PERTURBATIONS. Received September 20, 2004; Revised April 7, 2005

Transcription

1 International Journal of Bifurcation and Chaos, Vol. 16, No. 5 (2006) c World Scientific Publishing Company CHAOS INDUCEMENT AND ENHANCEMENT IN TWO PARTICULAR NONLINEAR MAPS USING WEAK PERIODIC/QUASIPERIODIC PERTURBATIONS JIE ZHANG and MICHAEL SMALL Department of Electronic and Information Engineering, Hong Kong Polytechnic University, Kowloon, Hong Kong enzhangjie@eie.polyu.edu.hk ensmall@polyu.edu.hk KAI ZHANG Department 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. In this paper, however, we demonstrate that weak periodic or quasiperiodic perturbation can also be used to induce chaos in nonchaotic parameter ranges of chaotic maps, or to enhance the already existing chaotic state. Two kinds of chaotic maps, the period doubling system and the Hopf bifurcation system, are employed as basic models to analyze and compare in detail the different mechanisms of inducing and enhancing chaos in them. In addition, a special kind of intermittency characterized by its periodicity is found for the first time in periodically perturbed Henon 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] on chaos control known as OGY method, a major thrust of investigation has focused on converting chaos found in various systems into periodic motion. However, chaos can be a desirable feature in areas such as fluid mixing [Rothstein et al., 1999], biology [Yang et al., 1995], electronics [Dhamala & Lai, 1999], and optics [VanWiggeren & Roy, 1998], where it is crucial for chaos to be sustained, induced, or sometimes enhanced. In addition, chaos could also be induced to prevent resonance in mechanics [Schwartz & Georgiou, 1998], and to facilitate diagnosing biological dynamics of pathological phenomena [Schiff et al., 1994]. An analytical scheme for generating chaos was presented in [Chen & Lai, 1998; Wang & Chen, 2000]. The author gave a rigorous proof that originally nonchaotic maps can be chaotified by introducing small amplitude state feedback controls. As for chaotic systems, however, things are some what different. Notice that periodic windows always dominate over large parameter ranges in chaotic maps, most work concerned is devoted to converting the transient chaos associated with the periodic windows to steady chaos. In [Yang et al., 1995; Dhamala et al., 1999], a method based on feedback control mechanism is presented, where a parameter or state variable is used to maintain system chaocity. In [Schwartz & Triandaf, 1996; Triandaf & Schwartz, 2000] the topology of the manifold 1585

2 1586 J. Zhang et al. of basin saddles is employed to design parameter control algorithms for sustaining chaos. As to nonfeedback methods, an open-loop approach was proposed in [Schwartz, et al., 2002], where the system is driven with a 1 : 2 resonance modulation with artificially changed phase and amplitude. Generally speaking, most literatures mentioned above dealt with transient chaos state, which is sustained or converted to steady chaos after the control scheme is applied. In contrast, what we manipulate in this paper is the stable periodic orbit that coexists with the transient chaos. The basic idea is that, by way of weak perturbation, the modulated periodic orbit will either go through new bifurcations (such as period doubling cascade to chaos) or be destablized to give way to chaotic regime, therefore chaos is induced naturally. The perturbation we use here is either periodic or quasiperiodic, which has been employed otherwise to suppress or eliminate chaos [Braiman & Goldhirsch, 1991; Meucci et al., 1994; González et al., 1998; Tereshko & Shchekinova, 1998; Chacón, 1999] in dynamical systems. We also address the problem of chaos enhancement, which is relatively less studied in present literatures. [Gupte et al., 1996] proposed a method by introducing small changes of parameters in phase space to enhance chaos. In this paper, we show for the first time that weak periodic/quasiperiodic perturbation 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 methodology of this paper is novel in that it exploits two different kinds of chaotic maps to analyze and compare the relevant mechanisms. And the passage is organized as follows. In Sec. 2, we make a general analysis of two chaotic maps, which gives necessary preparation for the following discussion. Then we use the two maps as the basic model to investigate how weak periodic and quasiperiodic perturbation can induce chaos in Secs. 3 and 4, respectively. In Sec. 5, we discuss chaos enhancement through weak periodic/quasiperiodic perturbations. And a special kind of intermittency found for the first time in perturbed Henon map, which we call periodic intermittency, is discussed in Sec. 6. Conclusions are made in Sec. 7. that undergoes the well-known period doubling route to chaos, and the other is a noninvertible map (2) which displays a quasiperiodicity route to chaos. { xn+1 = ax 2 n + 0.3y n + 1 y n+1 = x n, (1) { xn+1 = ay 2 n x n + 1 y n+1 = x n. (2) Their perturbed systems can be written as: { xn+1 = ax 2 n + 0.3y n ε cos(2πωn) y n+1 = x n, (3) { xn+1 = ay 2 n x n ε cos(2πωn) y n+1 = x n, (4) respectively, where ω is the perturbation frequency, and ε represents the perturbation amplitude. The Henon map (1) is a well studied chaotic map [Hao & Zheng, 1998; Murakami et al., 2002; Sonis, 1996]. As its control parameter a increases, the stable fixed point will undergo a series of period doubling bifurcation (P1 P2 P4 ) to chaos. For larger values of a, the trajectories are always chaotic. And within the parameter interval in which chaos can be observed, there are smaller intervals in which there exists periodic behavior, or periodic window. The map (2) demonstrates complicated dynamics as the control parameter a varies, and Fig. 1 gives a general view of its evolution. When a = , there is an invariant circle arising through Hopf bifurcation. Then it wrinkles and develops 2. Basic Model In this paper, we use two different kinds of chaotic maps as the basic models. One is Henon map (1) Fig. 1. Bifurcation diagram of map (2) versus a.

3 Chaos Inducement and Enhancement in Two Particular Nonlinear Maps 1587 quasiperiodic perturbation. And the situations in other periodic windows can all be analyzed likewise. Fig. 2. FFT transform for {x(n)} series of map (2) when a = 0.7, with a corresponding invariant circle in phase space. The two frequencies f 1 and f 2 are incommensurable. cusps and loops till breakup, where chaos arises in the presence of noninvertibility. Period-9 occurs afterwards, followed by period-14 and period-5 windows in the chaotic region. Further investigation demonstrates that these period-n orbits will turn into n coexistent chaotic attractors through period doubling cascade, or a second level of Hopf bifurcation. From a physical point of view, the invariant circle emerging via Hopf bifurcation is a typical quasiperiodic behavior, indicating two incommensurable frequencies intrinsic to the system governing the dynamics, see Fig. 2 for detail. These two fundamental frequencies exist throughout the evolution, and when they become commensurable (or locked), periodic windows such as period-9, period-14 and period-5 will appear. And the rotation numbers of these periodic lockings are on a Farey tree [Rademacher, 1964]. This property makes map (2) demonstrate quite a different mechanism when chaos is induced compared with map (1). For convenience of reference, we denote map (2) as Hopf bifurcation system, as opposed to the period doubling Henon map in (1) In the period doubling systems Notice that for map (1), there is a period-7 window when < a < , arising from a saddle-node bifurcation. So we fix a at 1.25 and vary the perturbing frequency ω in (3) for study. We plot the LLE (largest Lyapunov exponent) versus ω at two perturbation amplitude, ε = and ε = 0.05 in Fig. 3, where the frequency ω is confined to the interval [0, 0.5] since the cosine function is even symmetric. As can be seen, chaos does not arise under perturbation of extremely small amplitude (ε = 0.002), because the corresponding LLE remains negative. While for higher amplitude (ε = 0.05), chaos dominates over the entire interval of [0, 0.5]. Now we examine the system dynamics as the perturbation amplitude varies. Here we use p to denote the period of perturbation, i.e. p ω = 1. When ε is very small, the period-p perturbation will resonate with the period-7 of the unperturbed system, leading to resonant period-7p orbit. For example, when p = 3, the perturbed system will exhibit period-21 behavior at ε = When ε gradually increases, a period doubling cascade of period-21 occurs, giving rise to 21 separate chaotic attractors, see Fig. 4(a) for illustration. These chaotic attractors form seven groups, with each group composed of three attractors. If we plot every seventh iteration of map (3), we will obtain one group of these chaotic attractors, see Fig. 4(b) for illustration. 3. Periodic Perturbation that Induces Chaos Having made clear the general evolution of the dynamics in maps (1) and (2), we will focus on specific periodic windows within chaotic regions of the two systems, to investigate in detail how they can be converted to chaotic state through periodic/ Fig. 3. Two LLE curves versus ω for map (3) when ε = and ε = 0.05, where a = 1.25.

4 1588 J. Zhang et al. (a) Attractor for map (3) when ε = (b) Enlargement of (a). (c) Attractor for map (3) when ε = (d) Enlargement of (c). Fig. 4. Attractor for map (3) when ε varies, ω = 1/3, and a = Under further increase of ε, the seven groups will go through an interior crisis to form a global attractor, see Fig. 4(c)for illustration. In this case, the global attractor can be deemed as the combination of three similar chaotic attractors (see Fig. 4(d) for illustration), because the perturbed map (3) with period-p (p = 3) perturbation can be equivalently described by a set of p equations: x pn+i = ax 2 pn+i y pn+i [ + ε cos y pn+i = x pn+i 1 2π pn + i 1 p where i = 1, 2,..., p, and n = 0, 1, 2,.... ] (5) These p maps are topologically conjugated, and exhibit analogous features since map (3) is invertible and the amplitude of the perturbation is very small. Therefore the chaotic attractor of the period-p perturbed system can be deemed as p similar chaotic attractors congregating together, which is vividly illustrated in Fig. 4(d), and each attractor can be extracted through the plot of the pth iterate of the perturbed map. We examine other periodic perturbations such as period-4, period-5 and period-6..., and find similar results. Here a two-parameter (i.e. a and ɛ) bifurcation diagram of the period-21 window in map (3) can be made according to the procedures in [Hunt, 1999], and the fundamental structure of the period-21 window can

5 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 circumstances chaos may arise. In addition, we find that periodic perturbation can also induce chaos at the parameter ranges where map (1) exhibits period-2 n behavior (before the accumulating point), which cannot be achieved through the methods mentioned in Sec. 1, because they mainly deal with transient chaos, which does not exist at parameter ranges with period-2 n behavior. Now we consider the influence of the period-p perturbation on the overall structure of the bifurcation diagram, as well as the dissipative property of map (1). Take period-3 perturbation for example, we find that the original period doubling cascade of the stable fixed point of unperturbed map (1)(P1 P2 P4 ) is replaced by period doubling cascade of period-3 (P3 P6 P12 ), (see Fig. 5 for detail), with the distribution of periodic and chaotic regions after the accumulating point altered at the same time. For example, the period-7 window of the unperturbed map (1) is replaced by chaos in the perturbed map (3). Therefore the overall structure of the bifurcation diagram is qualitatively changed. However, the introduction of the periodic perturbation to autonomous system (1) does not change its originally dissipative property, because the perturbed nonautonomous map (3) can be converted into an autonomous system (6) by introducing a new variable z n. And the Jacobian determinant of map (6) is J = 0.3, exactly the same as that of the unperturbed map (1). Therefore the perturbed map (3) is still dissipative. x n+1 = ax 2 n + 0.3y n ε cos(2πω z n ) y n+1 = x n (6) z n+1 = z n In the Hopf bifurcation systems For map (2), a backward unstable periodic doubling bifurcation results in a period-5 window when < a < So we fix a at 1.15 to study how chaos is induced through periodic perturbation. In Fig. 6, we plot LLE versus perturbation frequency ω at two perturbation amplitudes (ε = and ε = 0.05), and find interestingly that when amplitude of perturbation is quite small (ε = 0.002), there still exits ranges of control parameter where the LLE is positive. We use period-2 perturbation to illustrate how chaos can be induced with very small ε in map (4). That is, the period-2 perturbation will react with the period-5 of the unperturbed system when ε is extremely small, leading to a period-10 orbit. As ε slightly increases to , the period-10 destablizes suddenly through a saddle-node bifurcation, giving rise to a chaotic attractor, see Fig. 7(b) for illustration. And near the transition, type-i intermittency is found. This phenomenon can also be explained from a physical point of view. As mentioned in Sec. 2, Fig. 5. (a) Bifurcation diagram of map (1); (b) Bifurcation diagram of map (3) with ω = 1/3 and ε = Fig. 6. Two LLE curves versus ω for map (4) at ε = and ε = 0.05, where a = 1.15.

6 1590 J. Zhang et al. (a) Fig. 7. (a) Attractor for map (2) at a = prior to period-5 window. (b) Attractor for map (4) at a = 1.15, ε = , ω = 0.5. (b) the period-5 window of unperturbed system (2) occurs due to the frequency locking of the two internal frequencies at ω = 0.2. When a period-2 (ω = 0.5) perturbation is added at very small amplitude, it will react with the originally locked two frequencies, resulting in a period-10 behavior. However, because the perturbation frequency is different from the originally locked frequencies, it destroys the original frequency locking regime as its amplitude slightly increases to Therefore the system is brought back to the chaotic regime prior to period-5 window. Note that the induced chaotic attractor within period-5 window (see Fig. 7(b)) is quite similar to the attractor (see Fig. 7(a)) of the unperturbed system (2) before period-5 window occurs. Furthermore, we investigate this phenomenon in the phase space. If we take the tenth iterate of map (4), the period-10 orbit which coexists with the unstable chaotic saddle (the transient chaos) can be deemed as ten attractors, each of which has the chaotic saddle lying on the boundary of the basin separating these ten attractors. As the amplitude of the perturbation increases, the attractor is found to approach the boundary and is finally made accessible to the boundary region. Therefore, the chaotic state is restored. In other words, the perturbation has included the chaotic saddle as part of the dynamics. The stability of the above mentioned period- 10 orbit depends on the amplitude of the period-2 perturbation. That is, period-10 is stable as long as Df (p) p (x) = Df(x i ) < 1, (7) i=1 where f is the autonomous system converted from the nonautonomous system (4), Df(x) is the Jacobian matrix of f at x, and p is the resonant period-p orbit (here p = 10). In this way the minimal amplitude of the perturbation sufficient to induce chaos can be obtained provided that the equation of the underlying system is available. We can be seen in Fig. 6, for higher values of perturbation amplitude (ε = 0.05), the LLE of perturbed system (4) versus ω remains positive (corresponding to chaos) except at ω = 0.2. This is due to the reinforcement of the original frequency locking regime in period-5 window of unperturbed map (2), since the perturbation frequency is the same as the resonant frequency of the unperturbed system. And we find that the resonance at ω = 0.2 will persist till the perturbation amplitude increases to ε = In addition, we find that periodic perturbation can also induce chaos at parameter ranges where map (2) exhibits quasiperiodic behavior. For example, let a = 0.7, where an invariant circle occurs in the phase space. When a perturbation with

7 frequency ω = 0.22 is exerted, the perturbed map (4) will demonstrate period-50 behavior, because the external frequency ω = 0.22 is locked with one internal frequency ω = of unperturbed system. And by increasing the perturbation amplitude, period doubling cascade of period-50 to chaos will be introduced. Chaos Inducement and Enhancement in Two Particular Nonlinear Maps Quasiperiodic Perturbation that Induces Chaos In the previous section, chaos inducement is achieved through periodic perturbation, whose frequency corresponds to rational number with zero measure over the interval of [0, 0.5]. Comparatively, there are more irrational numbers with nonzero measure that correspond to quasiperiodic frequencies. In this section we will demonstrate that quasiperiodic perturbation can also be used to induce chaos, and a quasiperiodic frequency ω = 5 1 is chosen for a detailed study In the period doubling systems In Fig. 8, we plot the LLE versus a for map (1) and its perturbed system (3), where the quasiperiodic perturbation has the parameter ω = 5 1, ε = Compare the two curves, we find that most periodic windows (with negative LLE) of the unperturbed system (1) are eliminated and replaced by chaos (with positive LLE) in the presence of the perturbation. Fig. 8. LLE versus a for map (1) (the top), and its perturbed system (3) with ω = 5 1, ε = 0.01 (the bottom). We still fix a = 1.25 for a detailed investigation, where map (1) demonstrates period-7 behavior. When the amplitude ε of the quasiperiodic perturbation is small, the period-7 orbit becomes a quasiperiodic attractor, or a torus attractor of 7 bands. As ε increases, the 7-torus attractor gets increasingly wrinkled and then transits to a 7- band SNA [Grebogi et al., 1984; Ding et al., 1989], characterized by its fractal geometry and nonpositive LLE, see Figs. 9(a) and 9(b) for details. For still larger ε, the 7-band SNA turns into a 1-band SNA (see Fig. 9(c)), and finally becomes a chaotic attractor. (a) SNA for map (3) when ε = , with corresponding LLE = (b) Enlargement view of (a). Fig. 9. Attractor of map (3) under different ε when a = 1.25, ω = 5 1.

8 1592 J. Zhang et al. (c) SNA for map (3) when ε = , with corresponding LLE = Fig. 9. (Continued ) Here the 7-band SNA is formed through the fractalization route [Nishikawa & Kaneko, 1996], and the following transition form 7-band SNA to 1-band SNA is due to an interior crisis. Now we examine the LLE curve in the whole evolution to chaos. As shown in Fig. 10, the LLE varies smoothly at the beginning, which correspond to the torus range marked in Fig. 10. Then it becomes nonmonotonic as ε increases, going through a series of transitions like torus SNA chaos SNA, and finally terminates in chaos. Fig. 10. LLE curve of map (3) versus ε with a = 1.25 and ω = 5 1. If we fix a = 1.26, where map (1) exhibits period-14 originating from period doubling of period-7, and gradually increase the perturbation amplitude, we will find another kind of transition to chaos via SNA known as Heagy Hammel route [Heagy & Hammel, 1994]. In fact, the transition from torus to chaos via SNA is typical in period doubling systems, and usually the amplitude sufficient for chaos inducement is quite small In the Hopf bifurcation systems Figure 11 plots the LLE versus control parameter a for map (2) and its quasiperiodically perturbed version (4), and two different mechanisms involved in inducing chaos are found here. One is similar to the mechanism in period doubling system, and the other is associated with the breakup of the frequency locking. We use period-5 and period-12 windows to explain them, respectively. We first fix a at 1.15, where map (2) exhibits period-5 behavior. And a perturbation with increasing amplitude ε is exerted. When ε is small, the period-5 orbit becomes a 5-torus. Then it wrinkles and transits to a 5-band SNA as ε increases, following a fractalization route [Nishikawa & Kaneko, 1996]. Figure 12 illustrates this phenomenon regarding only one band of the attractor. On increasing ε further, the 5-band SNA will become a 5-band chaotic attractor. This is different from

9 Chaos Inducement and Enhancement in Two Particular Nonlinear Maps 1593 Fig. 11. LLE curve versus a for map (2) (the top one), and its perturbed system (4) with ω = 5 1, ε = 0.01 (the bottom one). (a) LLE versus ε for map (4) when a = (b) LLE versus ε for map (4) when a = Fig. 13. LLE curve versus ε of perturbed map (4) (ω = 5 1) for period-5 and period-12 windows. Fig. 12. SNA of map (4) when a = 1.15, ε = , ω = 5 1, with corresponding LLE = the evolution in period doubling system, where the 7-band SNA first undergoes a crisis and then becomes a chaotic attractor. During the whole evolution, the LLE increases very slowly to positive (see Fig. 13(a)), because no crisis occurred. But there is a dramatic increase of the LLE later at ε = (denoted by an arrow in Fig. 13(a)), where the 5-band chaotic attractor undergoes attractor merging. Then we fix a at 1.098, where map (2) exhibits period-12 behavior. As a quasiperiodic perturbation ( ω = 5 1) is exerted with increasing amplitude ε, the period-12 becomes a 12-torus, and then disappears abruptly to be replaced by chaos as ε reaches , with type-ii intermittency found near the transition. During this process, the LLE increases from negative directly to positive and no SNA is found, which is illustrated in Fig. 13(b). The reason for the transition from 12-torus directly to chaos is the destruction of the quasiperiodic attractor (12-torus), or, from physical point of view, the breakup of frequency locking regime in the presence of the quasiperiodic perturbation, which is similar to the situation in Sec Quasiperiodic Perturbation that Enhances Chaos In addition to inducing chaos in nonchaotic parameter ranges of chaotic system, weak periodic/

10 1594 J. Zhang et al. quasiperiodic perturbations are also found to be able to enhance chaos. In this section we will demonstrate that by introducing different kinds of crisis, quasiperiodic perturbation produces a substantial enhancement of LLE for quite small perturbation amplitude In the period doubling systems For example, map (1) exhibits a 7-band chaotic attractor at a = 1.264, which arises from the period doubling cascade of period-7, see Fig. 14(a) for illustration. When a quasiperiodic perturbation (ω = 5 1) with increasing amplitude is exerted, each Fig. 15. LLE versus ε for map (3) when ω = 5 1 and a = The dash-dot line indicates the LLE of unperturbed map (1) at a = (a) Attractor for map (1) without perturbation. band of the attractor will grow and become somewhat fractalized. Then they merge to form a global attractor as the perturbation amplitude increases to , where an interior crisis occurs, see Fig. 14(b) for illustration. We plot the corresponding LLE curve in Fig. 15, and find that the LLE varies slightly at the beginning, but increases dramatically after the crisis. The reason for the abrupt increase of LLE is that the coexisting repeller at a = of map (3) has become a part of the whole attractor. This repeller is the remnant of the chaotic attractor which disappears at the saddle-node bifurcation, (b) Attractor for map (3) with perturbation ε = , ω = 5 1. Fig. 14. Attractor for map (1) and its perturbed system (3) at a = Fig. 16. LLE versus ε for map (3) when ω = 5 1 and a = , where three band mergings happen at 1(ω = ), 2(ω = 0.012), 3(ω = 0.064). And the dash-dot line indicates the LLE of unperturbed map (1) at a =

11 and it has a larger LLE than the 7-band chaotic attractor. Moreover, quasiperiodic perturbation can also enhance chaos by provoking band merging. For example, there is an 8-band attractor for map (1) at a = , When a quasiperiodic perturbation of frequency ω = 5 1 is exerted with gradually increased amplitude, this attractor will undergo three levels of band-merging in turn, i.e. from 8 bands 4 bands 2 bands 1 band. In Fig. 16 we can see that each time the band merging occurs, there will be a dramatic increase in LLE. And when Chaos Inducement and Enhancement in Two Particular Nonlinear Maps 1595 Fig. 18. LLE versus ε for map (4) when ω = 5 1 and a = 0.969, and the dash-dot line is LLE of unperturbed map (2) at a = ε = 0.1, the LLE reaches 0.198, well above the LLE of the unperturbed system. The increase of LLE here is due to the increase of the available phase space volume, and consequently the local rate of divergence of trajectories. (a) Attractor for map (2) without perturbation In the Hopf bifurcation systems Compared with the case discussed in Sec. 5.1, the attractor merging induced by perturbation in map (2) is somewhat different. Notice that there are nine separate chaotic attractors of map (2) when a = 0.969, which are plotted in Fig. 17(a). And these nine attractors undergo attractor merging when the amplitude of the quasiperiodic perturbation (ω = 5 1) reaches ε = , corresponding to a dramatic increase in LLE in Fig. 18. However, these nine separate chaotic attractors are characterized by two positive Lyapunov exponents, therefore differing manifestly from the attractor merging in strictly dissipative systems like map (1) and (3) [Grebogi et al., 1987]. And further investigation is needed to clarify the structure near basin boundaries of the separate attractors at the transition. (b) Attractor for map (4) with perturbation ε = , ω = 5 1. Fig. 17. Attractor for map (2) and its perturbed system (4) at a = Periodic Intermittency In this section, we will investigate a special kind of intermittency found in the perturbed Henon map (3). This intermittency has been reported by [Yang et al., 1996] in Duffing system when an extra forcing slightly detunes the original forcing. While our example shows that such intermittency can also

12 1596 J. Zhang et al. occur for discrete autonomous system with one periodic perturbation. Notice period-24 windows occurs in map (1) when a = If we exert a perturbation with fixed amplitude ε = 0.05 and varying frequency, many periodic orbits would be excited, which correspond to negative LLE in Fig. 19. Actually, these periodic orbits are unstable periodic orbits of the unperturbed system which resonate with the perturbation. And if perturbation frequency slightly misses one of the resonant frequencies, a special kind of intermittency will occur. For example, let the perturbation frequency be ω = , which differs slightly from one resonant frequency ω = We plot series x(n) versus n for iterations in Fig. 20 and find that the system moves regularly in certain time intervals, while goes chaotically in others. After a fixed time length T = , the motions are precisely repeated. This periodicity of such intermittency distinguishes itself from conventional intermittencies, in which periodic motion and chaotic motion appear randomly. Now we analyze this intermittency in terms of the geometrical structure of the attractor. In Fig. 20, we divide the periodic segment into three successive evolutions: E1, E2, and E3, and mark them correspondingly in the attractor plotted in Fig. 21. For reason of visualization, we only plot four of the 16 branches in the E1 in Fig. 21. We can see in Fig. 20 that as n increases, the 16 branches in E1 converge to the eight branches in E2, Fig. 20. x(n) versus n for map (3) when a = 1.073, ε = 0.05, ω = Fig. 21. Attractor for map (3) when a = 1.073, ε = 0.05, ω = , where E1, E2, E3 correspond to the three evolutions marked in Fig. 20, and P4 ( ), P8 ( ), P16 ( ) are saddle type periodic orbits. Fig. 19. LLE versus ω for map (3) when a = 1.073, ε = And the marked resonant frequencies are: 1(ω = 0.125), 2(ω = ), 3(ω = 0.25). and then to the four branches in E3. This lead us to speculate there are three saddle type periodic orbits (period-16, period-8, and period-4 in Fig. 21) near the converging points, and the branches that connect adjacent two periodic orbits are actually both the unstable manifold of the former periodic orbit, and the stable manifold of the latter periodic orbit. With this speculation, the evolution process from E1 through E2 to E3 is well explained: the iteration starts from P16, and leaves it gradually along its unstable manifold. Since P16 s unstable manifold is also P8 s stable manifold, the iteration will gradually approach P8 and converge near it. Similarly, the

13 Chaos Inducement and Enhancement in Two Particular Nonlinear Maps 1597 iteration continues to move away along the unstable manifold of P8, and approaches P4 along its stable manifold. Finally, the iteration leaves P4 along its unstable manifold towards the chaotic region, until it once again enters the periodic segment. Actually, we also find many such periodic orbits in the chaotic region in Fig. 21, or correspondingly, periodic pieces (branches) within the chaotic segment in Fig. 20. It is just the delicate configuration of the large number of saddle type periodic orbits that leads to the regularity of such intermittency. 7. Conclusion In this paper, we exert weak periodic/quasiperiodic perturbations on two particular chaotic maps, i.e. a period doubling system and a Hopf bifurcation system, and have found that the adopted perturbations can induce chaos in the nonchaotic parameter ranges of chaotic maps, or enhance the existing chaotic state. The periodic perturbation induces chaos in different ways for the two systems. In period doubling system, chaos is induced through a period doubling cascade of the resonant periodic orbit, while for Hopf bifurcation system, chaos arises due to breakup of the frequency locking regime. As to quasiperiodic perturbation, we find the inducement of chaos via a SNA in both systems. However, for certain periodic windows of the Hopf bifurcation system, chaos is also found to be induced through the breakup of the frequency locking regime. The reason why Hopf bifurcation system demonstrates a different mechanism from period doubling system when chaos is induced can be attributed to its dynamics governed by two fundamental frequencies. Therefore when the frequency locking is destroyed under external perturbation, chaotic state prior to frequency locking windows will recur. In terms of chaos enhancement, the mechanism for the two systems are similar, i.e. the introduction of interior crisis or attractor merging will increase the phase space volume, and consequently the rate of divergence of trajectories. In addition, we find the special periodic intermittency for the first time in the periodically perturbed Henon map. And we analyze its periodicity by giving reasonable speculations on the configuration of the large number of saddle type periodic orbits embedded in the attractor. References Barreto, E., Hunt, B. R., Grebogi, C. & Yorke, J. A. [1997] From high dimensional chaos to stable periodic orbits: The structure of parameter space, Phys. Rev. Lett. 78, Braiman, Y. & Goldhirsch, I. [1991] Taming chaotic dynamics with weak periodic perturbations, Phys. Rev. Lett. 66, Chacón, R. [1999] General results on chaos suppression for biharmonically driven dissipative systems, Phys. Lett. A 257, Chen, G. & Lai, D. [1998] Feedback anticontrol of discrete chaos, Int. J. Bifurcation and Chaos 8, Dhamala, M. & Lai, Y. C. [1999] Controlling transient chaos in deterministic flows with applications to electrical power systems and ecology, Phys. Rev. E 59, Ding, M., Grebogi, C. & Ott, E. [1989] Evolution of attractors in quasiperiodically forced systems: From quasiperiodic to strange nonchaotic to chaotic, Phys. Rev. A 39, González, J. A., Mello, B. A., Reyes, L. I. & Guerrero, L. E. [1998] Resonance phenomena of a solitonlike extended object in a bistable potential, Phys. Rev. Lett. 80, Grebogi, C., Ott, E., Pelikan, S. & Yorke, J. A. [1984] Strange attractors that are not chaotic, Physica D 13, Grebogi, C., Ott, E. & Yorke, J. A. [1987] Critical exponents for crisis-induced intermittency, Phys. Rev. A 36, Gupte, N. & Amritkar, R. E. [1996] Enhancing chaos in chaotic maps and flows, Phys. Rev. E 54, Hao, B. L. & Zheng, W. M. [1998] Applied Symbolic Dynamics and Chaos (World Scientific, Singapore). Heagy, J. F. & Hammel, S. M. [1994] The birth of strange nonchaotic attractors, Physica D 70, Hunt, B. R., Gallas, J., Grebogi, C., Yorke, J. A. & Kocak, H. [1999] Bifurcation rigidity, Physica D 129, Meucci, R., Gadomski, W., Ciofini, M. & Arecchi, F. T. [1994] Experimental control of chaos by means of weak parametric perturbations, Phys. Rev. E 49, Murakami, C., Murakami, W. & Hirose, K. [2002] Sequence of global period doubling bifurcation in the Henon maps, Chaos Solit. Fract. 14, Nishikawa, T. & Kaneko, K. [1996] Fractalization of a torus as a strange nonchaotic attractor, Phys. Rev. E 54, Ott, E., Grebogi, C. & Yorke, J. A. [1990] Controlling chaos, Phys. Rev. Lett. 64,

14 1598 J. Zhang et al. Rademacher, H. [1964] Lectures on Elementary Number Theory (Blaisdell, NY). Rothstein, D., Henry E. & Gollub, J. P. [1999] Persistent patterns in transient chaotic fluid mixing, Nature 401, Schiff, S. J., Jerger, K., Duong, D. H., Chang, T., Spano, M. L. & Ditto, W. L. [1994] Controlling chaos in the brain, Nature 370, Schwartz, I. B. & Triandaf, I. [1996] Sustaining chaos by using basin boundary saddles, Phys. Rev. Lett. 77, Schwartz, I. B. & Georgiou, I. T. [1998] Instant chaos and hysteresis in coupled linear-nonlinear oscillators, Phys. Lett. A 242, Schwartz, I. B., Triandaf, I., Meucci, R. & Carr, T. W. [2002] Open-loop sustained chaos and control: A manifold approach, Phys. Rev. E 66, Sonis, M. [1996] Once more on Hénon map: Analysis of bifurcations, Chaos Solit. Fract. 7, Tereshko, V. & Shchekinova, E. [1998] Resonant control of the Rössler system, Phys. Rev. E 58, Triandaf, I. & Schwartz, I. B. [2000] Tracking sustained chaos: A segmentation method, Phys. Rev. E 62, VanWiggeren, G. D. & Roy, R. [1998] Communication with chaotic lasers, Science 279, Wang, X. F. & Chen, G. [2000] Chaotification via arbitrarily small feedback controls: Theory, method, and applications, Int. J. Bifurcation and Chaos 10, Yang, J. Z., Qu, Z. L. & Hu, G. [1996] Duffing equation with two periodic forcings: The phase effect, Phys. Rev. E 53, Yang, W., Ding, M., Mandell, A. J. & Ott, E. [1995] Preserving chaos: Control strategies to preserve complex dynamics with potential relevance to biological disorders, Phys. Rev. E 51,