Harmonic Solutions Analysis and Floquet Theory in a Nonlinear Dissipative Hybrid Oscillator

The main objective is to study the simultaneously effects of the new nonlinear cubic, pure quadratic and hybrid dissipative terms which modify the classical Rayleigh-Duffing oscillator. Similarly, the parameters in presence are used to impose on the systems modeled by this oscillator, a regular, quasi-periodic or even chaotic behavior according to their field of application. In a nonlinear dissipative hybrid oscillator, hysteresis, quasiperiodicity, and chaoticity are investigated. It is decided to use a modified Rayleigh-Duffing oscillator. The new nonlinear cubic, pure quadratic, and hybrid dissipative terms that change the classical Rayleigh-Duffing oscillator are all taken into account at the same time. The impact of each of these new factors on the oscillator's dynamics has been thoroughly investigated, yielding some intriguing results. 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, depending on their field of use, these new parameters can be utilized to impose a regular, quasi-periodic, or even chaotic behavior on the systems depicted by this oscillator. The equation of the curve delimiting the zone of instability of the amplitudes of harmonic oscillations is one of the initial results obtained. 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. It is concluded that 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.