homoclinic bifurcation and chaos in coupled simple pendulum and ...

International Journal of Bifurcation and Chaos, Vol. 15, No. 1 (2005) 233–243 c World Scientific Publishing Company

HOMOCLINIC BIFURCATION AND CHAOS IN COUPLED SIMPLE PENDULUM AND HARMONIC OSCILLATOR UNDER BOUNDED NOISE EXCITATION W. Q. ZHU∗ and Z. H. LIU Department of Mechanics, Zhejiang University, Hangzhou 310027, P.R. China ∗[email protected] Received September 24, 2003; Revised January 6, 2004 The homoclinic bifurcation and chaos in a system of weakly coupled simple pendulum and harmonic oscillator subject to light dampings and weakly external and (or) parametric excitation of bounded noise is studied. The random Melnikov process is derived and mean-square criteria is used to determine the threshold amplitude of the bounded noise for the onset of chaos in the system. The threshold amplitude is also determined by vanishing the numerically calculated maximal Lyapunov exponent. The threshold amplitudes are further confirmed by using the Poincaré maps, which indicate the path from periodic motion to chaos or from random motion to random chaos in the system as the amplitude of bounded noise increases. Keywords: Homoclinic bifurcation; chaos; Melnikov method; Lyapunov exponent; Poincaré map; bounded noise.

1. Introduction

probability density was washed out for larger noise intensity. Ramesh and Narayanan [1999] considered the control of the chaotic response of Chua’s circuit and the Duffing–Ueda Oscillator under weak random perturbations. The Melnikov method [Melnikov, 1963] is an effective approach to detect chaotic dynamics near homoclinic of a system with deterministic or random perturbation. The method was first applied by Holmes [1979] to study a periodically forced Duffing oscillator with negative linear stiffness, and by Ariaratnam et al. [1989] to investigate the chaotic behavior of a parametrically excited system such as the transverse vibration of a buckled column under axial periodic excitation. For the case of random perturbation, Bulsara et al. [1990] considered the effect of weak additive noise on the homoclinic

In the last two decades there has been considerable interest in the effect of noise on chaos and the route to chaos. Kapitaniak [1986] 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 a multipeak structure of the probability density. Jung and H¨ anggi [1990] studied the effect of 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 noisefree system was chaotic and that, in the presence of noise, the multi-peaked structure of the

∗

Author for correspondence. 233

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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. The effect of noise on the chaotic behavior of a buckled column under parametric excitation was examined by Xie [1994] using a modified Melnikov method. It was found that the noise increases the homoclinic threshold. Lyapunov exponent was also studied and it was found that the critical value of periodic forcing amplitude for the onset of chaotic motion is reduced with increase of noise intensity. So no consistent conclusion has been drawn. Meanwhile, Melnikov method was extended by Frey and Simiu [1993] to study the effect of additive noise on nearintegrable second-order dynamical systems. The mean-square criterion for the Melnikov process was used to study the periodically forced Duffing system with additive random perturbation by Lin and Yim [1996]. Bounded noise is a harmonic function with constant amplitude and random frequency and phase. It has finite power and its spectral shape can be made to fit a target spectrum, such as the Dryden and von Karman spectra of wind turbulence, by adjusting its parameters [Lin & Cai, 1995]. Therefore, it can be a reasonable model for the random excitation or response in 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. Dimentberg [1988, 1991] used it to describe the structural response to traveling loads and the response of a traveling structure. Lin et al. [1993] used it as a model of wind turbulence and studied the motion stability of long-span bridges under the parametric excitation of bounded noise. The stability of viscoelastic system under parametric excitation of bounded noise was studied by Ariaratnam [1996]. Liu et al. [2001] has studied the effect of bounded noise on chaotic motion of Duffing oscillator under parametric excitation. Although the Melnikov method has been generalized to study the homoclinic bifurcation and chaos in multi-degree-of-freedom nonlinear systems with periodic or almost-periodic perturbation [Holmes & Marsden, 1982; Wiggins, 1988], the method has been applied only to single-degree-of-freedom nonlinear systems with random perturbation. In the present paper, the method is generalized to study

the homoclinic bifurcation and chaos in a system of coupled simple pendulum and harmonic oscillator under bounded noise perturbation. First, bounded noise is briefly introduced. Second, the Melnikov function for the Hamiltonian perturbation of coupled simple pendulum and harmonic oscillator is derived. Third, the random Melnikov process for the non-Hamiltonian perturbation of coupled simple pendulum and harmonic oscillator with dampings under bounded noise excitation is derived and the mean-square criterion is used to determine the threshold amplitude of bounded noise for the onset of chaos in the system. Finally, the maximal Lyapunov exponent is calculated and Poincaré maps are constructed to confirm the threshold amplitude.

2. Bounded Noise A bounded noise is a harmonic function with constant amplitude and random frequency and phase, which can be expressed as ξ(t) = µ sin(Ωt + ψ)

(1)

ψ = σB(t) + Γ

(2)

where µ and Ω are the amplitude and averaged frequency of bounded noise; B(t) is unit Wiener process; σ is an intensity parameter of random frequency; Γ is a random phase uniformly distributed in [0, 2π). ξ(t) is a stationary random process in a wide sense with zero mean [Lin & Cai, 1995]. Its covariance function is 2 σ |τ | µ2 exp − cos Ωτ cξ (τ ) = 2 2

(3)

and its two-side spectral density is 1 (µσ)2 1 + Sξ (ω) = 2π 4(ω − Ω)2 + σ 4 4(ω + Ω)2 + σ 4 =

ω 2 + Ω2 + σ 4 /4 µ2 σ 2 4π (ω 2 − Ω2 − σ 4 /4)2 + σ 4 ω 2

(4)

The variance of the bounded noise is c(0) =

µ2 2

(5)

Homoclinic Bifurcation and Chaos in Coupled Simple Pendulum and Harmonic Oscillator

is a bounded noise, εf (Q1 , P1 ) is a function of Q1 and P1 representing the amplitude of random excitation. ε is a small parameter. Thus, Eq. (6) describes an integrable Hamiltonian system subject to both Hamiltonian perturbation (coupling) and non-Hamiltonian perturbation (dampings and random excitation). The unperturbed system associated with system (6) is an integrable Hamiltonian system with Hamiltonian

1 0.9 0.8

Sξ(ω)

0.7 0.6 0.5 0.4 0.3

σ =1

0.2

σ =2

σ = 0.6 σ = 0.2 σ = 0.1

0.1 0 -3

-2

-1

0 ω

1

2

3

Fig. 1. The spectral density of bounded noise for several sets of parameters. σ = 0.1, 0.2, 0.6, 1.0, 2.0; µ = 1.0, Ω = 1.0.

H(q, p) =

1 p21 − cos q1 + (p22 + ω02 q22 ) 2 2

which implies that the noise has finite power. The two-side spectral density of the bounded noise for several sets of parameters is shown in Fig. 1. The position of the spectral density peak depends on Ω and the bandwidth of the noise depends mainly on σ. It is a narrow-band process when σ is small, and it approaches white noise when σ → ∞. It can be shown that the sample functions of the noise are continuous and bounded, required in the derivation of Melnikov function [Wiggins, 1988].

3. Coupled Simple Pendulum and Harmonic Oscillator Consider a system of coupled simple pendulum and harmonic oscillator subject to dampings and bounded noise excitation. The equations of motion of the system are Q˙ 1 = P1 P˙1 = −sin Q1 + ε[(Q2 − Q1 ) − βP1

(7)

q2 , p2 can be expressed in terms of action-angle variables I and θ as q2 = (2I/ω0 )1/2 sin θ,

+ f (Q1 , P1 )ξ(t)]

235

p2 = ω0 (2I/ω0 )1/2 cos θ (8)

So, the system (6) can be rewritten as Q˙ 1 = P1

2I ˙ sin θ − Q1 P1 = −sin Q1 + ε ω0 − βP1 + f (Q1 , P1 )ξ(t) 2I sin θ ˙θ = ω0 + ε sin θ − Q1 √ ω0 2Iω0 + γ sin θ cos θ

(9)

2I sin θ − Q1 2Iω0 cos θ I˙ = −ε ω0 2 + γ2I cos θ and Hamiltonian (7) can be rewritten as

(6)

Q˙ 2 = P2 P˙2 = −ω02 Q2 + ε[(Q1 − Q2 ) − γP2 ] where, Q1 , P1 are the generalized displacement and momentum of the simple pendulum; Q2 , P2 are the generalized displacement and momentum of the harmonic oscillator; ω0 is the natural frequency of the harmonic oscillator; ε is coupling coefficient; εβ and εγ are damping coefficients; ξ(t)

H(q1 , p1 , I) = F (q1 , p1 ) + G(I), F (q1 , p1 ) =

p21 − cos q1 , 2

(10) G(I) = ω0 I

The simple pendulum with Hamiltonian F possesses a saddle point (−π, 0) or (π, 0) and two homoclinic orbits q10 (t) = ±2 sin−1 (tanh t), which are shown in Fig. 2.

p10 (t) = ±2 sech t (11)


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Hamiltonian perturbed integrable system with Hamiltonian

3 2.5 2 1.5

H ε (q1 , p1 , θ, I) = F (q1 , p1 ) + G(I) + εH (q1 , p1 , θ, I)

1 p1

0.5 0

(12)

-0.5 -1

where

-1.5 -2

H (q1 , p1 , θ, I) = [(2I/ω0 )1/2 sin θ − q1 ]2 /2

-2.5 -3 -4

-3

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-1

0

1

2

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Fig. 2. Homoclinic orbits of the simple pendulum in plane (q1 , p1 ).

So, the equations of motion for the nonintegrable Hamiltonian system can be written as q˙1 =

When β = γ = µ = 0, system (6) becomes a nonintegrable Hamiltonian system, or a

M (t0 ) =

∞

−∞ ∞

= −∞

∂F ∂H +ε , ∂p1 ∂p1

∂H , θ˙ = ω0 + ε ∂I

4. Melnikov Function for Hamiltonian Perturbation

(13)

p˙1 = −

∂F ∂H −ε ∂q1 ∂q1

∂H I˙ = −ε ∂θ

(14)

The system (14) can be identified as System III in [Wiggins, 1988]. The Melnikov function is thus

{F, H }(t + t0 ) dt [−p10 (t)q10 (t) + (2I/ω0 )1/2 p10 (t) sin ω0 (t + t0 )] dt

= ±2π[2(h − 1)]1/2 sech(πω0 /2) sin ω0 t0 where h is the value of H ε . It is seen from Eq. (15) that the Melnikov function M (t0 ) may have a simple zero when the initial system energy h ≥ 1.

(15)

That is, the Poincaré map of the system has Smale horseshoes only when h ≥ 1. This is verified by the two Poincaré maps shown in Fig. 3.

2.5 2.5

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Fig. 3. Poincaré map of Hamiltonian perturbed system (14) in phase (q1 , p1 ). (a) ε = 0.003, q1 = 3.1, p1 = 0, q2 = 0.1, p2 = 0.1; (b) ε = 0.01, q1 = 3.1, p1 = 0, q2 = 0.1, p2 = 0.1.


It should be noted that the only initial conditions which are sufficiently close to the homoclinic orbit of the unperturbed system lead to chaotic motion. As the perturbation increases, the chaotic motion occurs in a larger region of the phase space. This effect of increasing

M (t0 ) =

∞

−∞

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perturbation can be seen by comparing Figs. 3(a) and 3(b).

5. Melnikov Process for Non-Hamiltonian Perturbation System (6) or (9) can be identified as system I in [Wiggins, 1988]. The Melnikov process is

2I sin ω0 (t + t0 ) − q10 (t) − βp10 (t) + f (q0 , p0 )ξ(t + t0 ) p10 (t) dt ω0

= ±2π[2(h − 1)]1/2 sech(πω0 /2) sin ω0 t0 − 8βB(1, 1) + Z(t0 ) where B(1, 1) is beta function and ∞ p10 (t)f (q10 , p10 )ξ(t + t0 ) dt. Z(t0 ) =

(17)

(16)

depends on the total energy of the system, which will be studied in the next section.

−∞

Note that Z(t0 ) and M (t0 ) are random processes. Thus, the simple zero of Melnikov process should be considered in mean square sense. Introduce the impulse response function h(t) = p10 (t)f (q10 , p10 )

(18)

The associated frequency response function is ∞ h(t)e−jωt dt (19) H(ω) = −∞

Thus, variance σZ2 can be obtained as follows: ∞ |H(ω)|2 Sξ (ω)dω (20) σZ2 = −∞

The mean-square criterion for simple zero is thus πω 0 sin2 ωt0 E[M 2 (t0 )] = ±8π(h − 1)sech2 2 − 64β 2 + σZ2 =0 (21) Note that the contribution of the coupling between the simple pendulum and harmonic oscillator to the mean square value of Melnikov process

N π/ω0

∆H =

−N π/ω0 N π/ω0

=

−N π/ω0 N π/ω0

= −N π/ω0

6. System Energy The energy h in nonintegrable Hamiltonian system (14) is determined by the initial condition while that in nonconservative system (6) or (9) by solving the energy equation derived from Eq. (6). The time rate of the energy of the system (6) near homoclinic orbit is ∂H ˙ ∂H ˙ ∂H ˙ ∂H ˙ Q1 + P1 + Q2 + P2 H˙ = ∂Q1 ∂P1 ∂Q2 ∂P2

= ε −βp210 (t) − γp220 (t) + p10 (t)f (q1 , p1 )ξ(t) (22) where q10 = ±2 sin−1 [tanh(t − t0 )] p10 = ±2 sech(t − t0 ) q20 = 2I/ω0 sin θ p20 = ω0 2I/ω0 cos θ

(23)

and 1 1 [H − (p210 /2 − cos θ)] = (H − 1) (24) ω0 ω0 Let ∆H denote the approximate change in energy from time −N π/ω0 to time N π/ω0 . I=

˙ Hdt ε[−βp210 (t) − γp220 (t) + p10 (t)f (q, p)ξ(t)] dt ε[−βp210 (t) − γ2Iω0 cos2 θ + p10 (t)f (q, p)ξ(t)] dt

(25)


238

where N is a fixed large integer [Holmes & Marsden, 1982]. The mean value of energy change is N π/ω0 2 2 ε[−βp10 (t) − γ2Iω0 cos θ + p10 (t)ξ(t)] dt E[∆H] = E −N π/ω0 2N π Ωπ Nπ cos Ωt1 − γ(H − 1) + 2πµ sech = ε −8β tanh ω0 2 ω0

(26)

The averaged system is stationary when E[∆H] = 0, from which energy of the averaged systems is obtained as 2N π Nπ Ωπ −γ + 2πµ sech (27) Hc = 1 + −8β tanh ω0 2 ω0 which approaches 1 as N → ∞. It can be seen that Hc > 1, ∆H < 0, if H < Hc , (28) ∆H > 0, if H > Hc , and

d (∆H)

= 0 dH H=Hc

(29)

Thus, Hc is a unique solution near Hc = 1 [Holmes & Marsden, 1982].

7. Numerical Example For example, when ε = 0.05, β = 2, γ = 2, N = 100, Ω = 0.7, ω0 = 1, Hc = 1 + 0.01273 − 0.003µ. Substituting these parameter values into Eq. (21), one obtains 3.99187(Hc − 1) − 64β 2 + σZ2 = 0

Suppose that the system is subjected to external excitation of bounded noise, i.e. f (q10 , p10 ) = 1. Then

h(t) = p10 (t) = ±2 sech t ∞ sech te−jωt dt = ±2π sech(πω/2) H(ω) = ±2 σZ2

(31) (32)

−∞

∞

|H(ω)|2 Sξ (ω) dω −∞ ∞ sech2 (πω/2) = πµ2 σ 2 =

−∞

60

ω 2 + Ω2 + σ 4 /4 (ω 2 − Ω2 − σ 4 /4)2 + σ 4 ω 2

dω

(33)

In this case, the threshold amplitude µcr for Melnikov process having simple zero in mean square sense determined by Eq. (30) is shown in Fig. 4 by solid line. Note that in Eq. (30), the first term is very small compared with the other two terms. Thus, the effect of the selection of the value N on the threshold amplitude µcr is negligible.

50 40 µcr

(30)

30 20 10 0 0

1

2

3

4

5

6

7

8

9

10

σ

Fig. 4. The threshold amplitude µcr of bounded noise excitation for the onset of chaos. ε = 0.05, β = 2, γ = 2, result by random Melnikov process with meanΩ = 0.7. result by vanishing the maximal square criterion; Lyapunov exponent.

8. Lyapunov Exponents 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 the most important characteristics of nonlinear dynamical systems. Exponential divergence of nearby orbits implies that the

Homoclinic Bifurcation and Chaos in Coupled Simple Pendulum and Harmonic Oscillator 0.16

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(e)

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35

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(f)

Fig. 5. The maximal Lyapunov exponent. ε = 0.05, β = 2, γ = 2, Ω = 0.7. (a) σ = 0; (b) σ = 0.1; (c) σ = 0.2; (d) σ = 0.5; (e) σ = 1; (f) σ = 2; (g) σ = 5; (h) σ = 10.

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(h) Fig. 5.

behavior of a dynamical system is sensitive to initial condition. A dynamical system with a positive maximal Lyapunov exponent is usually defined to be chaotic. Thus, the condition for the onset of chaos in a dynamical system can also be determined by numerically calculating the maximal Lyapunov exponent. The maximal Lyapunov exponent for system (6) is computed to check the threshold of bounded noise amplitude for the onset of chaos obtained by using random Melnilov process with mean-square criterion. The maximal Lyapunov exponent λ1 of systems (6) as function of bounded noise amplitude µ computed for some different noise intensity values is shown in Fig. 5. The threshold µcr of bounded noise amplitude for the onset of chaos in systems (6) obtained by vanishing λ1 is given in Fig. 4 using a dashed line. It is seen from Fig. 5 that for small values of the amplitude µ, the maximal Lyapunov exponent is negative. When the value of µ is increased, the maximal Lyapunov exponent changes from negative value to positive value, signifying the presence of chaotic motion. In the case of absence of noise, beyond the threshold µcr for the onset of chaotic motion, there are many “windows” or intervals, in which λ1 becomes negative again and the system then returns to periodic motion. For larger noise intensity values, there is no such “periodic window.” The effect of noise is to wash out or diminish these periodic windows or intervals.

(Continued)

9. Poincar´ e Map System (6) is also studied by using the Poincaré map which is defined as P : Σ → Σ, Σ = {q1 , p1 , | θ = 0, 2π/Ω, 4π/Ω, . . .} ∈ R2 . The differential equation (6) is solved and the solution is plotted for every T = 2π/Ω; for each initial point, 1000 iterative points are plotted. It is seen from Fig. 5 that when σ = 0, µ = 9, 10; σ = 0.1, µ = 7; σ = 0.2, µ = 6.5; σ = 0.5, µ = 6; σ = 1, µ = 7, system (6) is periodic or random, and when σ = 0, µ = 9.6, 13.5; σ = 0.1, µ = 9; σ = 0.2, µ = 8; σ = 0.5, µ = 7.5; σ = 1, µ = 10, system (6) is chaotic or random chaotic. These parameter values are selected for making Poincaré maps and the results are shown in Figs. 6–10. It is seen from Fig. 6 that in the case of harmonic excitation (σ = 0), the motion of system (6) is periodic when µ = 9, 10 and it is chaotic when µ = 9.6, 13.5. This is consistent with the maximal Lyapunov exponent shown in Fig. 5(a). Figures 7–10 show that under bounded noise excitation, the motion of system (6) is random for less µ and random chaotic for larger µ. These results are completely consistent with the threshold amplitude µcr predicted by vanishing the maximal Lyapunov exponent shown in Figs. 4 and 5. It is seen from Fig. 4 that the Melnikov process with mean square criterion generally underestimates the

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p1

Fig. 6. Poincaré maps. ε = 0.05, β = 2, γ = 2, Ω = 0.7. (a) σ = 0, µ = 9; (b) σ = 0, µ = 9.6; (c) σ = 0, µ = 10; (d) σ = 0, µ = 13.5.

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(a) Fig. 7.

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Poincaré maps. ε = 0.05, β = 2, γ = 2, Ω = 0.7. (a) σ = 0.1, µ = 7; (b) σ = 0.1, µ = 9.

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Poincaré maps. ε = 0.05, β = 2, γ = 2, Ω = 0.7. (a) σ = 0.5, µ = 5; (b) σ = 0.5,

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(a) Fig. 10.

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Poincaré maps. ε = 0.05, β = 2, γ = 2, Ω = 0.7. (a) σ = 1, µ = 6; (b) σ = 1, µ = 10.


threshold amplitude µcr . However, the relative error is generally less than 50%.

10. Concluding Remarks In the present paper, the homoclinic bifurcation and chaos in the system of coupled single pendulum and harmonic oscillator under perturbations of dampings and bounded noise has been studied analytically and numerically. Using Melnikov function and Poincaré map for Hamiltonian perturbation, it has been shown that the coupling between the simple pendulum and harmonic oscillator breaks the homoclinic orbits and leads to chaos in the system when the total energy of the system h ≥ 1. For the non-Hamiltonian perturbations (dampings and bounded noise excitation), the random Melnikov process has been derived, and the mean-square criterion has been used to establish the threshold amplitude of bounded noise for the onset of chaos. The analytical result is verified by using the threshold amplitude by vanishing the maximal Lyapunov exponent and using Poincaré maps. It has been concluded that the two threshold amplitudes obtained by vanishing the maximal Lyapunov exponent and using Poincaré maps agree well while the random Melnikov process with mean-square criterion underestimates the threshold amplitude. In the case of non-Hamiltonian perturbations, the contribution of the coupling between the simple pendulum and harmonic oscillator is generally small compared with those due to dampings and bounded noise excitation if the coupling coefficient is of the same small order of damping coefficients and amplitude of bounded noise.

Acknowledgments This work was supported by the National Science Foundation of China under Key Grant No. 10332030 and the Special Fund for Doctor Programs in Institutions of Higher Learning of China under Grant No. 20020335092.

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