Hysteresis, Quasiperiodicity and Chaoticity in a Nonlinear Dissipative Hybrid Oscillator

Abstract

Hysteresis, quasiperiodicity and chaoticity in a nonlinear dissipative hybrid oscillator are studied. The modified Rayleigh-Duffing oscillator is considered. We simultaneously take into account the new nonlinear cubic, pure quadratic and hybrid dissipative terms which modify the classical Rayleigh-Duffing oscillator. The influence of each of these new parameters on the dynamics of the oscillator has been seriously studied and interesting results are obtained. It is clear that each of these new dissipation terms can be used to control the dynamics of this oscillator. Some may be used to reduce or eliminate hysteresis, amplitude jump and resonance phenomena; others may accentuate them. Similarly, these new parameters can be used to impose on the systems modeled by this oscillator, a regular, quasi-periodic or even chaotic behavior according to their field of application. Thus, one of the original results obtained is the equation of the curve delimiting the zone of instabilities of the amplitudes of harmonic oscillations. This equation thus makes it possible to know the zone of amplitude permitted or of the amplitude jump for the systems and thus to control and predict the loss or gain of energy during this jump. Finally, the second stability of the oscillations of the system is studied as well as the influence of the dissipation parameters on this stability. It should be noted that the influence of some of these parameters depends on the simultaneous presence of these parameters.