Chaos Inducement and Enhancement in Two Particular Nonlinear Maps Using Weak Periodic/Quasiperiodic Perturbations

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International Journal of Bifurcation and Chaos, Vol. 16, No. 5 (2006) 1585–1598c! World Scientific Publishing Company

CHAOS INDUCEMENT AND ENHANCEMENTIN TWO PARTICULAR NONLINEAR MAPSUSING WEAK PERIODIC/QUASIPERIODIC

PERTURBATIONS

JIE ZHANG! and MICHAEL SMALL†

Department of Electronic and Information Engineering,Hong Kong Polytechnic University, Kowloon, Hong Kong

[email protected]†[email protected]

KAI ZHANGDepartment of Computer Science,

Hong Kong University of Science and Technology, Kowloon, Hong Kong

Received September 20, 2004; Revised April 7, 2005

Weak periodic perturbation has long been used to suppress chaos in dynamical systems. Inthis paper, however, we demonstrate that weak periodic or quasiperiodic perturbation canalso be used to induce chaos in nonchaotic parameter ranges of chaotic maps, or to enhancethe already existing chaotic state. Two kinds of chaotic maps, the period doubling system andthe Hopf bifurcation system, are employed as basic models to analyze and compare in detail thedi!erent mechanisms of inducing and enhancing chaos in them. In addition, a special kind ofintermittency characterized by its periodicity is found for the first time in periodically perturbedHenon map, and reasonable speculations are presented to explain its complicated dynamics.

Keywords : Two dimensional maps; chaos inducement; chaos enhancement; periodic perturbation;periodic intermittency.

1. Introduction

Since the pioneering work of [Ott et al., 1990] onchaos control known as OGY method, a majorthrust of investigation has focused on convertingchaos found in various systems into periodic motion.However, chaos can be a desirable feature in areassuch as fluid mixing [Rothstein et al., 1999], biol-ogy [Yang et al., 1995], electronics [Dhamala & Lai,1999], and optics [VanWiggeren & Roy, 1998], whereit is crucial for chaos to be sustained, induced,or sometimes enhanced. In addition, chaos couldalso be induced to prevent resonance in mechan-ics [Schwartz & Georgiou, 1998] , and to facilitatediagnosing biological dynamics of pathological phe-nomena [Schi! et al., 1994].

An analytical scheme for generating chaos waspresented in [Chen & Lai, 1998; Wang & Chen,2000]. The author gave a rigorous proof that origi-nally nonchaotic maps can be chaotified by intro-ducing small amplitude state feedback controls.As for chaotic systems, however, things are somewhat di!erent. Notice that periodic windows alwaysdominate over large parameter ranges in chaoticmaps, most work concerned is devoted to convert-ing the transient chaos associated with the peri-odic windows to steady chaos. In [Yang et al., 1995;Dhamala et al., 1999], a method based on feedbackcontrol mechanism is presented, where a parame-ter or state variable is used to maintain systemchaocity. In [Schwartz & Triandaf, 1996; Triandaf& Schwartz, 2000] the topology of the manifold

1585

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of basin saddles is employed to design parame-ter control algorithms for sustaining chaos. As tononfeedback methods, an open-loop approach wasproposed in [Schwartz, et al., 2002], where the sys-tem is driven with a 1 : 2 resonance modulation withartificially changed phase and amplitude.

Generally speaking, most literatures mentionedabove dealt with “transient chaos” state, whichis sustained or converted to “steady chaos” afterthe control scheme is applied. In contrast, whatwe manipulate in this paper is the stable peri-odic orbit that “coexists” with the transient chaos.The basic idea is that, by way of weak perturba-tion, the modulated periodic orbit will either gothrough new bifurcations (such as period doublingcascade to chaos) or be destablized to give wayto chaotic regime, therefore chaos is induced nat-urally. The perturbation we use here is either peri-odic or quasiperiodic, which has been employedotherwise to suppress or eliminate chaos [Braiman& Goldhirsch, 1991; Meucci et al., 1994; Gonzalezet al., 1998; Tereshko & Shchekinova, 1998; Chacon,1999] in dynamical systems.

We also address the problem of chaos enhance-ment, which is relatively less studied in present lit-eratures. [Gupte et al., 1996] proposed a method byintroducing small changes of parameters in phasespace to enhance chaos. In this paper, we show forthe first time that weak periodic/quasiperiodic per-turbation can also be used to achieve this goal.

Motivated to find out how weak periodic/quasiperiodic perturbation can be used to induce/enhance chaos in chaotic maps, the methodologyof this paper is novel in that it exploits two di!er-ent kinds of chaotic maps to analyze and comparethe relevant mechanisms. And the passage is orga-nized as follows. In Sec. 2, we make a general anal-ysis of two chaotic maps, which gives necessarypreparation for the following discussion. Then weuse the two maps as the basic model to investigatehow weak periodic and quasiperiodic perturbationcan induce chaos in Secs. 3 and 4, respectively. InSec. 5, we discuss chaos enhancement through weakperiodic/quasiperiodic perturbations. And a specialkind of intermittency found for the first time in per-turbed Henon map, which we call “periodic inter-mittency”, is discussed in Sec. 6. Conclusions aremade in Sec. 7.

2. Basic Model

In this paper, we use two di!erent kinds of chaoticmaps as the basic models. One is Henon map (1)

that undergoes the well-known period doublingroute to chaos, and the other is a noninvertiblemap (2) which displays a quasiperiodicity route tochaos.

!xn+1 = !ax2

n + 0.3yn + 1yn+1 = xn

, (1)

!xn+1 = !ay2

n + 0.35xn + 1yn+1 = xn

. (2)

Their perturbed systems can be written as:!

xn+1 = !ax2n + 0.3yn + 1 + ! cos(2"#n)

yn+1 = xn, (3)

!xn+1 = !ay2

n + 0.35xn + 1 + ! cos(2"#n)yn+1 = xn

, (4)

respectively, where # is the perturbation frequency,and ! represents the perturbation amplitude.

The Henon map (1) is a well studied chaoticmap [Hao & Zheng, 1998; Murakami et al., 2002;Sonis, 1996]. As its control parameter a increases,the stable fixed point will undergo a series of perioddoubling bifurcation (P1 " P2 " P4 · · ·) to chaos.For larger values of a, the trajectories are alwayschaotic. And within the parameter interval in whichchaos can be observed, there are smaller intervalsin which there exists periodic behavior, or periodicwindow.

The map (2) demonstrates complicated dynam-ics as the control parameter a varies, and Fig. 1gives a general view of its evolution. When a =0.5725, there is an invariant circle arising throughHopf bifurcation. Then it wrinkles and develops

Fig. 1. Bifurcation diagram of map (2) versus a.

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Fig. 2. FFT transform for {x(n)} series of map (2) whena = 0.7, with a corresponding invariant circle in phase space.The two frequencies f1 and f2 are incommensurable.

cusps and loops till breakup, where chaos arisesin the presence of noninvertibility. Period-9 occursafterwards, followed by period-14 and period-5windows in the chaotic region. Further investiga-tion demonstrates that these period-n orbits willturn into n coexistent chaotic attractors throughperiod doubling cascade, or a second level of Hopfbifurcation.

From a physical point of view, the invariantcircle emerging via Hopf bifurcation is a typicalquasiperiodic behavior, indicating two incommen-surable frequencies intrinsic to the system gov-erning the dynamics, see Fig. 2 for detail. Thesetwo fundamental frequencies exist throughout theevolution, and when they become commensurable(or locked), periodic windows such as period-9,period-14 and period-5 will appear. And the rota-tion numbers of these periodic lockings are ona Farey tree [Rademacher, 1964]. This propertymakes map (2) demonstrate quite a di!erent mech-anism when chaos is induced compared with map(1). For convenience of reference, we denote map(2) as Hopf bifurcation system, as opposed to theperiod doubling Henon map in (1).

3. Periodic Perturbation thatInduces Chaos

Having made clear the general evolution of thedynamics in maps (1) and (2), we will focus onspecific periodic windows within chaotic regions ofthe two systems, to investigate in detail how theycan be converted to chaotic state through periodic/

quasiperiodic perturbation. And the situations inother periodic windows can all be analyzed likewise.

3.1. In the period doubling systems

Notice that for map (1), there is a period-7 win-dow when 1.22662 < a < 1.27168, arising from asaddle-node bifurcation. So we fix a at 1.25 andvary the perturbing frequency # in (3) for study.We plot the LLE (largest Lyapunov exponent) ver-sus # at two perturbation amplitude, ! = 0.002 and! = 0.05 in Fig. 3, where the frequency # is con-fined to the interval [0, 0.5] since the cosine func-tion is even symmetric. As can be seen, chaos doesnot arise under perturbation of extremely smallamplitude (! = 0.002), because the correspondingLLE remains negative. While for higher amplitude(! = 0.05), chaos dominates over the entire intervalof [0, 0.5].

Now we examine the system dynamics as theperturbation amplitude varies. Here we use p todenote the period of perturbation, i.e. p # = 1.When ! is very small, the period-p perturbation willresonate with the period-7 of the unperturbed sys-tem, leading to resonant period-7p orbit. For exam-ple, when p = 3, the perturbed system will exhibitperiod-21 behavior at ! = 0.002. When ! graduallyincreases, a period doubling cascade of period-21occurs, giving rise to 21 separate chaotic attractors,see Fig. 4(a) for illustration. These chaotic attrac-tors form seven groups, with each group composedof three attractors. If we plot every seventh itera-tion of map (3), we will obtain one group of thesechaotic attractors, see Fig. 4(b) for illustration.

Fig. 3. Two LLE curves versus ! for map (3) when " = 0.002and " = 0.05, where a = 1.25.

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(a) Attractor for map (3) when " = 0.011. (b) Enlargement of (a).

(c) Attractor for map (3) when " = 0.015. (d) Enlargement of (c).

Fig. 4. Attractor for map (3) when " varies, ! = 1/3, and a = 1.25.

Under further increase of !, the seven groups will gothrough an interior crisis to form a global attractor,see Fig. 4(c)for illustration. In this case, the globalattractor can be deemed as the combination of threesimilar chaotic attractors (see Fig. 4(d) for illustra-tion), because the perturbed map (3) with period-p(p = 3) perturbation can be equivalently describedby a set of p equations:

"###$

###%

xpn+i = !ax2pn+i"1 + 0.3ypn+i"1 + 1

+ ! cos&2"

pn + i ! 1p

'

ypn+i = xpn+i"1

(5)

where i = 1, 2, . . . , p, and n = 0, 1, 2, . . . .

These p maps are topologically conjugated,and exhibit analogous features since map (3) isinvertible and the amplitude of the perturbation isvery small. Therefore the chaotic attractor of theperiod-p perturbed system can be deemed as p sim-ilar chaotic attractors congregating together, whichis vividly illustrated in Fig. 4(d), and each attrac-tor can be extracted through the plot of the pthiterate of the perturbed map. We examine otherperiodic perturbations such as period-4, period-5and period-6 . . . , and find similar results. Here atwo-parameter (i.e. a and $) bifurcation diagramof the period-21 window in map (3) can be madeaccording to the procedures in [Hunt, 1999], and thefundamental structure of the period-21 window can

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Chaos Inducement and Enhancement in Two Particular Nonlinear Maps 1589

be obtained by applying the conjecture in [Barreto,1997]. This will help decide under what circum-stances chaos may arise.

In addition, we find that periodic perturba-tion can also induce chaos at the parameter rangeswhere map (1) exhibits period-2n behavior (beforethe accumulating point), which cannot be achievedthrough the methods mentioned in Sec. 1, becausethey mainly deal with transient chaos, which doesnot exist at parameter ranges with period-2n

behavior.Now we consider the influence of the period-p

perturbation on the overall structure of the bifur-cation diagram, as well as the dissipative propertyof map (1). Take period-3 perturbation for exam-ple, we find that the original period doubling cas-cade of the stable fixed point of unperturbed map(1)(P1 " P2 " P4 · · ·) is replaced by period dou-bling cascade of period-3 (P3 " P6 " P12 · · ·), (seeFig. 5 for detail), with the distribution of periodicand chaotic regions after the accumulating pointaltered at the same time. For example, the period-7window of the unperturbed map (1) is replaced bychaos in the perturbed map (3). Therefore the over-all structure of the bifurcation diagram is qualita-tively changed.

However, the introduction of the periodic per-turbation to autonomous system (1) does notchange its originally dissipative property, becausethe perturbed nonautonomous map (3) can be

Fig. 5. (a) Bifurcation diagram of map (1); (b) Bifurcationdiagram of map (3) with ! = 1/3 and " = 0.05.

converted into an autonomous system (6) by intro-ducing a new variable zn. And the Jacobian deter-minant of map (6) is |J | = !0.3, exactly the sameas that of the unperturbed map (1). Therefore theperturbed map (3) is still dissipative.

"#$

#%

xn+1 = !ax2n + 0.3yn + 1 + ! cos(2"# · zn)

yn+1 = xn

zn+1 = zn + 1(6)

3.2. In the Hopf bifurcation systems

For map (2), a backward unstable periodic dou-bling bifurcation results in a period-5 window when1.14528< a< 1.16192. So we fix a at 1.15 to studyhow chaos is induced through periodic perturba-tion. In Fig. 6, we plot LLE versus perturbationfrequency # at two perturbation amplitudes (! =0.002 and ! = 0.05), and find interestingly thatwhen amplitude of perturbation is quite small (! =0.002), there still exits ranges of control parameterwhere the LLE is positive.

We use period-2 perturbation to illustrate howchaos can be induced with very small ! in map (4).That is, the period-2 perturbation will react withthe period-5 of the unperturbed system when !is extremely small, leading to a period-10 orbit.As ! slightly increases to 0.00166, the period-10destablizes suddenly through a saddle-node bifurca-tion, giving rise to a chaotic attractor, see Fig. 7(b)for illustration. And near the transition, type-Iintermittency is found.

This phenomenon can also be explained froma physical point of view. As mentioned in Sec. 2,

Fig. 6. Two LLE curves versus ! for map (4) at " = 0.002and " = 0.05, where a = 1.15.

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(a) (b)

Fig. 7. (a) Attractor for map (2) at a = 1.1452 prior to period-5 window. (b) Attractor for map (4) at a = 1.15, " = 0.00166,! = 0.5.

the period-5 window of unperturbed system (2)occurs due to the frequency locking of the twointernal frequencies at # = 0.2. When a period-2(# = 0.5) perturbation is added at very small ampli-tude, it will react with the originally locked twofrequencies, resulting in a period-10 behavior. How-ever, because the perturbation frequency is di!erentfrom the originally locked frequencies, it destroysthe original frequency locking regime as its ampli-tude slightly increases to 0.00166. Therefore the sys-tem is brought back to the chaotic regime priorto period-5 window. Note that the induced chaoticattractor within period-5 window (see Fig. 7(b))is quite similar to the attractor (see Fig. 7(a)) ofthe unperturbed system (2) before period-5 windowoccurs.

Furthermore, we investigate this phenomenonin the phase space. If we take the tenth iterate ofmap (4), the period-10 orbit which coexists withthe unstable chaotic saddle (the transient chaos)can be deemed as ten attractors, each of which hasthe chaotic saddle lying on the boundary of thebasin separating these ten attractors. As the ampli-tude of the perturbation increases, the attractor isfound to approach the boundary and is finally madeaccessible to the boundary region. Therefore, thechaotic state is restored. In other words, the per-turbation has included the chaotic saddle as part ofthe dynamics.

The stability of the above mentioned period-10 orbit depends on the amplitude of the period-2perturbation. That is, period-10 is stable as long as

|Df (p)(x)| =

(((((

p)

i=1

Df(xi)

((((( < 1, (7)

where f is the autonomous system converted fromthe nonautonomous system (4), Df(x) is the Jaco-bian matrix of f at x, and p is the resonant period-porbit (here p = 10). In this way the minimal ampli-tude of the perturbation su"cient to induce chaoscan be obtained provided that the equation of theunderlying system is available.

We can be seen in Fig. 6, for higher values ofperturbation amplitude (! = 0.05), the LLE of per-turbed system (4) versus # remains positive (corre-sponding to chaos) except at # = 0.2. This is due tothe reinforcement of the original frequency lockingregime in period-5 window of unperturbed map (2),since the perturbation frequency is the same as theresonant frequency of the unperturbed system. Andwe find that the resonance at # = 0.2 will persist tillthe perturbation amplitude increases to ! = 0.12.

In addition, we find that periodic perturbationcan also induce chaos at parameter ranges wheremap (2) exhibits quasiperiodic behavior. For exam-ple, let a = 0.7, where an invariant circle occursin the phase space. When a perturbation with

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Chaos Inducement and Enhancement in Two Particular Nonlinear Maps 1591

frequency # = 0.22 is exerted, the perturbed map(4) will demonstrate period-50 behavior, becausethe external frequency # = 0.22 is locked with oneinternal frequency # = 0.011 of unperturbed sys-tem. And by increasing the perturbation amplitude,period doubling cascade of period-50 to chaos willbe introduced.

4. Quasiperiodic Perturbation thatInduces Chaos

In the previous section, chaos inducement isachieved through periodic perturbation, whose fre-quency corresponds to rational number with zeromeasure over the interval of [0, 0.5]. Comparatively,there are more irrational numbers with nonzeromeasure that correspond to quasiperiodic frequen-cies. In this section we will demonstrate thatquasiperiodic perturbation can also be used toinduce chaos, and a quasiperiodic frequency # =#

5 ! 1 is chosen for a detailed study.

4.1. In the period doubling systems

In Fig. 8, we plot the LLE versus a for map (1)and its perturbed system (3), where the quasiperi-odic perturbation has the parameter # =

#5 ! 1,

! = 0.01. Compare the two curves, we find thatmost periodic windows (with negative LLE) of theunperturbed system (1) are eliminated and replacedby chaos (with positive LLE) in the presence of theperturbation.

Fig. 8. LLE versus a for map (1) (the top), and its perturbedsystem (3) with ! =

"5 # 1 , " = 0.01 (the bottom).

We still fix a = 1.25 for a detailed investiga-tion, where map (1) demonstrates period-7 behav-ior. When the amplitude ! of the quasiperiodicperturbation is small, the period-7 orbit becomesa quasiperiodic attractor, or a torus attractor of 7bands. As ! increases, the 7-torus attractor getsincreasingly wrinkled and then transits to a 7-band SNA [Grebogi et al., 1984; Ding et al., 1989],characterized by its fractal geometry and nonpos-itive LLE, see Figs. 9(a) and 9(b) for details. Forstill larger !, the 7-band SNA turns into a 1-bandSNA (see Fig. 9(c)), and finally becomes a chaoticattractor.

(a) SNA for map (3) when " = 0.006396, withcorresponding LLE = #0.0391.

(b) Enlargement view of (a).

Fig. 9. Attractor of map (3) under di!erent " when a = 1.25, ! ="

5 # 1.

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(c) SNA for map (3) when " = 0.00654, withcorresponding LLE = #0.0132.

Fig. 9. (Continued )

Here the 7-band SNA is formed through thefractalization route [Nishikawa & Kaneko, 1996],and the following transition form 7-band SNA to1-band SNA is due to an interior crisis. Now weexamine the LLE curve in the whole evolution tochaos. As shown in Fig. 10, the LLE varies smoothlyat the beginning, which correspond to the torusrange marked in Fig. 10. Then it becomes nonmono-tonic as ! increases, going through a series of tran-sitions like torus " SNA " chaos " SNA, andfinally terminates in chaos.

Fig. 10. LLE curve of map (3) versus " with a = 1.25 and! =

"5 # 1.

If we fix a = 1.26, where map (1) exhibitsperiod-14 originating from period doubling ofperiod-7, and gradually increase the perturbationamplitude, we will find another kind of transitionto chaos via SNA known as Heagy–Hammel route[Heagy & Hammel, 1994]. In fact, the transitionfrom torus to chaos via SNA is typical in perioddoubling systems, and usually the amplitude su"-cient for chaos inducement is quite small.

4.2. In the Hopf bifurcation systems

Figure 11 plots the LLE versus control parametera for map (2) and its quasiperiodically perturbedversion (4), and two di!erent mechanisms involvedin inducing chaos are found here. One is similarto the mechanism in period doubling system, andthe other is associated with the breakup of the fre-quency locking. We use period-5 and period-12 win-dows to explain them, respectively.

We first fix a at 1.15, where map (2) exhibitsperiod-5 behavior. And a perturbation with increas-ing amplitude ! is exerted. When ! is small, theperiod-5 orbit becomes a 5-torus. Then it wrinklesand transits to a 5-band SNA as ! increases, fol-lowing a fractalization route [Nishikawa & Kaneko,1996]. Figure 12 illustrates this phenomenon regard-ing only one band of the attractor. On increas-ing ! further, the 5-band SNA will become a5-band chaotic attractor. This is di!erent from

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Fig. 11. LLE curve versus a for map (2) (the top one), andits perturbed system (4) with ! =

"5 # 1, " = 0.01 (the

bottom one).

Fig. 12. SNA of map (4) when a = 1.15, " = 0.0045,! =

"5 # 1, with corresponding LLE = #0.0011.

the evolution in period doubling system, wherethe 7-band SNA first undergoes a crisis and thenbecomes a chaotic attractor. During the whole evo-lution, the LLE increases very slowly to positive(see Fig. 13(a)), because no crisis occurred. Butthere is a dramatic increase of the LLE later at ! =0.004728 (denoted by an arrow in Fig. 13(a)), wherethe 5-band chaotic attractor undergoes attractormerging.

Then we fix a at 1.098, where map (2) exhibitsperiod-12 behavior. As a quasiperiodic perturbation( # =

#5! 1) is exerted with increasing amplitude

!, the period-12 becomes a 12-torus, and thendisappears abruptly to be replaced by chaos as !

(a) LLE versus " for map (4) when a = 1.15.

(b) LLE versus " for map (4) when a = 1.098.

Fig. 13. LLE curve versus " of perturbed map (4) (! ="5 # 1) for period-5 and period-12 windows.

reaches 0.0006907, with type-II intermittency foundnear the transition. During this process, the LLEincreases from negative directly to positive and noSNA is found, which is illustrated in Fig. 13(b). Thereason for the transition from 12-torus directly tochaos is the destruction of the quasiperiodic attrac-tor (12-torus), or, from physical point of view, thebreakup of frequency locking regime in the presenceof the quasiperiodic perturbation, which is similarto the situation in Sec. 3.2.

5. Quasiperiodic Perturbation thatEnhances Chaos

In addition to inducing chaos in nonchaotic para-meter ranges of chaotic system, weak periodic/

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1594 J. Zhang et al.

quasiperiodic perturbations are also found to beable to enhance chaos. In this section we will demon-strate that by introducing di!erent kinds of crisis,quasiperiodic perturbation produces a substantialenhancement of LLE for quite small perturbationamplitude.

5.1. In the period doubling systems

For example, map (1) exhibits a 7-band chaoticattractor at a = 1.264, which arises from the perioddoubling cascade of period-7, see Fig. 14(a) for illus-tration. When a quasiperiodic perturbation (# =#

5 ! 1) with increasing amplitude is exerted, each

(a) Attractor for map (1) without perturbation.

(b) Attractor for map (3) with perturbation " = 0.0026,! =

"5 # 1.

Fig. 14. Attractor for map (1) and its perturbed system (3)at a = 1.264.

Fig. 15. LLE versus " for map (3) when ! ="

5 # 1 anda = 1.264. The dash-dot line indicates the LLE of unper-turbed map (1) at a = 1.264.

band of the attractor will grow and become some-what fractalized. Then they merge to form a globalattractor as the perturbation amplitude increases to0.0026, where an interior crisis occurs, see Fig. 14(b)for illustration. We plot the corresponding LLEcurve in Fig. 15, and find that the LLE variesslightly at the beginning, but increases dramaticallyafter the crisis. The reason for the abrupt increaseof LLE is that the coexisting repeller at a = 1.264 ofmap (3) has become a part of the whole attractor.This repeller is the remnant of the chaotic attractorwhich “disappears” at the saddle-node bifurcation,

Fig. 16. LLE versus " for map (3) when ! ="

5 # 1and a = 1.0595, where three band mergings happen at1(! = 0.0019), 2(! = 0.012), 3(! = 0.064). And the dash-dotline indicates the LLE of unperturbed map (1) at a = 1.0595.

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Chaos Inducement and Enhancement in Two Particular Nonlinear Maps 1595

and it has a larger LLE than the 7-band chaoticattractor.

Moreover, quasiperiodic perturbation can alsoenhance chaos by provoking band merging. Forexample, there is an 8-band attractor for map (1)at a = 1.0595, When a quasiperiodic perturbationof frequency # =

#5 ! 1 is exerted with gradu-

ally increased amplitude, this attractor will undergothree levels of band-merging in turn, i.e. from 8bands " 4 bands " 2 bands " 1 band. In Fig. 16we can see that each time the band merging occurs,there will be a dramatic increase in LLE. And when

(a) Attractor for map (2) without perturbation.

(b) Attractor for map (4) with perturbation " = 0.0005,! =

"5 # 1.

Fig. 17. Attractor for map (2) and its perturbed system (4)at a = 0.969.

Fig. 18. LLE versus " for map (4) when ! ="

5 # 1 anda = 0.969, and the dash-dot line is LLE of unperturbed map(2) at a = 0.969.

! = 0.1, the LLE reaches 0.198, well above the LLEof the unperturbed system. The increase of LLEhere is due to the increase of the available phasespace volume, and consequently the local rate ofdivergence of trajectories.

5.2. In the Hopf bifurcation systems

Compared with the case discussed in Sec. 5.1, theattractor merging induced by perturbation in map(2) is somewhat di!erent. Notice that there arenine separate chaotic attractors of map (2) whena = 0.969, which are plotted in Fig. 17(a). Andthese nine attractors undergo attractor mergingwhen the amplitude of the quasiperiodic perturba-tion (# =

#5!1) reaches ! = 0.0005, corresponding

to a dramatic increase in LLE in Fig. 18. However,these nine separate chaotic attractors are charac-terized by two positive Lyapunov exponents, there-fore di!ering manifestly from the attractor mergingin strictly dissipative systems like map (1) and (3)[Grebogi et al., 1987]. And further investigation isneeded to clarify the structure near basin bound-aries of the separate attractors at the transition.

6. Periodic Intermittency

In this section, we will investigate a special kind ofintermittency found in the perturbed Henon map(3). This intermittency has been reported by [Yanget al., 1996] in Du"ng system when an extra forc-ing slightly detunes the original forcing. While ourexample shows that such intermittency can also

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1596 J. Zhang et al.

occur for discrete autonomous system with one peri-odic perturbation.

Notice period-24 windows occurs in map (1)when a = 1.073. If we exert a perturbation withfixed amplitude ! = 0.05 and varying frequency,many periodic orbits would be excited, whichcorrespond to negative LLE in Fig. 19. Actually,these periodic orbits are unstable periodic orbits ofthe unperturbed system which resonate with theperturbation. And if perturbation frequency slightlymisses one of the resonant frequencies, a specialkind of intermittency will occur.

For example, let the perturbation frequency be# = 0.25001, which di!ers slightly from one reso-nant frequency # = 0.25. We plot series x(n) ver-sus n for 50 000 iterations in Fig. 20 and find thatthe system moves “regularly” in certain time inter-vals, while goes “chaotically” in others. After a fixedtime length T = 25000, the motions are preciselyrepeated. This “periodicity” of such intermittencydistinguishes itself from conventional intermitten-cies, in which periodic motion and chaotic motionappear randomly.

Now we analyze this intermittency in termsof the geometrical structure of the attractor. InFig. 20, we divide the periodic segment into threesuccessive evolutions: E1, E2, and E3, and markthem correspondingly in the attractor plotted inFig. 21. For reason of visualization, we only plotfour of the 16 branches in the E1 in Fig. 21.

We can see in Fig. 20 that as n increases, the 16branches in E1 converge to the eight branches in E2,

Fig. 19. LLE versus ! for map (3) when a = 1.073, " = 0.05.And the marked resonant frequencies are: 1(! = 0.125),2(! = 0.1875), 3(! = 0.25).

Fig. 20. x(n) versus n for map (3) when a = 1.073, " = 0.05,! = 0.25001.

Fig. 21. Attractor for map (3) when a = 1.073, " = 0.05,! = 0.25001, where E1, E2, E3 correspond to the three evo-lutions marked in Fig. 20, and P4 ( ), P8 ( ), P16 ( ) aresaddle type periodic orbits.

and then to the four branches in E3. This lead us tospeculate there are three saddle type periodic orbits(period-16, period-8, and period-4 in Fig. 21) nearthe converging points, and the branches that con-nect adjacent two periodic orbits are actually boththe unstable manifold of the former periodic orbit,and the stable manifold of the latter periodic orbit.With this speculation, the evolution process fromE1 through E2 to E3 is well explained: the iterationstarts from P16, and leaves it gradually along itsunstable manifold. Since P16’s unstable manifold isalso P8’s stable manifold, the iteration will gradu-ally approach P8 and converge near it. Similarly, the

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Chaos Inducement and Enhancement in Two Particular Nonlinear Maps 1597

iteration continues to move away along the unstablemanifold of P8, and approaches P4 along its stablemanifold. Finally, the iteration leaves P4 along itsunstable manifold towards the chaotic region, untilit once again enters the periodic segment.

Actually, we also find many such periodic orbitsin the chaotic region in Fig. 21, or correspond-ingly, periodic pieces (branches) within the chaoticsegment in Fig. 20. It is just the delicate config-uration of the large number of saddle type peri-odic orbits that leads to the “regularity” of suchintermittency.

7. Conclusion

In this paper, we exert weak periodic/quasiperiodicperturbations on two particular chaotic maps, i.e.a period doubling system and a Hopf bifurcationsystem, and have found that the adopted perturba-tions can induce chaos in the nonchaotic parame-ter ranges of chaotic maps, or enhance the existingchaotic state.

The periodic perturbation induces chaos in dif-ferent ways for the two systems. In period doublingsystem, chaos is induced through a period dou-bling cascade of the resonant periodic orbit, whilefor Hopf bifurcation system, chaos arises due tobreakup of the frequency locking regime.

As to quasiperiodic perturbation, we find theinducement of chaos via a SNA in both sys-tems. However, for certain periodic windows of theHopf bifurcation system, chaos is also found tobe induced through the breakup of the frequencylocking regime. The reason why Hopf bifurcationsystem demonstrates a di!erent mechanism fromperiod doubling system when chaos is induced canbe attributed to its dynamics governed by two fun-damental frequencies. Therefore when the frequencylocking is destroyed under external perturbation,chaotic state prior to frequency locking windowswill recur.

In terms of chaos enhancement, the mechanismfor the two systems are similar, i.e. the introductionof interior crisis or attractor merging will increasethe phase space volume, and consequently the rateof divergence of trajectories.

In addition, we find the special “periodic”intermittency for the first time in the periodicallyperturbed Henon map. And we analyze its “peri-odicity” by giving reasonable speculations on theconfiguration of the large number of saddle typeperiodic orbits embedded in the attractor.

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