HOMOCLINIC BIFURCATION AND CHAOS IN Φ⁶-RAYLEIGH OSCILLATOR WITH THREE WELLS DRIVEN BY AN AMPLITUDE MODULATED FORCE

Abstract

With amplitude modulated excitation, the effect on chaotic behavior of Φ6-Rayleigh oscillator with three wells is investigated in this paper. The Melnikov theorem is used to detect the conditions for possible occurrence of chaos. The results show that the domain of the appearance of chaos is enlarged as both amplitudes of modulated and unmodulated forces increase. The effect of these two amplitudes, when both frequencies of modulated and unmodulated forces are different, on bifurcation diagram and Poincaré map is also investigated, in addition to the surface of Maximal Lyapunov exponent versus modulated and unmodulated parameters for suppressing chaos being shown.

Global orbit of a complicated nonlinear system with the global dynamic frequency method

“…Actually, the hyperbolic type of triple-well orbits has been analytically obtained to find the critical values of chaos through Melnikov integrations. 33,34 Thus, people may directly implement the global dynamic frequency method to solve such tristable or even more complicated systems.…”

Tropomyosin dynamics during cardiac muscle contraction as governed by a multi-well energy landscape

“…In our analysis, parameters such as the damping coefficient ζ, excitation frequency ω, and driving amplitude T o are expected to have influence on the Tm transitions from one position to another and its general dynamical responses. In order to study the effects of these parameters, Melnikov [33] theory is used to find bifurcation conditions and transitions to chaos [34]. Melnikov function M is considered to be a fundamental theory in bifurcation analysis and commonly used tool in nonlinear dynamics studies to determine the existence of chaos induced by a small perturbation to a smoothed Hamiltonian system, similar to the one derived in this study.…”

“…Considering the unperturbed system (i.e., ϵ = 0), expressions for both homoclinic and heteroclinic orbits [34] can be derived and given respectively as…”

Chaotic Motions of the Duffing-Van der Pol Oscillator with External and Parametric Excitations

“…Recently, chaotic motions of many systems, for example, 6 -Rayleigh oscillator [18], Duffing oscillator [19], Gylden's problem [20], and nonsmooth systems [21], have been investigated by the Melnikov method. In this section, we use the Melnikov method to investigate the chaotic motions of system (4).…”