Nonlinear dynamics of plasma oscillations modeled by an anharmonic oscillator

Abstract

This paper considers nonlinear dynamics of plasma oscillations modeled by an anharmonic oscillator. These plasma oscillations are described by a nonlinear differential equation of the form ẍ + ͑1+x 2 ͒ẋ + x + x 2 + ␦x 3 = F cos ⍀t. The amplitudes of the forced harmonic, superharmonic, and subharmonic oscillatory states are obtained using the harmonic balance technique and the multiple time scales method. Admissible values of the amplitude of the external strength are derived. Bifurcation sequences displayed by the model for each type of oscillatory states are performed numerically through the fourth-order Runge-Kutta scheme.

Hysteresis, resonant oscillations and bifurcation mode of a system modeled by a forced modified Van der Pol-Duffing oscillator

“…When the friction term vanishes (є = 0), then the equation reduces to forced modified Duffing oscillator equation (Enjieu et al, 2007) and є ≠ 0, α = 0 leads to an anharmonic oscillator (Enjieu et al, 2008). Assuming that the fundamental component of the solution and the external excitation have the same period, the amplitude of harmonic oscillations can be tackled using the harmonic balance method (Hayashi, 1964).…”

“…Through these studies, we found the effects of parameters in general and in particular the effect of the hybrid quadratic parameter α which shown the difference between this equation and anharmonic equation obtained in (Enjieu et al, 2008).…”

“…Experiments suggest that some plasma behavior is approximately described by anharmonic oscillations. Thus, it has been shown experimentally (Ostrikov and Xu, 2007) and theoretically (Enjieu et al, 2007) that in plasma physics, the electron beam surfaces, the Tonks-Datter resonances of mercury vapor and low frequency ion sound waves, oscillations are described by the following anharmonic Equations (1) or (2) theoretically (Enjieu et al, 2008): …”

“…F stands for the amplitude of the external excitation whileβ; and γ are the quadratic, cubic nonlinearities and damping parameters. Enjieu et al (2008) studied with a rigorous theoretical consideration through a method based on the harmonic balance formalism the amplitude of the forced harmonic oscillations states. It is found the resonance states, admissible values of F and bifurcation structures numerically.…”

Analysis of Multiresonance and Chaotic Behavior of the Polarization in Materials Modeled by a Duffing Equation with Multifrequency Excitations

“…By varying the parameters of system, we draw the resulting bifurcation diagram and the variation of the corresponding largest Lyapunov exponent versus amplitude F=F 1 =F 2 . The Lyapunov exponent is defined as (Enjieu, Chabi Orou & P. Woafo, 2008a, Enjieu, Nana, Chabi Orou & Talla, 2008b:…”