A Van der Pol–Mathieu equation for the dynamics of dust grain charge in dusty plasmas

Abstract

The chaotic profile of dust grain dynamics associated with dust-acoustic oscillations in a dusty plasma is considered. The collective behaviour of the dust plasma component is described via a multi-fluid model, comprising Boltzmann distributed electrons and ions, as well as an equation of continuity possessing a source term for the dust grains, the dust momentum and Poisson's equations. A Van der Pol-Mathieu-type nonlinear ordinary differential equation for the dust grain density dynamics is derived. The dynamical system is cast into an autonomous form by employing an averaging method. Critical stability boundaries for a particular trivial solution of the governing equation with varying parameters are specified. The equation is analysed to determine the resonance region, and finally numerically solved by using a fourth-order Runge-Kutta method. The presence of chaotic limit cycles is pointed out.

Ionization effects and linear stability in a coaxial plasma device

“…There exist various forms for source terms in literature. Since we consider the plasma pattern as a complex and nonlinear structure, Q + and Q − can be determined as αN and −1/3βN 3 (see in [18,19]). Meanwhile we only deal with the linear analysis in this study, we do not take into account the term Q − , which includes a higher order of N. The subscripts and refer to electrons, ions and excited atoms respectively.…”

Two Parameter Families of Global Solutions for Some Classes of Nonlinear Second Order Ordinary Differential Equations

“…Let λ(t), µ(t) and ν(t) be continuous functions on the interval [t 0 ; +∞) and let λ(t) > > 0, µ(t) ≥ 0, ν(t) ≥ 0, t ≥ t 0 . Consider the following Van der Pol's type equation (see [12]).…”

Dynamics of a delayed van der Pol-Mathieu oscillator

“…Assuming the static equilibrium, we can write the nonlinear continuity, momentum balance, and the Poisson equation for dust-charge fluctuation as [8,7],…”

“…Further, we have approximated the term ∇ · (nv v v) with n 0 ∇ · v v v, assuming a uniform distribution of the charged dust particles in space (∇n ≈ 0). The dust-charge q(t) is assumed to be changing harmonically with time with a frequency ν [8,7],…”

“…However, the findings usually throws some lights on the complex dynamics plasmas can have. Momeni, Kourakis, and Shukla [7] has studied a simplified model of nonlinear dustcharge fluctuation based on a vdPM equation, where they have discussed the stability regions of the vdPM equation and have shown the existence of stable and unstable periodic orbits in different parameter space. This vdPM equation is originally derived by Saitou and Honzawa [8], where they have shown that in a very restricted region of parameter space, the system exhibits chaotic behavior.…”