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Chaos control by harmonic excitation with proper random phase
Chaos, Solitons and Fractals 21 (2004) 1175–1181 www.elsevier.com/locate/chaos
Chaos control by harmonic excitation with proper random phase Youming Lei a
a,*
, Wei Xu a, Yong Xu a, Tong Fang
b
Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an 710072, PR China Department of Applied Mechanics, Northwestern Polytechnical University, Xi’an 710072, PR China
b
Accepted 8 December 2002
Abstract Chaos control may have a dual function: to suppress chaos or to generate it. We are interested in a kind of chaos control by exerting a weak harmonic excitation with random phase. The dual function of chaos control in a nonlinear dynamic system, whether a suppressing one or a generating one, can be realized by properly adjusting the level of random phase and determined by the sign of the top Lyapunov exponent of the system response. Two illustrative examples, a Duffing oscillator subject to a harmonic parametric control and a driven Murali-Lakshmanan-Chua (MLC) circuit imposed with a weak harmonic control, are presented here to show that the random phase plays a decisive role for control function. The method for computing the top Lyapunov exponent is based on Khasminskii’s formulation for linearized systems. Then, the obtained results are further verified by the Poincare map analysis on dynamical behavior of the system, such as stability, bifurcation and chaos. Both two methods lead to fully consistent results. Ó 2004 Elsevier Ltd. All rights reserved.
1. Introduction The control of chaos is a hot topic in scientific research today. One of the basic characteristics of chaos in a nonlinear dynamic system is the repellency of any two adjacent trajectories of chaos, which results in a positive top Lyapunov exponent of the system. Broadly speaking, there are two kinds of chaos control. One is the so-called OGY method, first introduced in 1990 by Ott, Grebogi and Yorke [1–3], which uses some weak feedback control to make the chaotic trajectory approach and settle down finally to a desired stabilized periodic orbit, formerly unstably embedded in the chaotic manifold. The OGY method is a kind of feedback control. And the other kind of chaos control belongs to nonfeedback control, which usually uses given external or parametric excitations to control system behavior. Both two kinds of chaos control are still in developing [4–10]. A detailed review on chaos control updated to the end of last century was written by Boccaletti et al. [4]. Chen and Dong [5] gave a general survey on evolution of chaos and its control. Chen and Lai suggested a feedback anti-control for discrete chaos [6], and suggested also a feedback control for Lyapunov exponents to generate chaos in a dynamical system [7]. Ramesh and Narayanan [9] explored the robustness in non-feedback chaos control in presence of uniform noise and found that the system would lose control while noise amplitude was raised to a threshold level. Wei and Leng [11] studied the chaotic behavior in Duffing oscillator in presence of white noise by the Lyapunov exponent. Liu et al. [12] investigated the generation of chaos in a kind of Hamiltonian system subject to bounded noise by the criterion of stochastic Melnikov function and Lyapunov exponent. Qu, Hu et al. [13] further applied weak harmonic excitations to investigate the chaos control of non-autonomous systems, and especially observed that the phase control in weak harmonic excitation may greatly affect taming nonautonomous chaos, which arousing our further concerning on the effect of random phase control.
*
Corresponding author.
0960-0779/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2003.12.086
1176
Y. Lei et al. / Chaos, Solitons and Fractals 21 (2004) 1175–1181
As a matter of fact, chaos control may have a dual function: to suppress or to generate chaos. We are interested in a kind of non-feedback chaos control by exerting a weak harmonic excitation with random phase. The dual function of chaos control in a nonlinear dynamic system, whether a suppressing one or a generating one, can be realized by properly adjusting the level of random phase. And the change of sign for the top Lyapunov exponent of the system response can be taken as an effective criterion for transition of chaos in a stochastic nonlinear dynamic system. Two illustrative examples, a Duffing oscillator subject to a harmonic parametric control and a Murali-Lakshmanan-Chua (MLC) circuit imposed with a weak harmonic control, are presented here to show that the random phase plays a decisive role for control function. The method for computing the top Lyapunov exponent is based on Khasminskii’s formulation for linearized systems [14], later on which was realized in programming by Wedig [15]. Based on the extension of Wedig’s algorithm for linear stochastic systems, top Lyapunov exponents of two kinds of Duffing systems were calculated by Ariaratnam [16] and Wei [11] respectively. In this paper, we first explore the effect of random phase on the function of chaos control in the two examples by the Lyapunov exponent criterion. Then the obtained results are further verified by corresponding Poincare map analysis. The two results are found to be fully consistent.
2. Stochastic duffing system controlled by parametric excitation Consider a nonlinear stochastic Duffing System in the form, €x þ a_x x þ x3 ¼ lx sin½Xt þ DnðtÞ
ð1Þ
where a and l denote the damping coefficient and the amplitude of the parametric excitation respectively, nðtÞ represents a standard Gaussian white noise with Dirac-Delta function as its correlation, namely, EnðtÞ ¼ 0;
EnðtÞnðt þ sÞ ¼ dðsÞ
ð2Þ
Equation (1) can be rewritten as, x_ 1 ¼ x2 x_ 2 ¼ x1 x31 þ lx1 sin½Xt þ DnðtÞ ax2
ð3Þ
The way to produce the sample excitation is based on the work of Shinozuka [17]. The sample responses of system (3) can be obtained by any effective numerical method. So the derivative dx31 =dx1 ð¼ 3x21 Þ can be readily obtained, too. The Khasminskii’s formulation for the top Lyapunov exponent is based on the linearized system, which can be expressed as y_ 1 ¼ y2 ð4Þ y_ 2 ¼ f1 3x21 þ l sin½Xt þ DnðtÞgy1 ay2 Let f ðtÞ ¼ 3x21 l sin½Xt þ DnðtÞ, we have 0 y1 ; b¼ y_ ¼ ½b þ f ðtÞry; y ¼ y2 1
1 ; a
r¼
0 0 1 0
ð5Þ
Assume that f ðtÞ is ergodic and E½kb þ f ðtÞrk < 1, where the norm kAk is defined as the square root of the largest eigenvalue of the matrix AT A. By Oseledec multiple ergodic theorem [18], there exists two real numbers k1 , k2 and two random subspaces E 1 , E 2 ðE E 2 ¼ U d ðOÞ R2 , U d ðOÞ denotes the neighborhood of Oð0; 0Þ), such that, k1 ¼ lim
t!þ1
1 log kyðtÞk if and only if t
y0 2 E i n fOg; i ¼ 1; 2
ð6Þ
where kyðtÞk ¼ ðy12 þ y22 Þ1=2 , ki ði ¼ 1; or 2Þ is the Lyapunov exponent, representing the rate of exponential convergence or divergence of nearby orbits in a specific direction in E i . The Oseledec multiple ergodic theorem states that for almost all random initial values in random subset U d ðOÞ there holds k ¼ maxi ki ¼ limt!þ1 1t log kyðtÞk, and k is defined as the largest (or top) Lyapunov exponent. Using Khasminskii’s technique, the computation of the largest Lyapunov exponent of system (3) can be presented as follows: Let yi si ¼ ; i ¼ 1; 2; a ¼ kyk ð7Þ a It follows that
Y. Lei et al. / Chaos, Solitons and Fractals 21 (2004) 1175–1181
X s0i ¼ ½bij aðtÞdij þ ðrij bðtÞdij Þf ðtÞsj
1177
ð8Þ
j
where aðtÞ ¼
P
k;l
bkl sk sl , bðtÞ ¼
P
k;l
rkl sk sl , dij ¼ 1ði ¼ jÞ, dij ¼ 0ði 6¼ jÞ, and
0
a ¼ ½aðtÞ þ bðtÞf ðtÞa
ð9Þ
Then, the largest Lyapunov exponent can be expressed as, k ¼ lim
t!þ1
1 1 log a ¼ lim t!þ1 t t
Z
t
½aðsÞ þ bðsÞf ðsÞ ds
ð10Þ
0
Now the top Lyapunov exponent can be obtained by numerical integration of equation (10). Let a ¼ 0:2, l ¼ 0:5, X ¼ 1:0, the relation between top Lyapunov exponent and the amplitude of the stochastic phase is shown in Fig. 1. The original system is chaotic when D ¼ 0, as shown in Fig. 2a. As the amplitude of noise D increases slightly the top Lyapunov exponent still remains positive. However, when the amplitude arrives at a critical value DC 0:0275, the sign of top Lyapunov exponent suddenly turns from positive to negative, namely the behavior of system (1) turns from chaotic to stable abruptly. From then on, the increase of the amplitude of stochastic phase would not affect the sign of top Lyapunov exponent any longer in the interested parameter range. This suggests the random phase noise effectively stabilizes the system for the parameter range, D > DC . Now we apply Poincare map of Equation (1) to verify the above results. Set Poincare map as P:
X
!
X ;
X
¼ fðxðtÞ; x_ ðtÞÞjt ¼ 0; 2p=X; 4p=X; . . .g R2 :
One hundred initial points are randomly chosen on phase plane. For each initial condition, the differential equation (1) is solved by the sixth order Runge-Kutta-Verner method and the solution is plotted for every T ¼ 2p=X; for each initial point, after deleting the first 500 transient points, the succeeded 200 iteration points are used to plot the Poincare map. It is already known from the analysis of the top Lyapunov exponent shown in Fig. 1 that for D ¼ 0 and D ¼ 0:01, system (1) is chaotic; and for D ¼ 0:05 and D ¼ 0:10 the system is stable almost everywhere. These parameter values are selected for plotting Poincare map and the results are shown in Fig. 2. It is seen from Fig. 2 that when D < Dc the original system has a strange attractor, and when D > Dc there appear two stable stochastic attractors and an unstable saddle. That implies, for almost all initial conditions, the Poincare points will fall on the union of two stable stochastic attractors with probability 1.
Fig. 1. Top Lyapunov exponents k vs. noise amplitude (stochastic phase) D.
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Y. Lei et al. / Chaos, Solitons and Fractals 21 (2004) 1175–1181
Fig. 2. Poincare maps of Eq. (1) vs. increase of amplitude of noise D (a) Poincare map for the excitation without stochastic noise, i.e. D ¼ 0 (b) Poincare map for D ¼ 0:01 (c) Poincare map for D ¼ 0:05 (d) Poincare map for D ¼ 0:10.
3. Harmonically driven Murali-Lakshmanan-Chua circuit This section is devoted to investigating the generation of chaos in the MLC circuit [9] in the form, x_ 1 ¼ x2 gðx1 Þ x_ 2 ¼ rx2 bx1 þ F sinðxtÞ þ Fc sinðxc tÞ
ð11Þ
where gðxÞ ¼ bx þ 0:5ða bÞ½jx þ 1j jx 1j For b ¼ 1:0, r ¼ 1:015, x ¼ 0:75, F ¼ 0:15, a ¼ 1:02, b ¼ 0:55, xc ¼ x, and Fc ¼ 0:06, the Poincare map of system (11), Fig. 4(a), shows system (11) is stable with an attractor of period 3T. Now the effect of stochastic phase is to be explored by adding a random phase to the second harmonic excitation in system (11), then, the corresponding system is x_ 1 ¼ x2 gðx1 Þ ð12Þ x_ 2 ¼ rx2 bx1 þ F sinðxtÞ þ Fc sinðxc t þ DnðtÞÞ where nðtÞ is a standard Gaussian white noise. The linearized equation of (12) can be expressed as y_ 1 ¼ y2 g0 ðx1 Þy1 y_ 2 ¼ ry2 by1
ð13Þ
Y. Lei et al. / Chaos, Solitons and Fractals 21 (2004) 1175–1181
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Fig. 3. Top Lyapunov exponents k vs. amplitude of noise (stochastic phase) D.
Fig. 4. (a) Poincare map for excitation without stochastic phase, i.e. D ¼ 0 (b) Poincare map for D ¼ 0:005 (c) Poincare map for D ¼ 0:01 (d) Poincare map for D ¼ 0:05 (e) Local magnification of Fig. 4(d) (f) Poincare map for D ¼ 0:10; (g) Local magnification of (f).
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Y. Lei et al. / Chaos, Solitons and Fractals 21 (2004) 1175–1181
Fig. 4 (continued)
with 0
g ðxÞ ¼
a; b;
1 < x < 1 otherwise:
Like in Section 2, the top Lyapunov exponent is computed and its connection with amplitude of stochastic phase is shown in Fig. 3. One can see in Fig. 3, when the stochastic phase is inactive, namely D ¼ 0, the state of system (11) is controlled on periodic trajectory; when the amplitude increases gradually, there will appear unstable tendency of our system (13), and the critical value of amplitude is about 0.040, that is, if D > 0:040 the chaos will be observed for system (12). Obviously, when D ¼ 0:0005 and D ¼ 0:01 the top Lyapunov exponent is negative and the system is stable, but when D ¼ 0:05 and D ¼ 0:10 the top Lyapunov exponent is positive and the system is chaotic. That is to say, chaos will appear suddenly when the noise amplitude exceeds a critical value, Dc 0:040. Here Poincare map approach is applied again to check the results mentioned above and the excellent agreement has been observed. For several given values of amplitude, we plot Poincare maps as Fig. 4.
4. Conclusion and discussion Based on the work of Wedig [15] and Khasminskii [14], the top Lyapunov exponents are computed for two kinds of nonlinear stochastic systems. And at the same time we investigate the effect of stochastic phase on the function of nonfeedback control. Opposite results about the effect of stochastic phase are observed for two different stochastic systems, the parametrically excited Duffing oscillator and the driven MLC circuit. That is to say, the increase of stochastic phase can either quench chaos or generate chaos. The related Poincare map analyses fully confirm these results.
Y. Lei et al. / Chaos, Solitons and Fractals 21 (2004) 1175–1181
1181
Now we have to point out that in a stochastic dynamic system the phenomenon of transition of a chaotic state to a regular state or vise versa is called stochastic bifurcation, which usually can be defined by the criterion for change of sign of the top Lyapunov exponent of the system. Hence, the transition phenomena in our two examples should be taken as stochastic bifurcation [19]. An alternative definition for stochastic bifurcation has been proposed by Xu et al. [19], where stochastic bifurcation is defined as the sudden change of topological character of stochastic attractors. The results of the top Lyapunov exponent analysis and the corresponding Poincare map analysis in this paper show that these two definitions are consistent in the cases we considered.
Acknowledgements The authors are grateful for the support of National Science Foundation of China (Grant No. 10332030). The third author would also like to express his gratitude to the Doctorate Creation Foundation of Northwestern Polytechnical University (Grant No. CX200326).
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
[16] [17] [18] [19]
Ott E, Grebogi C, Yorke JA. Controlling chaos. Phys Rev Lett 1990;64:1196–9. Shinbrot T, Ott E, Grebogi C, Yorke JA. Using small perturbations to control chaos. Nature 1993;363:411–7. Shinbrot T, Ott E, Grebogi C, Yorke JA. Using chaos to direct trajectories. Phys Rev Lett 1990;65:3215–8. Boccaletti S et al. The control of chaos: theory and applications. Physics Reports 2000;329:103–97. Chen G, Dong X. From Chaos to Order: Methodologies, Perspectives, and Applications. Singapore: World Scientific; 1998. Chen G, Lai D. Feedback of anticontrol of discrete chaos. Int J Bifurcation and Chaos 1998;8(7):1585–90. Chen G, Lai D. Making a dynamical system chaotic: feedback control of Lyapunov exponents for discrete-time dynamical systems. IEEE Trans Circuits Syst I 1997;44:250–3. Lima R, Pettini M. Suppression of chaos by resonant parametric perturbations. Phys Rev A 1990;41:726–33. Ramesh M, Narayanan S. Chaos control by nonfeedback methods in the presence of noise. Chaos, Soliton and Fractals 1999;10:1473–89. Braiman Y et al. Taming chaotic dynamics with weak periodic perturbations. Phys Rev Lett 1991;20:2545–8. Wei JG, Leng G. Lyapunov exponent and chaos of Duffing’s equation perturbed by white noise. Applied Mathematics and Computation 1997;88:77–93. Liu WY, Zhu WQ, Huang ZL. Effect of bounded noise on chaotic motion of a Duffing oscillator under parametric excitation. Chaos, Solitons and Fractals 2001;12:527–37. Qu Z, Hu G, et al. Phase effect in taming nonautonomous chaos by weak harmonic perturbations. Phys Rev Lett 1995;74:1736–9. Khasminskii RZ. Necessary and sufficient conditions for the asymptotic stability of linear stochastic systems. Theory Probability Appl 1967;12:44–147. Wedig WV. Dynamic stability of beams under axial forces – Lyapunov exponents for general fluctuating loads, in Proceedings Eurodyn’90, Conference on Structure Dynamics (Edited by Kratzig W.B. et al.), Vol. 1, pp 57-64, Ruhr-University Bochum, SFB 151 (1990). Ariaratnam ST, Xie WC. Almost-sure stochastic stability of coupled non-linear oscillators. Int J Non-Linear Mechanics 1994;29:197–204. Shinozuka M. Digital simulation of random processes and its applications. Journal of Sound and Vibration 1972;25:111–28. Oseledec YI. A multiplicative ergodic theorem: Lyapunov characteristic number for dynamical systems. Trans Moscow Math Soc 1968;19:197–231 (English translation). Xu W, He Q, Rong HW, Fang T. Global analysis of stochastic bifurcation in Ueda system, in Proceedings of the Fifth International Conference on Stochastic Structural Dynamics-SSD03 (Edited by Zhu W.Q. et al.), pp 509-515, CRC Press (2003).
Chaos control by harmonic excitation with proper random phase Youming Lei a
a,*
, Wei Xu a, Yong Xu a, Tong Fang
b
Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an 710072, PR China Department of Applied Mechanics, Northwestern Polytechnical University, Xi’an 710072, PR China
b
Accepted 8 December 2002
Abstract Chaos control may have a dual function: to suppress chaos or to generate it. We are interested in a kind of chaos control by exerting a weak harmonic excitation with random phase. The dual function of chaos control in a nonlinear dynamic system, whether a suppressing one or a generating one, can be realized by properly adjusting the level of random phase and determined by the sign of the top Lyapunov exponent of the system response. Two illustrative examples, a Duffing oscillator subject to a harmonic parametric control and a driven Murali-Lakshmanan-Chua (MLC) circuit imposed with a weak harmonic control, are presented here to show that the random phase plays a decisive role for control function. The method for computing the top Lyapunov exponent is based on Khasminskii’s formulation for linearized systems. Then, the obtained results are further verified by the Poincare map analysis on dynamical behavior of the system, such as stability, bifurcation and chaos. Both two methods lead to fully consistent results. Ó 2004 Elsevier Ltd. All rights reserved.
1. Introduction The control of chaos is a hot topic in scientific research today. One of the basic characteristics of chaos in a nonlinear dynamic system is the repellency of any two adjacent trajectories of chaos, which results in a positive top Lyapunov exponent of the system. Broadly speaking, there are two kinds of chaos control. One is the so-called OGY method, first introduced in 1990 by Ott, Grebogi and Yorke [1–3], which uses some weak feedback control to make the chaotic trajectory approach and settle down finally to a desired stabilized periodic orbit, formerly unstably embedded in the chaotic manifold. The OGY method is a kind of feedback control. And the other kind of chaos control belongs to nonfeedback control, which usually uses given external or parametric excitations to control system behavior. Both two kinds of chaos control are still in developing [4–10]. A detailed review on chaos control updated to the end of last century was written by Boccaletti et al. [4]. Chen and Dong [5] gave a general survey on evolution of chaos and its control. Chen and Lai suggested a feedback anti-control for discrete chaos [6], and suggested also a feedback control for Lyapunov exponents to generate chaos in a dynamical system [7]. Ramesh and Narayanan [9] explored the robustness in non-feedback chaos control in presence of uniform noise and found that the system would lose control while noise amplitude was raised to a threshold level. Wei and Leng [11] studied the chaotic behavior in Duffing oscillator in presence of white noise by the Lyapunov exponent. Liu et al. [12] investigated the generation of chaos in a kind of Hamiltonian system subject to bounded noise by the criterion of stochastic Melnikov function and Lyapunov exponent. Qu, Hu et al. [13] further applied weak harmonic excitations to investigate the chaos control of non-autonomous systems, and especially observed that the phase control in weak harmonic excitation may greatly affect taming nonautonomous chaos, which arousing our further concerning on the effect of random phase control.
*
Corresponding author.
0960-0779/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2003.12.086
1176
Y. Lei et al. / Chaos, Solitons and Fractals 21 (2004) 1175–1181
As a matter of fact, chaos control may have a dual function: to suppress or to generate chaos. We are interested in a kind of non-feedback chaos control by exerting a weak harmonic excitation with random phase. The dual function of chaos control in a nonlinear dynamic system, whether a suppressing one or a generating one, can be realized by properly adjusting the level of random phase. And the change of sign for the top Lyapunov exponent of the system response can be taken as an effective criterion for transition of chaos in a stochastic nonlinear dynamic system. Two illustrative examples, a Duffing oscillator subject to a harmonic parametric control and a Murali-Lakshmanan-Chua (MLC) circuit imposed with a weak harmonic control, are presented here to show that the random phase plays a decisive role for control function. The method for computing the top Lyapunov exponent is based on Khasminskii’s formulation for linearized systems [14], later on which was realized in programming by Wedig [15]. Based on the extension of Wedig’s algorithm for linear stochastic systems, top Lyapunov exponents of two kinds of Duffing systems were calculated by Ariaratnam [16] and Wei [11] respectively. In this paper, we first explore the effect of random phase on the function of chaos control in the two examples by the Lyapunov exponent criterion. Then the obtained results are further verified by corresponding Poincare map analysis. The two results are found to be fully consistent.
2. Stochastic duffing system controlled by parametric excitation Consider a nonlinear stochastic Duffing System in the form, €x þ a_x x þ x3 ¼ lx sin½Xt þ DnðtÞ
ð1Þ
where a and l denote the damping coefficient and the amplitude of the parametric excitation respectively, nðtÞ represents a standard Gaussian white noise with Dirac-Delta function as its correlation, namely, EnðtÞ ¼ 0;
EnðtÞnðt þ sÞ ¼ dðsÞ
ð2Þ
Equation (1) can be rewritten as, x_ 1 ¼ x2 x_ 2 ¼ x1 x31 þ lx1 sin½Xt þ DnðtÞ ax2
ð3Þ
The way to produce the sample excitation is based on the work of Shinozuka [17]. The sample responses of system (3) can be obtained by any effective numerical method. So the derivative dx31 =dx1 ð¼ 3x21 Þ can be readily obtained, too. The Khasminskii’s formulation for the top Lyapunov exponent is based on the linearized system, which can be expressed as y_ 1 ¼ y2 ð4Þ y_ 2 ¼ f1 3x21 þ l sin½Xt þ DnðtÞgy1 ay2 Let f ðtÞ ¼ 3x21 l sin½Xt þ DnðtÞ, we have 0 y1 ; b¼ y_ ¼ ½b þ f ðtÞry; y ¼ y2 1
1 ; a
r¼
0 0 1 0
ð5Þ
Assume that f ðtÞ is ergodic and E½kb þ f ðtÞrk < 1, where the norm kAk is defined as the square root of the largest eigenvalue of the matrix AT A. By Oseledec multiple ergodic theorem [18], there exists two real numbers k1 , k2 and two random subspaces E 1 , E 2 ðE E 2 ¼ U d ðOÞ R2 , U d ðOÞ denotes the neighborhood of Oð0; 0Þ), such that, k1 ¼ lim
t!þ1
1 log kyðtÞk if and only if t
y0 2 E i n fOg; i ¼ 1; 2
ð6Þ
where kyðtÞk ¼ ðy12 þ y22 Þ1=2 , ki ði ¼ 1; or 2Þ is the Lyapunov exponent, representing the rate of exponential convergence or divergence of nearby orbits in a specific direction in E i . The Oseledec multiple ergodic theorem states that for almost all random initial values in random subset U d ðOÞ there holds k ¼ maxi ki ¼ limt!þ1 1t log kyðtÞk, and k is defined as the largest (or top) Lyapunov exponent. Using Khasminskii’s technique, the computation of the largest Lyapunov exponent of system (3) can be presented as follows: Let yi si ¼ ; i ¼ 1; 2; a ¼ kyk ð7Þ a It follows that
Y. Lei et al. / Chaos, Solitons and Fractals 21 (2004) 1175–1181
X s0i ¼ ½bij aðtÞdij þ ðrij bðtÞdij Þf ðtÞsj
1177
ð8Þ
j
where aðtÞ ¼
P
k;l
bkl sk sl , bðtÞ ¼
P
k;l
rkl sk sl , dij ¼ 1ði ¼ jÞ, dij ¼ 0ði 6¼ jÞ, and
0
a ¼ ½aðtÞ þ bðtÞf ðtÞa
ð9Þ
Then, the largest Lyapunov exponent can be expressed as, k ¼ lim
t!þ1
1 1 log a ¼ lim t!þ1 t t
Z
t
½aðsÞ þ bðsÞf ðsÞ ds
ð10Þ
0
Now the top Lyapunov exponent can be obtained by numerical integration of equation (10). Let a ¼ 0:2, l ¼ 0:5, X ¼ 1:0, the relation between top Lyapunov exponent and the amplitude of the stochastic phase is shown in Fig. 1. The original system is chaotic when D ¼ 0, as shown in Fig. 2a. As the amplitude of noise D increases slightly the top Lyapunov exponent still remains positive. However, when the amplitude arrives at a critical value DC 0:0275, the sign of top Lyapunov exponent suddenly turns from positive to negative, namely the behavior of system (1) turns from chaotic to stable abruptly. From then on, the increase of the amplitude of stochastic phase would not affect the sign of top Lyapunov exponent any longer in the interested parameter range. This suggests the random phase noise effectively stabilizes the system for the parameter range, D > DC . Now we apply Poincare map of Equation (1) to verify the above results. Set Poincare map as P:
X
!
X ;
X
¼ fðxðtÞ; x_ ðtÞÞjt ¼ 0; 2p=X; 4p=X; . . .g R2 :
One hundred initial points are randomly chosen on phase plane. For each initial condition, the differential equation (1) is solved by the sixth order Runge-Kutta-Verner method and the solution is plotted for every T ¼ 2p=X; for each initial point, after deleting the first 500 transient points, the succeeded 200 iteration points are used to plot the Poincare map. It is already known from the analysis of the top Lyapunov exponent shown in Fig. 1 that for D ¼ 0 and D ¼ 0:01, system (1) is chaotic; and for D ¼ 0:05 and D ¼ 0:10 the system is stable almost everywhere. These parameter values are selected for plotting Poincare map and the results are shown in Fig. 2. It is seen from Fig. 2 that when D < Dc the original system has a strange attractor, and when D > Dc there appear two stable stochastic attractors and an unstable saddle. That implies, for almost all initial conditions, the Poincare points will fall on the union of two stable stochastic attractors with probability 1.
Fig. 1. Top Lyapunov exponents k vs. noise amplitude (stochastic phase) D.
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Y. Lei et al. / Chaos, Solitons and Fractals 21 (2004) 1175–1181
Fig. 2. Poincare maps of Eq. (1) vs. increase of amplitude of noise D (a) Poincare map for the excitation without stochastic noise, i.e. D ¼ 0 (b) Poincare map for D ¼ 0:01 (c) Poincare map for D ¼ 0:05 (d) Poincare map for D ¼ 0:10.
3. Harmonically driven Murali-Lakshmanan-Chua circuit This section is devoted to investigating the generation of chaos in the MLC circuit [9] in the form, x_ 1 ¼ x2 gðx1 Þ x_ 2 ¼ rx2 bx1 þ F sinðxtÞ þ Fc sinðxc tÞ
ð11Þ
where gðxÞ ¼ bx þ 0:5ða bÞ½jx þ 1j jx 1j For b ¼ 1:0, r ¼ 1:015, x ¼ 0:75, F ¼ 0:15, a ¼ 1:02, b ¼ 0:55, xc ¼ x, and Fc ¼ 0:06, the Poincare map of system (11), Fig. 4(a), shows system (11) is stable with an attractor of period 3T. Now the effect of stochastic phase is to be explored by adding a random phase to the second harmonic excitation in system (11), then, the corresponding system is x_ 1 ¼ x2 gðx1 Þ ð12Þ x_ 2 ¼ rx2 bx1 þ F sinðxtÞ þ Fc sinðxc t þ DnðtÞÞ where nðtÞ is a standard Gaussian white noise. The linearized equation of (12) can be expressed as y_ 1 ¼ y2 g0 ðx1 Þy1 y_ 2 ¼ ry2 by1
ð13Þ
Y. Lei et al. / Chaos, Solitons and Fractals 21 (2004) 1175–1181
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Fig. 3. Top Lyapunov exponents k vs. amplitude of noise (stochastic phase) D.
Fig. 4. (a) Poincare map for excitation without stochastic phase, i.e. D ¼ 0 (b) Poincare map for D ¼ 0:005 (c) Poincare map for D ¼ 0:01 (d) Poincare map for D ¼ 0:05 (e) Local magnification of Fig. 4(d) (f) Poincare map for D ¼ 0:10; (g) Local magnification of (f).
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Y. Lei et al. / Chaos, Solitons and Fractals 21 (2004) 1175–1181
Fig. 4 (continued)
with 0
g ðxÞ ¼
a; b;
1 < x < 1 otherwise:
Like in Section 2, the top Lyapunov exponent is computed and its connection with amplitude of stochastic phase is shown in Fig. 3. One can see in Fig. 3, when the stochastic phase is inactive, namely D ¼ 0, the state of system (11) is controlled on periodic trajectory; when the amplitude increases gradually, there will appear unstable tendency of our system (13), and the critical value of amplitude is about 0.040, that is, if D > 0:040 the chaos will be observed for system (12). Obviously, when D ¼ 0:0005 and D ¼ 0:01 the top Lyapunov exponent is negative and the system is stable, but when D ¼ 0:05 and D ¼ 0:10 the top Lyapunov exponent is positive and the system is chaotic. That is to say, chaos will appear suddenly when the noise amplitude exceeds a critical value, Dc 0:040. Here Poincare map approach is applied again to check the results mentioned above and the excellent agreement has been observed. For several given values of amplitude, we plot Poincare maps as Fig. 4.
4. Conclusion and discussion Based on the work of Wedig [15] and Khasminskii [14], the top Lyapunov exponents are computed for two kinds of nonlinear stochastic systems. And at the same time we investigate the effect of stochastic phase on the function of nonfeedback control. Opposite results about the effect of stochastic phase are observed for two different stochastic systems, the parametrically excited Duffing oscillator and the driven MLC circuit. That is to say, the increase of stochastic phase can either quench chaos or generate chaos. The related Poincare map analyses fully confirm these results.
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Now we have to point out that in a stochastic dynamic system the phenomenon of transition of a chaotic state to a regular state or vise versa is called stochastic bifurcation, which usually can be defined by the criterion for change of sign of the top Lyapunov exponent of the system. Hence, the transition phenomena in our two examples should be taken as stochastic bifurcation [19]. An alternative definition for stochastic bifurcation has been proposed by Xu et al. [19], where stochastic bifurcation is defined as the sudden change of topological character of stochastic attractors. The results of the top Lyapunov exponent analysis and the corresponding Poincare map analysis in this paper show that these two definitions are consistent in the cases we considered.
Acknowledgements The authors are grateful for the support of National Science Foundation of China (Grant No. 10332030). The third author would also like to express his gratitude to the Doctorate Creation Foundation of Northwestern Polytechnical University (Grant No. CX200326).
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