Resonant oscillation and homoclinic bifurcation in a Φ6-Van der Pol oscillator

Siewe, M. Siewe; Kakmeni, F. M. Moukam; Tchawoua, Clément

doi:10.1016/j.chaos.2003.12.014

Cited by 62 publications

(21 citation statements)

References 20 publications

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“…We identify this as the Melnikov function for our model. Notice that it reduces to the Melnikov function for a single Φ 6 oscillator [24,25] if either γ 1 or γ 2 is set to zero. From this function, we find that the threshold (F 0cr ) for homoclinic (or heteroclinic) bifurcation is given by…”

Section: Apply the Melnikov Theory To Control Instabilitymentioning

confidence: 99%

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Chaos controlling self-sustained electromechanical seismograph system based on the Melnikov theory

et al. 2010

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This paper deals with problems for which it is necessary to represent the oscillations of the electromechanical seismograph about their position of equilibrium as regards the synchronous condition. The loss of stability of the system occurs through a saddle-node bifurcation, where there is a collision of the stable orbit with an unstable one. Then, global bifurcations and chaotic dynamics of an electromechanical seismograph are the aims of this study. The electrical part of the model is described by an extended force Rayleigh M. Siewe Siewe ( ) oscillator with Φ 6 -potential, while the mechanical part is described by a damped and driven linear oscillator. By using the direct perturbation technique, we analytically obtain the general solution of the first-order equation. Through the boundedness condition of the general solution we get the famous Melnikov function predicting the onset of chaos in the case where the Φ 6 -potential is three wells, which are complemented by numerical simulations by which we illustrate the bifurcation curves and the fractality of the basins of attraction. The results show that the threshold amplitude of harmonic excitation for the onset of instability will move upwards as the amplitude intensity of the ground motion increases. These results suggest that much attention should be paid to controlling the increase of the amplitude of the ground motion, especially when the harmonic excited electromechanical seismograph system as a main device is applied to some practical systems.

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Section: Apply the Melnikov Theory To Control Instabilitymentioning

confidence: 99%

“…It should be pointed out here that several investigations of homoclinic/heteroclinic chaos in the Φ 6 oscillator's dynamics [24][25][26] did not provide good analytical expressions for these orbits.…”

Section: Apply the Melnikov Theory To Control Instabilitymentioning

confidence: 99%

Chaos controlling self-sustained electromechanical seismograph system based on the Melnikov theory

et al. 2010

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“…To verify the effectiveness of proposed adaptive controller (15) and (16) for the 6 -Van der Pol system with both external and parametric excitation, we have shown a series of numerical simulation where parameter values are selected as in the simulations of Fig. 4f.…”

Section: Adaptive Chaos Synchronization For 6 -Van Der Pol Systemmentioning

confidence: 99%

Chaos control and synchronization of the Φ6-Van der Pol system driven by external and parametric excitations

2007

Nonlinear Dyn

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The dynamical behavior of the 6 -Van der Pol system subjected to both external and parametric excitation is investigated. The effect of parametric excitation amplitude on the routes to chaos is studied by numerical analysis. It is found that the probability of chaos happening increases along with the parametric excitation amplitude increases while the external excitation amplitude fixed. Based on the invariance principle of differential equations, the system is lead to desirable periodic orbit or chaotic state (synchronization) with different control techniques. Numerical simulations are provided to validate the proposed method.

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“…Classical Melnikov method is used in many cases to predict the occurrence of chaotic orbits in nonautonomous smooth one-degree-of-freedom nonlinear systems [Litak et al, 2008;Sanjuán, 1999;Siewe Siewe et al, 2004, 2005Soliman & Thompson, 1992]. It involves transverse intersection of stable and unstable manifolds that represent the starting point for a successive route to chaotic dynamics.…”

Section: Melnikov Analysismentioning

confidence: 99%

“…In this case, the fixed points are connected by the heteroclinic trajectories given by (see [Siewe Siewe et al, 2004])…”

Section: A Single Well Potential Casementioning

confidence: 99%

Melnikov Chaos in a Modified Rayleigh–Duffing Oscillator with ϕ6 Potential

Miwadinou

Monwanou

Hinvi

et al. 2016

Int. J. Bifurcation Chaos

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The chaotic behavior of the modified Rayleigh-Duffing oscillator with φ 6 potential and external excitation is investigated both analytically and numerically. The so-called oscillator models, for example, ship rolling motions. The single well and triple well potential cases are considered. Melnikov method is applied and the conditions for the existence of homoclinic and heteroclinic chaos are obtained. The effects of nonlinear damping on roll motion of ships are analyzed in detail. As it is known, nonlinear roll damping is a very important parameter in estimating ship responses. It is noted that the pure and unpure quadratic damping parameters affect the Melnikov criterion in the heteroclinic and homoclinic cases respectively while the pure cubic parameter affects the amplitude in both cases. The predictions have been tested with numerical simulations based on the basin of attraction. It is pointed out that certain quadratic damping effects are contrary to cubic damping effect.

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Resonant oscillation and homoclinic bifurcation in a Φ6-Van der Pol oscillator

Cited by 62 publications

References 20 publications

Chaos controlling self-sustained electromechanical seismograph system based on the Melnikov theory

Chaos controlling self-sustained electromechanical seismograph system based on the Melnikov theory

Chaos control and synchronization of the Φ6-Van der Pol system driven by external and parametric excitations

Melnikov Chaos in a Modified Rayleigh–Duffing Oscillator with ϕ6 Potential

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