Homoclinic bifurcation and chaos in simple pendulum under bounded noise excitation

Chaos, Solitons and Fractals 20 (2004) 593–607 www.elsevier.com/locate/chaos

Homoclinic bifurcation and chaos in simple pendulum under bounded noise excitation Z.H. Liu, W.Q. Zhu

*

Department of Mechanics, Zhejiang University, Hangzhou 310027, People’s Republic of China Accepted 31 July 2003

Abstract The homoclinic bifurcation and chaos in a simple pendulum subject to bounded noise excitation is studied. The random Melnikov process is derived and the mean-square criterion is used to determine the threshold amplitude of the bounded noise excitation for the onset of the chaos in the system. The threshold amplitude is also determined by vanishing the numerically calculated largest Lyapunov exponent. It is found that the two values of the threshold amplitude are comparable over a large range of the intensity value of the random frequency. Finally, the Poincar
e maps are constructed to show the route from periodic motion to chaos or from random motion to random chaos as the amplitude of the bounded noise excitation increases and to further verify the two threshold amplitudes. Ó 2003 Elsevier Ltd. All rights reserved.

1. Introduction In the last two decades there has been considerable interest in the effect of noise on the chaos and the route to chaos. Kapitaniak [1] solved the Fokker–Planck–Kolmogorov (FPK) equation for Duffing oscillator and simple pendulum under external excitations of both periodic force and Gaussian white noise using path-integral method and found multipeaks structure of the probability density. Jung and H€ anggi [2] studied the effect external Gaussian white noise on the invariant measure of a periodically driven damped simple pendulum through solving the FPK equation numerically. They found that the probability density had the characteristics of multiple maxima if the noise-free system was chaotic and that, in the presence of noise, the multi-peaked structure of the probability density was washed out for larger noise intensity. Bulsara et al. [3] considered the effect of weak additive noise on the homoclinic threshold of a periodically driven dissipative nonlinear system. A ‘‘generalized’’ Melnikov function was derived for the system, which was the Melnikov function for the corresponding noise-free system plus a correct term that depends on the second-order noise characteristics. Schieve and Bulsara [4] studied the effect of multiplicative noise on the homoclinic threshold of an externally excited system. Xie [5] applied the ‘‘generalized’’ Melnikov function’s method [3] to study the effect of weak noise on the homoclinic threshold of Duffing oscillator under parametric excitation of Gaussian white noise. The ‘‘generalized’’ Melnikov function’s method predicted that weak noise increased the homoclinic threshold while vanishing the largest Lyapunov exponent predicted that weak noise reduced the critical amplitude for the onset of chaos. Frey and Simiu [6] proposed a method to calculate the variance of random Melnikov process of a nonlinear system subject to external excitations of both periodic force and Shinozuka noise. Lin and Yim [7] applied the random Melnikov process with mean-square criterion to study Duffing oscillator subject to external excitations of both periodic force and Gaussian white noise. They concluded that the presence of noise lowered the homoclinic threshold, which is

*

Corresponding author. Tel.: +86-571-7991150; fax: +86-571-87952651. E-mail address: [email protected] (W.Q. Zhu).

0960-0779/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2003.08.010

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Z.H. Liu, W.Q. Zhu / Chaos, Solitons and Fractals 20 (2004) 593–607

contradictory to the conclusion of Xie [5] using the ‘‘generalized’’ Meknikov function’s method and consistent with that by vanishing the largest Lyapunov exponent. Bounded noise is a harmonic function with constant amplitude and random frequency and phase. It has finite power and its spectral density shape can be adjusted to fit a target one, such as the Drydon and von Karman spectra of wind turbulence [8]. Thus, it can be a reasonable model for random excitation or response of engineering systems. This model has been used for a long time in electrical engineering but it was only recently used in mechanical and structural engineering [9–12]. The effect of bounded noise excitation on the chaos of Duffing oscillator under parametric excitation has been studied recently [13]. In the present paper, the homoclinic bifurcation and chaos in a simple pendulum under external and (or) parametric excitations of bounded noise is investigated. Random Melnikov process with mean-square criterion and vanishing the largest Lyapunov exponent are used to determine the threshold amplitude of bounded noise excitation for the onset of chaos in the system and Poincar
e maps are constructed to further verify the threshold amplitude and to show the route from periodic or random motion to chaos or random chaos.

2. Bounded noise A bounded noise is a harmonic function with constant amplitude and random frequency and phase, which can be expressed as nðtÞ ¼ l sinðXt þ wÞ

ð1Þ

w ¼ rBðtÞ þ C

ð2Þ

where l and X are constants representing the amplitude and averaged frequency of bounded noise, respectively; BðtÞ is unit Wiener process; r is a constant representing intensity of random frequency; C is a random variable uniformly distributed in ½0; 2pÞ representing random phase angle. nðtÞ is a stationary random process in wide sense with zero mean [8]. Its covariance is
2  l2 r jsj Cn ðsÞ ¼ cos Xs ð3Þ exp
2 2 and its spectral density is " # l2 r2 1 1 l2 r2 x2 þ X2 þ r4 =4 Sn ðxÞ ¼ þ ¼ 2p 4ðx
XÞ2 þ r4 4ðx þ XÞ2 þ r4 4p ðx2
X2
r4 =4Þ2 þ r4 x2

ð4Þ

The variance of the bounded noise is Cn ð0Þ ¼ l2 =2

ð5Þ

Sξ (ω )

1 0.9 0.8 0.7 0.6 0.5 0.4

σ=0.6

σ=1

0.3

σ=0.2

σ=2

0.2

σ=0.1

0.1 0 -3

-2

-1

0

1

2

3

ω

Fig. 1. The spectral density of bounded noise for several sets of parameters. r ¼ 0:1, 0.2, 0.6, 1.0, 2.0; l ¼ 1:0, X ¼ 1:0.

Z.H. Liu, W.Q. Zhu / Chaos, Solitons and Fractals 20 (2004) 593–607

595

which implies that the noise has finite power. The spectral density of bounded noise for several sets of parameters is shown in Fig. 1. The position of peak of the spectral density depends on X, and the bandwidth of the noise depends mainly on r. It is a narrow-band process when r is small and it approaches to white noise as r ! 1. It can be shown that the sample functions of the noise are continuous and bounded which are required in the derivation of Melnikov function [14].

3. Random Melnikov process Consider a single degree-of-freedom Hamiltonian system subject to light damping and weakly external and (or) parametric excitations of bounded noise. The equation of motion of the system is of the form oH Q_ ¼ oP _P ¼
oH
ecðQ; P Þ oH þ ef ðQ; P ÞnðtÞ oQ oP

ð6Þ

where Q and P are generalized displacement and momentum, respectively; H ¼ H ðQ; P Þ is Hamiltonian with continuous first-order derivatives; e is a small positive parameter; cðQ; P Þ denotes the coefficient of quasi-linear damping; f ðQ; P Þ denotes the amplitude of excitation. f nðtÞ is an external excitation of bounded noise if f is a constant. Otherwise, it is a parametric excitation. It is assumed that the Hamiltonian system associated with Eq. (6) possesses a hyperbolic fixed point connected to itself by homoclinic orbits ðq0 ðtÞ; p0 ðtÞÞ. A bounded noise with spectral density (4) can be approximated by a sum of many harmonic functions with different frequency and random phases. As in [13], the random Melnikov process for system (6) is   Z 1 oH oH Mðt0 Þ ¼
cðQ; P Þ þ f ðQ; P Þnðt þ t0 Þ dt ¼ Md þ Zðt0 Þ ð7Þ oP
1 oP where Md is the component of Melnikov process due to damping and Zðt0 Þ is that due to bounded noise excitation. Note that QðtÞ and P ðtÞ in the integrand of Eq. (7) are taken as q0 ðtÞ and p0 ðtÞ of homoclinic orbit. Obviously, the mean of random Melnikov process is
2 Z 1 oH cðQ; P Þ dt ð8Þ E½Mðt0 Þ ¼ Md ¼
oP
1 where E½ denotes the expectation operator. For positive damping, Eq. (8) always yields negative value, which implies that system (6) cannot be chaotic in mean sense. The mean-square values of the two components of random Melnikov process (7) are
2 !2 Z 1 oH 2 Md ¼ cðQ; P Þ dt ð9Þ oP
1 "
Z 2 # 1 oH 2 2 f ðQ; P ÞðtÞnðt þ t0 Þ dt rZ ¼ E½Z ðt0 Þ ¼ E ð10Þ
1 oP To calculate r2Z , Zðt0 Þ in Eq. (7) is rewritten as a convolution integral Z 1 oH f ðQ; P ÞðtÞnðt þ t0 Þ dt ¼ hðtÞ  nðtÞ Zðt0 Þ ¼
1 oP

ð11Þ

where hðtÞ ¼ ðoH =oP Þf ðQ; P ÞjQ¼q0 ðtÞ;P ¼p0 ðtÞ can be regarded as the impulse response function of a time-invariant linear system while nðtÞ as an input to the system. Thus, the variance of the system can be obtained in frequency domain as follows: Z 1 r2Z ¼ jH ðxÞj2 Sn ðxÞ dx ð12Þ
1

where H ðxÞ is the frequency response function of the system, which is the Fourier transformation of the impulse response function hðtÞ.

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Z.H. Liu, W.Q. Zhu / Chaos, Solitons and Fractals 20 (2004) 593–607

Random Melnikov process (7) has a simple zero in the mean-square sense when Md2 ¼ r2Z

ð13Þ

which yields the threshold amplitude of bounded noise excitation for the onset of chaos in system (6).

4. Threshold amplitude by random Melnikov method Now let us apply the random Melnikov process with mean-square criterion to a simple pendulum with light linear damping and subject to weakly external and (or) parametric excitations of bounded noise. The Hamilton equations of the system is of the form Q_ ¼ P P_ ¼
sin Q
ebP þ ef ðQ; P ÞnðtÞ

ð14Þ

where Q denotes the angle of the pendulum and b is a constant representing damping coefficient. The Hamiltonian associated with system (14) is 1 H ðq; pÞ ¼ p2
cos q 2

ð15Þ

The Hamiltonian system with Hamiltonian (15) possesses a saddle point ð
p; 0Þ or ðp; 0Þ and two homoclinic orbits q0 ðtÞ ¼ 2 sin
1 ðtanh tÞ;

p0 ðtÞ ¼ 2sech t

ð16Þ

Following Eq. (7), the random Melnikov process for system (14) is obtained as follows: Z 1 ½
bp02 ðtÞ þ p0 ðtÞf ðq0 ðtÞ; p0 ðtÞÞnðt þ t0 Þ dt ¼ Md þ Zðt0 Þ Mðt0 Þ ¼

ð17Þ


1

where Md ¼
b

Z

1


1

p02 ðtÞ dt ¼
8bBð1; 1Þ

ð18Þ

in which Bð1; 1Þ is Beta function, and Zðt0 Þ can be written as a convolution integral in Eq. (11) where hðtÞ ¼ p0 ðtÞf ðq0 ðtÞ; p0 ðtÞÞ

ð19Þ

The frequency response function associated with impulse response function (19) is Z 1 H ðxÞ ¼ p0 ðtÞf ðq0 ðtÞ; p0 ðtÞÞ expð
jxtÞ dt

ð20Þ


1

which depends on function f ðq; pÞ. In the following, three cases of f ðq; pÞ are considered. Case 1: f ðq; pÞ ¼ 1. In this case, hðtÞ ¼ p0 ðtÞ ¼ 2sech t Z 1 sech t expð
jxtÞ dt ¼ 2psechðpx=2Þ H ðxÞ ¼ 2

ð21Þ ð22Þ


1

r2Z ¼ pl2 r2

Z

1


1

sech2

 px  2

x2 þ X2 þ r4 =4 ðx2


X2
r4 =4Þ2 þ r4 x2

dx

ð23Þ

Case 2: f ðq; pÞ ¼ q. In this case, hðtÞ ¼ p0 ðtÞq0 ðtÞ ¼ 4sech t sin
1 ðtanh tÞ Z 1 H ðxÞ ¼ 4 sech t sin
1 ðtanh tÞ exp½
jxt dt
1

ð24Þ ð25Þ

Z.H. Liu, W.Q. Zhu / Chaos, Solitons and Fractals 20 (2004) 593–607

r2Z ¼

l2 r2 4p

Z

1

jH ðxÞj2


1

x2 þ X2 þ r4 =4 ðx2
X2
r4 =4Þ2 þ r4 x2

597

ð26Þ

dx

Case 3: f ðq; pÞ ¼ 1
p. In this case, hðtÞ ¼ p0 ðtÞð1
p0 ðtÞÞ ¼ sech tð1  2 sin
1 ðtanh tÞÞ Z 1 sech tð1  2sech tÞ exp½
jxt dt ¼ 2p½sech ðpx=2Þ  2xcschðpx=2Þ H ðxÞ ¼ 

ð27Þ ð28Þ


1

r2Z ¼

l2 r2 4p

Z

1

jH ðxÞj2


1

x2 þ X2 þ r4 =4 ðx2
X2
r4 =4Þ2 þ r4 x2

ð29Þ

dx

The threshold amplitude lcr of bounded noise excitation for the onset of chaos in system (14) is determined by r2Z ¼ 64b2 B2 ð1; 1Þ

ð30Þ

where r2Z is given by Eqs. (23), (26) or (29). Some numerical results of lcr as function of r for three cases are shown in Figs. 2–4 by using solid lines. µ

5

2.2

µ

4.5

2

4

1.8

3.5

1.6 1.4

3

1.2

2.5

1

2

0.8

1.5

0.6

1

0.4

0.5

0.2 0

0 0

2

4

(a)

6

8

0

10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

σ

(b)

σ

Fig. 2. The threshold amplitude lcr of bounded noise excitation for the onset of chaos in system (14) in case 1. b ¼ 0:2, X ¼ 1:0. (a) r 2 ½0; 10; (b) r 2 ½0; 1. (––) Result by random Melnikov process with mean-square criterion; ( ) result by vanishing the largest Lyapunov exponent.



µ

7

µ

2 1.8

6

1.6 5

1.4 1.2

4

1 3

0.8 0.6

2

0.4 1

0.2

0

0 0

(a)

1

2

3

4

5

σ

6

7

8

9

10

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

σ

(b)

Fig. 3. The threshold amplitude lcr of bounded noise excitation for the onset of chaos in system (14) in case 2. b ¼ 0:2, X ¼ 1:0. (a) r 2 ½0; 10; (b) r 2 ½0; 1. (––) Result by random Melnikov process with mean-square criterion; ( ) result by vanishing the largest Lyapunov exponent.



598 µ

Z.H. Liu, W.Q. Zhu / Chaos, Solitons and Fractals 20 (2004) 593–607

3.5

µ

1

3 0.8

2.5 0.6

2 1.5

0.4

1 0.2

0.5 0

0

0

1

2

3

4

5

σ

(a)

6

7

8

9

0

10

0.2

0.4

0.6

0.8

1

σ

(b)

Fig. 4. The threshold amplitude lcr of bounded noise excitation for the onset of chaos in system (14) in case 3. b ¼ 0:2, X ¼ 1:0. (a) r 2 ½0; 10; (b) r 2 ½0; 1. (––) result by random Melnikov process with mean-square criterion; ( ) result by vanishing the largest Lyapunov exponent.



5. Threshold amplitude by vanishing the largest Lyapunov exponent Lyapunov exponent represents the asymptotic rate of exponential convergence or divergence of nearby orbits of a dynamical system in phase space and is one of most important characteristics of system behavior. Exponential divergence of nearly orbit implies that the behavior of a dynamical system is sensitive to initial condition, which is a characteristics of chaotic dynamical system. A dynamical system with a positive largest Lyapunov exponent is usually identified as chaotic. Thus, the threshold amplitude for the onset of chaos in a dynamical system can be determined by vanishing the largest Lyapunov exponent. So far, there is no analytical method to evaluate the largest Lyapunov exponent of a dynamical system near homoclinic or heteroclinic orbits. However, Wolf et al. [15] developed a powerful algorithm for systematically computing all the Lyapunov exponents of a dynamical system. Here, their algorithm is used to calculate the largest Lyapunov exponent of system (14). Denote the largest Lyapunov exponents by k1 . k1 as function of bounded noise amplitude l for a serious of r values in case 1 is shown in Fig. 5. It is seen from the figures that k1 is negative for small value of l. As l increases, k1 changes from negative to positive, which signifying the presence of chaotic motion in the system. In the case of periodic excitation (r ¼ 0), k1 becomes negative again after it first reaches positive value as l increases. This interval of l value is called ‘‘periodic window’’. In the case of bounded noise excitation, the ‘‘periodic window’’ is diminished or washed out by the random frequency as r increases. The threshold amplitude of bounded noise for the onset of chaos in system (14) determined by k1 ¼ 0 is shown in Figs. 2–4 by using dotted lines. It is seen from the figures that, over a large range of r value, the random Melnikov process with mean-square criterion and vanishing k1 yield comparable threshold amplitudes. The largest difference in the two threshold amplitudes occurs at r ¼ 0, i.e. for periodic excitation. In this case, the random Melnikov process with mean-square criterion always underestimates the threshold amplitude. This is reasonable since simple zero of Melnikov function is only the necessary condition for onset of chaos. 6. Poincar
e map The behavior of a non-autonomous two-dimensional system such as system (14) can be depicted by using Poincar
e map, which is defined as P : R ! R;

R ¼ fQ; P jh ¼ 2np=X; n ¼ 0; 1; 2; . . .g 2 R2

One hundred initial points are selected on the phase plane. For each initial condition, Eq. (14) is solved numerically and the solution is plotted for every T ¼ 2p=X. For each initial point, 500 iteration points are plotted. The Poincar
e maps for system (14) with externally periodic excitation (r ¼ 0) are shown in Fig. 6. It is seen from the figures that the motion of the system is periodic when l 6 2:1 and chaotic when l P 2:2. This implies that the threshold amplitude lcr for onset of chaos should be in the interval (2.1, 2.2). This confirms the result obtained by vanishing the largest Lyapunov exponent (see Fig. 2(b)). The Poincar
e maps for system (14) under external excitation of bounded

Z.H. Liu, W.Q. Zhu / Chaos, Solitons and Fractals 20 (2004) 593–607

599

noise with r ¼ 0:1 is shown in Fig. 7. It is seen from the figures that the motion of the system is random when l 6 0:8 and random chaotic when l P 1:2 and something in between when l ¼ 1:0 . This implies that the threshold amplitude should be around lcr ¼ 1:0, which confirms both the results obtained by using random Melnikov process with meansquare criterion and by vanishing the largest Lyapunov exponent (see Fig. 2(b)). The former method slightly underestimates lcr while the later method slightly overestimates lcr . The Poincar
e maps for system (14) subject to external excitation of bounded noise with r ¼ 0:2, 0.4, 0.6, 0.8, 1.0 are also shown in Figs. 8–12. All these figures show how the motion of system (14) goes from random to random chaos as l increases and confirm both the threshold amplitudes obtained by the two methods. It should be noted that the transition of the motion of the system from random to

λ1

0.2

0.2

λ1

0.15

0.15

0.1 0.1

0.05 0.05

0 0

-0.05 -0.05

-0.1

-0.1

-0.15 0

0.5

1

(a) λ1

1.5

2

2.5

0

3

0.2

1.5

2

2.5

3

2

2.5

3

2

2.5

σ

0.15

0.15

0.1

0.1

0.05

0.05

0

0

-0.05

-0.05 -0.1

0

0.5

1

(c)

1.5

2

2.5

0

3

0.5

1

(d)

σ 0.2

λ1

1.5

σ 0.2

0.15

0.15

0.1

0.1

0.05

0.05

0

0

-0.05

-0.05 -0.1

-0.1

(e)

1

0.2

λ1

-0.1

λ1

0.5

(b)

σ

0

0.5

1

1.5

σ

2

2.5

3

0

(f)

0.5

1

1.5

3

σ

Fig. 5. The largest Lyapunov exponent of system (14) in case 1. b ¼ 0:2, X ¼ 1:0. (a) r ¼ 0; (b) r ¼ 0:02; (c) r ¼ 0:05; (d) r ¼ 0:1; (e) r ¼ 0:2; (f) r ¼ 0:4; (g) r ¼ 0:6; (h) r ¼ 0:8; (i) r ¼ 1:0.

600 λ1

Z.H. Liu, W.Q. Zhu / Chaos, Solitons and Fractals 20 (2004) 593–607 0.15

λ1

0.15

0.1

0.1

0.05

0.05

0

0

-0.05

-0.05

-0.1

-0.1 0

0.5

1

1.5

(g)

2

2.5

0

3

σ

λ1

0.5

1

(h)

1.5

2

2.5

3

σ

0.15

0.1

0.05

0

-0.05

-0.1 0

(i)

0.5

1

1.5

2

2.5

3

σ

Fig. 5 (continued)

random chaos is gradual and the threshold amplitude lcr for the onset of chaos should be an interval rather than a single value. Thus, it can be concluded that over a large range of r value both the random Melnikov process with meansquare criterion and vanishing the largest Lyapunov exponent can correctly predict the threshold amplitude for the onset of chaos in system (14) under external excitation of bounded noise. It also can be observed from Figs. 6–12 that when l < lcr , the motion of system (14) goes from periodic to random as r increases from zero and diffuses in a larger and larger domain of phase plane as r further increases. Similarly, when l > lcr , the motion of system (14) goes from chaos to random chaos as r increases from zero and becomes more and more random and less and less chaotic as r further increases. A few Poincar
e maps for system (14) under parametric excitation (case 2), and both external and parametric excitations (case 3) of periodic force and bounded noise are shown in Figs. 13–16. Similar observations as those in the case of external excitation can be made. The difference in threshold amplitude obtained by random Melnikov process with mean-square criterion and by vanishing the largest Lyapunov exponent is larger for case 3. It is noted that, in the case of pure parametric excitation of periodic force or bounded noise, the trivial solution of system (14) is stable and the system stays in origin when l < lcr and k1 < 0 is also the condition for stability. It is also noted that the strange attractors for three different excitations are different when l > lcr .

7. Conclusions In the present paper, the homoclinic bifurcation and chaos in a simple pendulum subject to external and (or) parametric excitations of bounded noise have been studied by using random Melnikov process, the largest Lyapunov exponent and Poincar
e map. Over a large range of random frequency intensity, the random Melnikov process with mean-square criterion, vanishing the largest Lyapunov exponent and Poincar
e map yield comparable threshold

Z.H. Liu, W.Q. Zhu / Chaos, Solitons and Fractals 20 (2004) 593–607 p

2

p

2

1

1

0

0

-1

-1

-2

-2 -4

-3

-2

-1

0

1

2

3

-4

4

q

(a) p

601

-3

-2

-1

4

p

1

2

3

4

1

2

3

4

4

3

3

2

2

1

1

0

0

-1

-1

-2

-2

-3

-3

-4

0

q

(b)

-4

-4

-3

-2

-1

0

1

2

3

4

-4

q

(c) p

-3

-2

-1

(d)

0

q

4 3 2 1 0 -1 -2 -3 -4 -4

(e)

-3

-2

-1

0

1

2

3

4

q

Fig. 6. Poincar
e maps of system (14) in case 1. b ¼ 0:2, X ¼ 1:0. (a) r ¼ 0, l ¼ 0:8; (b) r ¼ 0, l ¼ 1:0; (c) r ¼ 0, l ¼ 2:1; (d) r ¼ 0, l ¼ 2:2; (e) r ¼ 0, l ¼ 2:3.

amplitude of bounded noise excitation for the onset of chaos in the system. When the amplitude of bounded noise is less than the threshold value, the motion of the system is periodic (in the case of external excitation of periodic force), equilibrium at origin (in the case of pure parametric excitation of periodic force or bounded noise), or random (in the case of external excitation or both external and parametric excitations of bounded noise). When the amplitude of bounded noise is larger than the threshold value, the motion of the system is chaotic (under periodic excitation) or random chaotic (under excitation of bounded noise). The strange attractors are different in pattern for three different (external, parametric of both) excitations of periodic forces. The randomness in the excitation frequency makes the

602

Z.H. Liu, W.Q. Zhu / Chaos, Solitons and Fractals 20 (2004) 593–607 p

3

3

p 2

2

1

1

0

0

-1

-1

-2

-2

-3

-3

-4

-3

-2

-1

0

(a)

1

2

3

-4

4

-3

-2

-1

(b)

q

0

1

2

3

4

q

3

p

2 1 0 -1 -2 -3 -4

-3

-2

-1

0

(c)

1

2

3

4

q

Fig. 7. Poincar
e maps of system (14) in case 1. b ¼ 0:2, X ¼ 1:0. (a) r ¼ 0:1, l ¼ 0:8; (b) r ¼ 0:1, l ¼ 1:0; (c) r ¼ 0:1, l ¼ 1:2.

p

3

p

2

2

1

1

0

0

-1

-1

-2

-2 -3

-3

(a)

3

-4

-3

-2

-1

0

1

2

3

-4

4

p

-3

-2

-1

(b)

q

0

1

2

3

4

q

3 2 1 0 -1 -2 -3 -4

(c)

-3

-2

-1

0

1

2

3

4

q

Fig. 8. Poincar
e maps of system (14) in case 1. b ¼ 0:2, X ¼ 1:0. (a) r ¼ 0:2, l ¼ 0:7; (b) r ¼ 0:2, l ¼ 0:8; (c) r ¼ 0:2, l ¼ 1:0.

Z.H. Liu, W.Q. Zhu / Chaos, Solitons and Fractals 20 (2004) 593–607 p

3

p

603

3

2

2

1

1

0

0

-1

-1

-2

-2 -3

-3 -4

-3

-2

-1

0

(a)

1

2

3

-4

4

-3

-2

-1

(b)

q

0

1

2

3

4

q

3

p

2 1 0 -1 -2 -3 -4

-3

-2

-1

0

(c)

1

2

3

4

q

Fig. 9. Poincar
e maps of system (14) in case 1. b ¼ 0:2, X ¼ 1:0. (a) r ¼ 0:4, l ¼ 0:6; (b) r ¼ 0:4, l ¼ 0:8; (c) r ¼ 0:4, l ¼ 1:0.

p

3

p

3

2

2

1

1

0

0

-1

-1

-2

-2 -3

-3 -4

-3

-2

-1

0

(a)

1

2

3

-4

4

q

p

-3

-2

-1

(b)

0

1

2

3

4

q

3 2 1 0 -1 -2 -3 -4

(c)

-3

-2

-1

0

1

2

3

4

q

Fig. 10. Poincar
e maps of system (14) in case 1. b ¼ 0:2, X ¼ 1:0. (a) r ¼ 0:6, l ¼ 0:6; (b) r ¼ 0:6, l ¼ 0:8; (c) r ¼ 0:6, l ¼ 1:0.

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Z.H. Liu, W.Q. Zhu / Chaos, Solitons and Fractals 20 (2004) 593–607 3

p

p

2

2

1

1

0

0

-1

-1

-2

-2 -3

-3 -4

-3

-2

-1

(a)

0

1

2

3

-4

4

p

1

1

0

0

-1

-1

-2

-2

-3 -3

-2

-1

0

-1

1

2

3

1

2

3

4

1

2

3

4

-3

4

-4

-3

-2

-1

(d)

q

0

3 2

-4

-2

q

2

(c)

-3

(b)

q 3

p

3

0

q

Fig. 11. Poincar
e maps of system (14) in case 1. b ¼ 0:2, X ¼ 1:0. (a) r ¼ 0:8, l ¼ 0:4; (b) r ¼ 0:8, l ¼ 0:6; (c) r ¼ 0:8, l ¼ 0:8; (d) r ¼ 0:8, l ¼ 1:0.

p

3

p

2

2

1

1

0

0

-1

-1

-2

-2 -3

-3 -4

-3

-2

-1

(a) p

0

1

2

3

-4

4

-3

-2

-1

(b)

q 3

p

0

1

2

3

4

q 3

2

2

1

1

0

0

-1

-1

-2

-2 -3

-3 -4

(c)

3

-3

-2

-1

0

q

1

2

3

-4

4

(d)

-3

-2

-1

0

1

2

3

4

q

Fig. 12. Poincar
e maps of system (14) in case 1. b ¼ 0:2, X ¼ 1:0. (a) r ¼ 1:0, l ¼ 0:4; (b) r ¼ 1:0, l ¼ 0:6; (c) r ¼ 1:0, l ¼ 0:8; (d) r ¼ 1:0, l ¼ 1:0.

Z.H. Liu, W.Q. Zhu / Chaos, Solitons and Fractals 20 (2004) 593–607 p

4

p

4

3

3

2

2

1

1

0

0

-1

-1

-2

-2

-3

-3

-4

605

-4 -4

-3

-2

-1

0

(a)

1

2

3

4

-4

-3

-2

-1

(b)

q

0

1

2

3

4

q

4

p

3 2 1 0 -1 -2 -3 -4 -4

-3

-2

-1

0

(c)

1

2

3

4

q

Fig. 13. Poincar
e maps of system (14) in case 2. b ¼ 0:2, X ¼ 1:0. (a) r ¼ 0, l ¼ 0:6; (b) r ¼ 0, l ¼ 1:0; (c) r ¼ 0, l ¼ 1:7.

p

4

p

4

3

3

2

2

1

1

0

0

-1

-1

-2

-2

-3

-3 -4

-4 -4

-3

-2

-1

0

(a)

1

2

3

-4

4

q

p

-3

-2

-1

(b)

0

1

2

3

4

q

4 3 2 1 0 -1 -2 -3 -4 -4

(c)

-3

-2

-1

0

1

2

3

4

q

Fig. 14. Poincar
e maps of system (14) in case 2. b ¼ 0:2, X ¼ 1:0. (a) r ¼ 0:2, l ¼ 0:6; (b) r ¼ 0:2, l ¼ 1:0; (c) r ¼ 0:2, l ¼ 1:4.

606

Z.H. Liu, W.Q. Zhu / Chaos, Solitons and Fractals 20 (2004) 593–607 p

4

p

4

3

3

2

2

1

1

0

0

-1

-1

-2

-2

-3

-3 -4

-4 -4

-3

-2

-1

0

(a)

1

2

3

-4

4

p

-3

-2

-1

(b)

q

0

1

2

3

4

q

4 3 2 1 0 -1 -2 -3 -4 -4

-3

-2

-1

0

(c)

1

2

3

4

q

Fig. 15. Poincar
e maps of system (14) in case 3. b ¼ 0:2, X ¼ 1:0. (a) r ¼ 0, l ¼ 0:2; (b) r ¼ 0, l ¼ 0:4; (c) r ¼ 0, l ¼ 0:8.

p

4

p

4

3

3

2

2

1

1

0

0

-1

-1

-2

-2

-3

-3 -4

-4 -4

-3

-2

-1

0

(a)

1

2

3

-4

4

p

-3

-2

-1

(b)

q

0

1

2

3

4

q

4 3 2 1 0 -1 -2 -3 -4 -4

(c)

-3

-2

-1

0

1

2

3

4

q

Fig. 16. Poincar
e maps of system (14) in case 3. b ¼ 0:2, X ¼ 1:0. (a) r ¼ 0:2, l ¼ 0:2; (b) r ¼ 0:2, l ¼ 0:4; (c) r ¼ 0:2, l ¼ 0:6.

Z.H. Liu, W.Q. Zhu / Chaos, Solitons and Fractals 20 (2004) 593–607

607

strange attractors diffusive or smeary, and the larger the random frequency intensity is, the more diffusive or smeary the strange attractor is. These conclusions are consistent with those for Duffing oscillator subject to parametric excitation of bounded noise [13].

Acknowledgements This work is supported by the National Natural Science Foundation of China under a key grant and grant no. 19972059 and the special fund for doctor programs in Institutions of Higher Learning of China under grant no. 20020335092.

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