Melnikov chaos in a periodically driven Rayleigh–Duffing oscillator

Mechanics Research Communications 37 (2010) 363–368

Contents lists available at ScienceDirect

Mechanics Research Communications journal homepage: www.elsevier.com/locate/mechrescom

Melnikov chaos in a periodically driven Rayleigh–Duffing oscillator M. Siewe Siewe a,b,∗ , C. Tchawoua b , P. Woafo c a

Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 0002, South Africa Université de Yaoundé I, Faculté des sciences, Département de Physique, Laboratoire de Mécanique, BP: 812 Yaoundé-Cameroun c Université de Yaoundé I, Faculté des sciences, Département de Physique, Laboratory of Nonlinear Modelling and Simulation in Engineering and Biological Physics, BP: 812 Yaoundé-Cameroun b

a r t i c l e

i n f o

Article history: Received 15 December 2009 Received in revised form 3 April 2010 Available online 13 May 2010 Keywords: Homoclinic orbit Melnikov chaos Rayleigh oscillator Bifurcation

a b s t r a c t The chaotic behavior of Duffing–Rayleigh oscillator under harmonic external excitation is investigated. Melnikov technique is used to detected the necessary conditions for chaotic motion of this deterministic system. The results show that the shape of the basin boundaries of attraction are fractals as the damping increases above the threshold of Melnikov chaos. The effect of damping parameter on phase portraits and Poincaré maps, in addition to the numerical simulations of bifurcation diagram and maximum Lyapunov exponents is also investigated. Crown Copyright © 2010 Published by Elsevier Ltd. All rights reserved.

1. Introduction The nonlinear dynamics in forced systems have received a substantial amount of attention since the prediction of the scaling routes of chaos. Three of the fundamental forced oscillators, Duffing, Van der Pol, and Rayleigh oscillators, have been extensively examined because a number of dynamic features embedded in the physical systems can be realized from these two systems (Ueda and Kawakami, 1996; Kao, 1993; Venkatesan and Lakshmanan, 1997). The Rayleigh oscillator is much like the Van der Pol oscillator save one key difference: as voltage increases, the Van der Pol oscillator increases in frequency while the Rayleigh oscillator increases in amplitude. A quite important class of dynamical systems is that of Duffing–Rayleigh forced systems driven with onewell or two-well potential depending on the set of the parameters. These systems are of interest because they can occur in physical situations (Yamapi and Woafo, 2005), or in situations where some chemical or biological oscillators are driven by one periodic signals with frequencies (Winfree, 1980; Kuramoto, 1980), and also in engineering (Li et al., 2004). Much work has been done on the mechanism by which strange attractors arise and change as a parameter of the system is varied. These mechanisms include period-doubling cascades, intermittency, crisis, etc. Recently it has been shown that basin boundaries for typical nonlinear dynamical systems can be strange (in the sense of having fractal dimension) and they occur in

∗ Corresponding author. Tel.: +237 70254631. E-mail addresses: [email protected], [email protected] (M. Siewe Siewe), [email protected] (C. Tchawoua), [email protected] (P. Woafo).

the simplest nonlinear systems (e.g., Josephson junctions, clamped beams, electrical circuits, etc...) (Grebogi et al., 1986). The effect of the Rayleigh damping term on homoclinic transition to chaos was earlier considered in the context of pendulum oscillations (Litak et al., 1999) and vehicle vibrations with the inverted Duffing potential (Litak et al., 2008). The Melnikov criterion has been introduced by Melnikov in Melnikov (1963) and studied extensively in Guckenheimer and Holmes (1983). Particular interest has been devoted to apply the Melnikov method in the parametrically modified Sine–Gordon (Kenfack and Kofane, 1998) for estimating the transition threshold which indicates the change from oscillation (or liberation) to rotation solutions, generated by a periodic spatial driven field. This method was also applied by considering a one-dimensional Sine–Gordon model under the influence of dcdriven field to predict the threshold of irregular behavior (Kenfack and Kofane, 1995). It is also important to notice the paper of Taki et al. (1988) who were the first to study temporal chaos in ac- and dc-driven Sine–Gordon systems with breathers or bosons for initial data. The resultant equation of Duffing–Rayleigh forced system is given by x¨ − (1 − x˙ 2 )x˙ − ˛x + ˇx3 = F cos ωt.

(1)

where , ˛, ˇ are respectively nonlinear damping, linear and nonlinear restoring parameters, F and ω are respectively amplitude and frequency of the external force. We consider here the effect of the external periodic excitation and nonlinear damping on the regular and chaotic dynamics of model Eq. (1) with double-well potential. Our goal in this paper is to show explicitly how the perturbation method base on the Melnikov theory can be applied to the above strongly nonlinear oscillator Eq. (1) and to show how fractal basin

0093-6413/$ – see front matter. Crown Copyright © 2010 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.mechrescom.2010.04.001

364

M. Siewe Siewe et al. / Mechanics Research Communications 37 (2010) 363–368

boundaries arise and change as the damping coefficient varied. In particular, the case of the two-well potential is considered. The rest of the paper is organized as follows. In Section 2, we deal with description and analysis of the model. In Section 3, the conditions of existence of Melnikov’s chaos under damping resulting from the homoclinic bifurcation are performed. Finally in Section 4, a convenient demonstration of the accuracy of the method is obtained from the fractal basin boundaries, bifurcation diagram, maximum Lyapunov exponent, Poincaré map. We end up in Section 5 with conclusion. 2. Description and analysis of the model It is well known that in the absence of the damping and the external force, because of energy conservation one can clearly never get chaos from the motion of a single degree of freedom. Since we are concern with the Melnikov’s chaos, we therefore add both a driving force and damping, in order to remove energy conservation. When ˇ = 0, we obtain the Rayleigh equation which has been extensively studied, and it is known to be the most important equation to investigate the bifurcation phenomena in a driven oscillator (Sekikawa et al., 2001), from a viewpoint of its basic importance. It is known that the forced Rayleigh equation is the first differential equation for which the existence of a non-periodic solution is theoretically demonstrated (Levinson, 1949). We have plotted in Fig. 1(a) the phase portrait corresponding to the limit cycle in the unforce Rayleigh oscillator. Fig. 1(b) represents the phase portrait of the unforce Rayleigh–Duffing oscillator in the case of the two-well potential. We next derive the fixed points and the phase portrait corresponding to the unperturbed system Eq. (1). In the left hand of Eq. (1), the second term containing nonlinear dissipation, and the force term in the right hand side are considered as perturbations terms. In the forthcoming part of the paper, parameter values are ˛ = 0.5, ˇ = 0.05. If we let  = F = 0, Eq. (1) is considered as an unperturbed system and can be written as x˙ = y, y˙ = ˛x − ˇx3 .

(2)

The system of Eq. (2) corresponds to an integrable Hamiltonian system with a potential function ˛ ˇ V (x) = − x2 + x4 2 4

(3)

and the associated Hamiltonian function which corresponds to the total energy is H(x, y) =

1 2 ˛ 2 ˇ 4 y − x + x . 2 2 4

(4)

From Eqs. (2)–(4), the unperturbed system has three equilibrium points: two centers (c1 , 0) and (c2 , 0) and one saddle (0, 0). The saddle is connected to itself by two symmetric homoclinic orbits as shown in Fig. 2(b). 3. Taming chaotic behavior in the Rayleigh-Duffing oscillator The generalized Melnikov method developed by Kovacic and Wiggins (1992) consists of studying a system in which the unperturbed problem is an integrable Hamiltonian system having a normally hyperbolic invariant set whose stable and unstable manifolds intersect non-transversally. The structure of the unperturbed system is that of two uncoupled one-degree of freedom Hamiltonian systems. In this section, we discuss the chaotic behaviors of the system of Eq. (1) in which  and F are assumed to be small parameters. A transformation of  → ε, F → εF is done in order to

Fig. 1. (a) Limit cycle of the unforce Rayleigh oscillator ˛ = 0.5; (b) phase portrait of the unforce two-well Rayleigh–Duffing oscillator.

apply the ε first-order perturbation scheme of the Melnikov theory. Hence, the system of Eq. (1) may be written as

⎧ ⎨ x˙ = y,





y˙ = ˛x − ˇx3 + ε (1 − y2 )y + F cos 1 ,

(5)

⎩ ˙ = ω 1

When the perturbations are added, the closed homoclinic breaks, and may have transverse homoclinic orbits. By Smale–Birkhoff Theorem (Wiggins, 1990), the existence of such orbits results in chaotic dynamics. As is well known, their predictions for the appearance of chaos are both limited (only valid for orbits starting at points sufficiently near the separatrix) and approximate. Although the chaos does not manifest itself in form of permanent chaos, it does in terms of the fractal basin boundaries, as it was shown by Trueba et al. (2003). Satisfying the conditions for a double-well potential gives rise to a homoclinic orbit in the system’s phase space for ε = 0. The homoclinic trajectory can be found by setting H(x, y) = 0. Solving for the resulting displacement and differentiating to determine velocity, the homoclinic trajectory is given as follows:



(xh , yh ) =



√ 2˛ sech( ˛t), − ˇ

√ √ 2˛ √ ˛sech( ˛t) tanh ˛t ˇ



. (6)

M. Siewe Siewe et al. / Mechanics Research Communications 37 (2010) 363–368

Fig. 3. Homoclinic bifurcation in the

365



 

/F , ω plane.

for the existence of chaos. Since the Melnikov function theory measures the distance between the perturbed stable and unstable manifolds in the Poincaré section, to preserve the homoclinic loops under a perturbations requires that at t0 , if M(t0 ) has a simple zero, then a homoclinic bifurcation occurs, signifying the possibility of chaotic behavior. This means that only necessary conditions for the appearance of strange attractors are obtained from Poincaré–Melnikov–Arnold analysis, and therefore one always has the chance of finding sufficient conditions for the elimination of even transient chaos. Then the general necessary condition for which the invariant manifolds intersect is given by



Fig. 2. (a) The two-well potential function of the system; (b) the corresponding phase space portraits.

Note the characteristic of unique saddle point that is (0, 0) is going to be reached in exactly defined albeit infinite time t corresponding to +∞ and −∞ for stable and unstable orbits, respectively. We therefore apply the Melnikov method to system Eq. (5) for finding the criteria of the existence of homoclinic bifurcation and chaos.





±

yh2 dt

M (t0 ) = 

−



yh4 dt

+F

yh cos ω(t − t0 )dt.

(7)

where t0 is the cross-section time of the Poincaré map and t0 can be interpreted as the initial time of the forcing term. This Melnikov expression comprises in a compact way a lot of particular results that can be found in the literature. After substituting xh and yh by formulate given in Eq. (6) into Eq. (7) and evaluating the integral, we obtain the Melnikov function M ± (t0 ) = I0 − I2 + FI3 sin ωt0 where I0 =

2˛2 ˇ

I2 =

4˛4 ˇ2



I3 = −



+∞



−∞

√ √ sech4 ( ˛t) tanh4 t ˛dt +∞



(10)

this implies that if ε > 0 is sufficiently small, the reduced Eq. (5) has transverse homoclinic orbits resulting in possible chaotic dynamic. With ˛ and ˇ constant, we study chaotic threshold as a function of only the frequency parameter ω. A typical plot of (/F) is shown in Fig. 3. The qualitative form of this function remains the same as ˛ and ˇ take the values for which the potential is two-well. Another point is that, when the ratio (/F) tend to zero, this means that the external amplitude F tend to infinity, then the Melnikov theory is not valid for these values. From Eq. (8), chaotic behavior is guaranteed for the trajectories whose initial data are sufficiently near the unperturbed separatrix Eq. (6) if  ≤ F

 F

cr

   3ωˇ 2ˇsech ω √   2 ˛  = √ ,  4˛2 (3˛ − ˇ) ˛ 

(11)

where (/F)cr is the threshold function. 4. Bifurcation analysis and fractal basins

√ √ sech2 ( ˛t) tanh2 t ˛dt

−∞ +∞ 2˛2 ˇ

(8)



ω     3ωˇ 2ˇsech 2√˛  = √ , F  4˛2 (3˛ − ˇ) ˛ 

4.1. Bifurcation diagram and Lyapunov exponent (9)

√ √ sech( ˛t) tanh ˛t sin ωtdt

−∞

After evaluation of these elementary integrals (see Appendix A), the homoclinic Melnikov function is derived. Let us study the intersections of the invariant manifolds of the saddle point. It is known, that these intersections are the necessary conditions

We draw the bifurcation diagram of Eq. (1) in (x, ) plane and the corresponding maximal Lyapunov exponents corresponding in Fig. 4. Numerical calculations have been made for the selected parameter values F = 0.8 and ω = 1. Eq. (1) can be readily solved numerically using the fourth-order Runge–Kutta algorithm. Since the equation is nonlinear, its solution therefore admits the possibility of periodic and chaotic orbits. In Fig. 4(a), after a large band of chaotic regime for  ∈ [0, 0.1], one can find a sequence of back-

366

M. Siewe Siewe et al. / Mechanics Research Communications 37 (2010) 363–368

Fig. 4. (a) Bifurcation diagram of Eq. (1) with varying parameter  and (b) corresponding maximum Lyapunov exponent. The other parameters are ω = 1, and F = 0.8.

ward period-doubling bifurcations as a route to periodic motion and after a periodic regime, another bifurcation takes place at a critical value  = 0.12 where another large band of chaotic regime with small periodic window occurs. At  ≥ 0.275, the system displays periodic behavior after backward period-doubling bifurcations. Such maps can be used to suppress chaotic dynamics of the system. A positive top Lyapunov exponent for a bounded attractor is usually a sign of chaos. The top Lyapunov exponents of Eq. (1) are also calculated. The results are shown in Fig. 4(b) for  ∈ [0, 1]. Meanwhile, the thresholds of damping coefficient for onset of chaos are obtained by letting the top Lyapunov exponents vanish, which are also shown in Fig. 4(b). From Fig. 4(b) one can see that for smaller values of damping amplitude , the top Lyapunov exponent is positive. As  increases, the top Lyapunov exponent changes from positive value to negative value, signifying the suppressing of homoclinic chaos motion. Beyond the threshold for onset of chaotic motion, there are some “periodic windows”, this can be the feature of the transient chaos. However, for larger damping, there are no longer “periodic windows” presented up to the cutoff value of  in simulation. To support our analytical prediction of the Melnikov chaos, we have performed computer simulations on the perturbed system (1).

Fig. 5. Phase trajectories; (a) chaotic regime  = 0.02, (b) period-2 motion  = 0.095, (c) period-1 motion  = 0.11, and (d) chaotic regime  = 0.205, with ω = 1, and F = 0.8.

M. Siewe Siewe et al. / Mechanics Research Communications 37 (2010) 363–368

367

Fig. 6. Poincaré section for (a)  = 0.02, (b)  = 0.205, with ω = 1 and F = 0.8.

This system is integrated numerically by using the fourth-order Runge–Kutta method. First, we choose across the bifurcation curve in Fig. 3 several parameter values of . We show that for lower values of i.e.,  ≤ 0.275, both chaotic or regular oscillations may appear. For this purpose, the phase portraits given by Fig. 5 were simulated for the external amplitude F = 0.8 and the frequency ω = 1, with the initial condition of (x0 , y0 ) = (0, 0). We found that, for the values less than the analytical threshold of the homoclinic bifurcation either chaos or periodic motion can arise and are shown in Fig. 5. A bifurcation diagram shows the different types of attractors to which the system settles to as the bifurcation parameter is varied. However, a bifurcation diagram provides very little information concerning the shape of the attractor in the state space. In order to gain further insight into the geometry of the attractors, one may use the so-called Poincaré maps. As presented in Fig. 6 for two values of the damping coefficient chosen in different attractors of the bifurcation diagram of Fig. 4(a), the evolution of the attractors of the Poincaré map shows the increasing loss of smoothness by accentuated fractal geometric shape, and thus confirm the chaotic behavior of the system. 4.2. Basin of attraction A basin of attraction is an important means of checking whether a regulation of the fractal boundaries of basin of attraction has been realized (Cao and Chen, 2005). We want to study how is the effect of using nonlinear damping terms on the equation of the oscillator and how the basins of attraction are affect as the coefficient parameter  is varied. To show the fractal structure, we consider the case of the bifurcation of motion close to the resonance since it may undergo the limit cycles in the system. Hence

Fig. 7. Basin of attraction for (a)  = 0.02, (b)  = 0.06, and F = 0.8.

we fix F = 0.8 and ω = 1, the Melnikov threshold value is given analytically by he 0.025 but the numerical one is he 0.12 which is greater than the numerical value. This is not surprising since we have used the first-order Melnikov method approximation. Figs. 7 and 8 show the basins of three coexisting attractors of the forced Duffing–Rayleigh oscillator. The picture was made by starting with 500 × 500 grid of points in [−5, 5] × [−2, 2] as initial values, and by solving the differential equation for initial condition representing each pixel, and coloring the pixel blue, black, and green depending on which sink orbit attracts the orbit. The three attractors in Fig. 7(a) and (b) (with green color, blue color and black color) are two stable fixed points and one unstable fixed point. In fact the green color (and the blue color) corresponds to the left well (and the right well) potential given by Fig. 2(a), respectively. These two colors may display two periodic attractors in the basin of attraction while the black color which correspond to the unstable point (middle of potential well given by Fig. 2(a)) may display chaotic attractor in the basin of attraction. From Fig. 7(a) and (b), the fractal shape of the basin boundaries of attraction is observed. In Fig. 8(a), it is shown that under the Melnikov control, the shape of the basin is smooth and the boundary of the basin consists of the smooth curve.

368

M. Siewe Siewe et al. / Mechanics Research Communications 37 (2010) 363–368

Acknowledgments The authors are grateful to the anonymous referees for their valuable comments. The first author M. Siewe Siewe is also indebted to the University of Pretoria for its financial support to do research work as a Postdoctoral Fellow and also indebted to the Department of Mathematics and Applied Mathematics for hosting him to undertake part of this work. Appendix A. Then computation of the Melnikov integrals



2˛2 I0 = ˇ I2 =

−∞



4˛4 ˇ2

=−

+∞

−∞

 I3 = −

+∞

2˛2 ˇ

ω

√ √ √ 4˛ ˛ sech2 ( ˛t) tanh2 t ˛dt = 3ˇ

(A.1)

√ √ √ 16˛ ˛ sech4 ( ˛t) tanh4 t ˛dt = 35ˇ2

(A.2)





+∞

−∞

2ˇ

ˇ

√ √ sech( ˛t) tanh ˛t sin ωtdt

ω sech √ 2 ˛

(A.3)

References

Fig. 8. Basin of attraction for (a)  = 0.09, (b)  = 0.24, and F = 0.8.

5. Conclusion Using Melnikov’s method, chaos has been predicted to occur in Rayleigh-Duffing oscillator. The result of this analysis was an inequality describing the set of parameters where chaos occurs, which is a useful design tool for tailoring the oscillator’s parameters so homoclinic chaos either occurs or does not occur as desired. For a representative parameter set, numerical simulations showed that chaos does occur in regions of parameter space satisfying the criterion from Melnikov’s method and helped to better understand the complicated dynamics of the oscillator. By varying damping coefficient  of the system, many interesting attractors are found, including periodic orbits and chaos. The phase portraits, the Poincaré maps, the basins of attraction, the maximum Lyapunov exponent and the bifurcation diagrams show that the stability of the Rayleigh-Duffing oscillator will lose once the damping term exceeds a critical value. All these results strongly demonstrate that the damping term can produce a considerable effect on the Rayleigh-Duffing oscillator.

Cao, H., Chen, G., 2005. Global and local control of homoclinic and heteroclinic bifurcations. Int. J. Bifurcat. Chaos 15, 2411–2432. Grebogi, C., Ott, E., Yorke, J.A., 1986. Methamorphoses of basin boundaries in nonlinear dynamical systems. Phys. Rev. Lett. 59, 1011–1014. Guckenheimer, J., Holmes, P., 1983. Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vectorfields. Springer, New York. Kao, Y.H., 1993. Analog study of bifurcation structures in Van der Pol oscillator with nonlinear restoring force. Phys. Rev. E, 2514–2520. Kenfack, A., Kofane, T.C., 1998. Melnikov’s method for spatial periodic field and bifurcation in a modified Sine–Gordon model. Physica Scripta 58, 659– 663. Kenfack, A., Kofane, T.C., 1995. Melnikov’s method for spatial periodic field and bifurcation in a modified Sine–Gordon model. Phys. Rev. B 58, 10359–10363. Kovacic, G., Wiggins, S., 1992. Orbits homoclinic to resonances with an application to chaos in a model of the forced and damped Sine–Gordon equation. Physica D 57, 185–225. Kuramoto, 1980. Chemical Oscillations, Wave and Turbulence. Springer, Berlin. Levinson, N., 1949. A second order differential equation with singular solutions. Ann. Math. 50, 127–153. Li, S., Yang, S., Guo, W., 2004. Investigation on chaotic motion in histeresis nonlinear suspension system with multi-frequency excitation. Mechan. Res. Commun. 31, 229–236. Litak, G., Spuz-Szpos, K., Szabelski, J., Warminski, J., 1999. Vibration of externally forced froude pendulum. Int. J. Bifurcat. Chaos 9, 561–570. Litak, G., Borowiec, M.I., Friswell, K., 2008. Chaotic vibration of a quarter-car model excited by the road surface profile. Commun. Nonlinear Sci. Numer. Simul. 13, 1373–1383. Melnikov, V.K., 1963. On the stability of the center for time periodic perturbations. Trans. Moscow Math. Soc. 12, 1–56. Sekikawa, M., Inaba, N., Yoshinagat, T., Kawakamit, H., 2001. The bifurcation structure of fractional-harmonic entrainments in the forced Rayleigh oscillator. IEEE 31, 1269–1272. Taki, M., Fernandez, J.C., Reinisch, G., 1988. Collective-coordinate description of chaotic Sine–Gordon breathers on zero frequency breathers: the non dissipative case. Phys. Rev. A 38, 3086–3097. Trueba, J.L., Baltanás, J.P., Sanjuá, M.A.F., 2003. A generalized perturbed pendulum. Chaos Soliton Fract. 15, 911–924. Ueda, T., Kawakami, H., 1996. Unstable saddle-node connecting orbits in the averaged Duffing–Rayleigh equation. ISCAS 3, 288–291. Venkatesan, A., Lakshmanan, M., 1997. Bifurcation and chaos in the double-well Duffing-Van der Pol oscillator: numerical and analytical studies. Phys. Rev. E 56, 6321–6330. Wiggins, S., 1990. Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer-Verlag, New York, Heidelberg, Berlin. Winfree, A.T., 1980. The Geometry of Biological Time. Springer, New York. Yamapi, R., Woafo, P., 2005. Dynamics and synchronization of coupled self-sustained electromechanical devices. J. Sound Vibr. 285, 1151–1170.