IOP PUBLISHING JOURNAL OF PHYSICS A: MATHEMATICAL AND THEORETICAL J. Phys. A: Math. Theor. 40 (2007) F473–F481 doi:10.1088/1751-8113/40/24/F06 FAST TRACK COMMUNICATION A Van der Pol–Mathieu equation for the dynamics of dust grain charge in dusty plasmas M Momeni 1 , I Kourakis 2 , M Moslehi-Fard 1 and P K Shukla 2 1 Faculty of Physics, Tabriz University, Tabriz 51664, Iran 2 Institut f ¨ ur theoretische Physik IV, Fakult¨ at f¨ ur Physik und Astronomie, Ruhr-Universit¨ at Bochum, 44780 Bochum, Germany E-mail: mehdy momeny@yahoo.com and ioannis@tp4.rub.de Received 9 March 2007, in final form 5 May 2007 Published 30 May 2007 Online at stacks.iop.org/JPhysA/40/F473 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. PACS numbers: 52.27.Lw, 52.30.−q, 82.40.bj, 05.45.−a 1. Introduction The properties of dusty plasmas have recently been attracting growing interest. A dusty (or complex) plasma, compared to an electron–ion plasma, i.e. a large ensemble of electrons and positive ions, is characterized by an additional charged component of micron or submicron- sized dust particulates. The dust grain charge, q d , in contrast to the electron and ion charge, is not constant. When dust grains are immersed in a gaseous plasma, the charge residing on dust particles varies as a result of the flow of plasma particles onto their surface [1, 2]. Dust grains are considerably massive (typically a billion times heavier than protons), and their size ranges from nanometres to millimetres. The presence of massive charged dust particles in a plasma can drastically affect its dispersive and nonlinear properties [3, 4]. Dust grains are an important component of astrophysical systems where self-gravitational effects are dominant 1751-8113/07/240473+09$30.00 © 2007 IOP Publishing Ltd Printed in the UK F473