn.1 � un/ C .1 � b1/un C 0:1exn ;

vnC1 D a2un.1 � vn/ C .1 � b2/vn C 0:3 � 0:15yn � 0:02y3n: (7.17)

Let us use the values a1 D 0:05; a2 D 0:02; b1 D 0:94, and b2 D 0:96 such thatthe origin is an asymptotically stable equilibrium for (7.13).

Figure 7.7 shows the solution of (7.16) with x0 D �0:0191; y0 D �0:0084. Thefigure confirms that the solution is chaotic. Using the solution represented in Fig. 7.7as the perturbation, we depict the solution of (7.17) with u0 D 0:2257; v0 D 0:4608

in Fig. 7.8. The chaotic behavior seen in Fig. 7.8 is a manifestation of the chaosextension. If one considers the problem from an input–output point of view, weshall say that the chaotic sequence (input), which is achieved through the Ikedamap (7.16), acts on (7.13) such that a chaotic solution (output) is generated. In other

0 10 20 30 40 50 60 70 80 90 100

0

0.5

1

1.5

n

x n

0 10 20 30 40 50 60 70 80 90 100−2

−1

0

1

n

y n

Fig. 7.7 Chaotic solution of Ikeda map (7.16)

0 10 20 30 40 50 60 70 80 90 1000.10.20.30.40.5

n

u n

0 10 20 30 40 50 60 70 80 90 100

0.2

0.4

0.6

n

v n

Fig. 7.8 Extension of chaos in Eq. (7.17). The figure confirms that the chaotic solution (input) ofthe Ikeda map (7.16) acts on Eq. (7.13) such that a chaotic solution (output) is generated

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words, supporting our mathematically approved results, the perturbation influencesthe gonorrhea model in such a way that chaotic behavior emerges.

In our next discussion we will utilize the logistic map (7.7), which can beemployed to model species with nonoverlapping generations [30], with � D 4, asthe source for chaotic perturbations. Let us consider the system of discrete equations

unC1 D a1vn.1 � un/ C .1 � b1/un;

vnC1 D a2un.1 � vn/ C .1 � b2/vn C 0:5xn C 0:2; (7.18)

where a1 D 0:15; a2 D 0:12; b1 D 0:92, and b2 D 0:85, and fxng is a solutionof (7.7).

To illustrate the formation of chaos, we make use of the solution fxng of (7.7) withx0 D 0:4209 in Eq. (7.18). The solution of (7.18) with u0 D 0:0628; v0 D 0:7056 isshown in Fig. 7.9. Moreover, the chaotic attractor of (7.18) is depicted in Fig. 7.10.The figures support our theoretical results such that Eq. (7.18) behaves chaotically.

0 10 20 30 40 50 60 70 80 90 1000.04

0.06

0.08

0.1

0.12

n

u n

0 10 20 30 40 50 60 70 80 90 100

0.4

0.6

0.8

n

v n

Fig. 7.9 Chaotic behavior in Eq. (7.18). The chaotic structure reflects the extension of chaos inthe gonorrhea model through the prior chaos of the logistic map (7.7)

Fig. 7.10 Chaotic attractorof Eq. (7.18). The figureconfirms the presence ofchaos in (7.18)

0.04 0.06 0.08 0.1 0.120.2

0.3

0.4

0.5

0.6

0.7

0.8

un

v n

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7.8 Conclusion

In this chapter we considered the extension of chaotic behavior from one discreteequation to another using the input–output mechanism. To study the subject rigor-ously, we considered collections of bounded sequences and provided definitionsof the ingredients of Devaney chaos for such collections. The extension of aperiod-doubling cascade, homoclinic and heteroclinic orbits, and the control ofchaos were also discussed. Various numerical examples are presented to supportthe theoretical results. The presented techniques can be used to obtain chaoticdiscrete-time dynamical systems with arbitrarily high dimensions. We believe thatour approach could be developed in application to the security of communicationsystems on the basis of works that consider chaotic discrete equations as instrumentsof ciphering and deciphering [18].

Acknowledgements M.O. Fen is supported by the 2219 scholarship programme of TÜBITAK,the Scientific and Technological Research Council of Turkey.

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