Nonlinear dust charge fluctuations in dusty (complex) plasmas: a Van der Pol-Mathieu model equation
M. Momeni¤, I. Kourakis a;† and P. K. Shukla b;†
¤Faculty of Physics, Tabriz University, Tabriz 51664, Iran
†Ruhr-Universität Bochum, Institut fürTheoretische Physik IV, D-44780 Bochum, Germany
a www.tp4.rub.de/»ioannis ; b www.tp4.rub.de/»ps
Abstract. The parametric excitation of dust acoustic oscillations due to dust grain charge fluctuations in a dusty plasma (DP)
is investigated. A three-component model DP is considered, consisting of negative inertial dust grains (of constant size, mass
and charge, for simplicity), in addition to a thermalized (Maxwellian) background of electrons and ions. By employing a
fluid plasma description, and assuming a periodic fluctuation of the dust charge, a Van der Pol-Mathieu-type hybrid nonlinear
oscillator model ordinary differential equation is obtained for the dust number density. An averaging technique provides the
framework for an analysis of the dust density evolution in time, via an analytical reduction to an autonomous set of equations
for a slowly varying pair of perturbation amplitudes. A phase space portrait is determined analytically and numerically, in
terms of the DP frequency wd;p and the parametric excitation (charge fluctuation) frequency wd f .
Keywords: Dust-acoustic plasma waves, Van der Pol equation, Mathieu equation, phase space dynamics, chaos.
PACS: 52.30.-q, 82.40.bj, 05.45.-a
Dusty (or complex) plasmas (DP) are characterized by the presence of massive mesoscopic (micron-sized, typically)
particulates (“dust grains”), which is known to modify plasma properties substantially. The electric charge qd which
resides on dust grains is acquired dynamically (when dust grains are immersed in a gaseous plasma) and may fluctuate
in time via a variety of charging processes, as a result of plasma particle flow onto their surface . New modes and
associated instabilities occur in DP, some of which are absent in ordinary e-i plasmas. A recent study was devoted to a
dust charging instability modelled via the chaotic behavior of charged dust in the plasma . A Van der Pol-Mathieu
equation was introduced therein in order to model the dynamical behavior of the dust grain charge. Generic nonlinear
oscillator model differential equations of this kind can be studied by existing analytical methods [3, 4]. The Van der Pol
equation has been studied by many researchers [5, 6]. Siewe  has investigated a system consisting of an extended
Duffin-Van der Pol oscillator, in which resonance and off-resonance oscillations are analyzed using the multiple time
scale method, while Maccari  introduced a new asymptotic perturbation method in search of an exact solution.
The present work aims at investigating the dynamical behavior of dust dynamics near parametric resonance. Relying
on a dust fluid model, a Van der Pol – Mathieu nonlinear equation is shown to govern the dust dynamics. The equation
is analyzed in the vicinity of resonance, and a phase portrait is drawn, both analytically and numerically.
A VAN DER POL-MATHIEU (VDPM) EQUATION FOR DUST FLUCTUATIONS
We consider an unmagnetized collisionless dusty plasma, consisting of electrons (mass me, charge ¡e), ions (mass mi,
charge +Zie) and dust grains. The dust grain mass md is assumed to be constant, for simplicity, whilst the dust grain
charge is a time-dependent variable qd(t) = ¡Zd(t)e.
The cold inertial dust fluid density nd and velocity vd are governed by the density evolution equation where ni and ne denote the ion and electron number density, respectively. The right-hand side of Eq. (3) cancels at
equilibrium, thanks to the charge neutrality condition Zini0¡ne0¡Zdnd0 =0, where ns0, for s=i;e;d, denotes the ion,
electron and dust particle number density at equilibrium, respectively. We shall assume a harmonic potential variation
in space, characterized by a wave length l ´ 2p=k (and a wave number k), i.e. f (z; t) = fˆ(t)exp(ikz). Being much
lighter than dust particles, both electrons and ions are assumed to be in local thermodynamic equilibrium, so their number
densities, ne and ni, obey a Boltzmann distribution, viz. ne = ne0 exp
and ni = ni0 exp
Assuming f ¿fkBTe=e; kBTi=Zieg, and considering a harmonic dust charge fluctuation, i.e. qd =qd0(1+hcosgt)1=2
(where the real parameter h ¿ 1 and the frequency g are real constants), Eqs. (1) to (3) are combined into a closed
evolution equation for the dust density, in the form of a HYBRID MATHIEU-VAN DER POL EQUATION
dt2 ¡(a ¡b x2)
0 (1+hcosgt)x = 0; (4)
where we have defined the dimensionless parameter x = nd=nd0¡1, and the characteristic oscillation frequency w0 =
)1=2; the dust plasma frequency reads wpd = (4pnd0q2d
0=md)1=2. The Debye wave number kD is defined
via the effective Debye length lDe f f = (l ¡2
De +l ¡2
Di )¡1=2, where lDe = (kBTe=4pne0e2) 12
and lDi = (kBTi=4pZini0e2) 12
are the electron and ion Debye radii, respectively. The inertialess electrons and ions affect the dust acoustic oscillations
via a dynamical charge balance. In the limit kÀkD, w0 ¼wpd. The dust density is assumed to be uniform in space.
The second term in the left-hand side of equation (4) is characteristic of the Van der Pol (VdP) nonlinear oscillator
model equation; indeed, the VdP Equation, which generically describes a self-sustained nonlinear oscillation, is exactly
recovered for h = 0. On the other hand, for a = b = 0, one recovers a Mathieu-type equation, which describes a
parametric type oscillation. The ordinary differential equation (ODE) (4) is a hybrid equation, combining the features
of the Van der-Pol and the Mathieu equations.
REDUCTION TO A SET OF COUPLED AMPLITUDE ODE’S
The averaging method. In this section, we shall investigate the dynamical behavior of the nonlinear Van der Pol-
Mathieu (VdPM) oscillator (4), under the effect of parametric resonance. The dynamical profile is determined by an
interplay among the parameters a, b and w2(t) = w2
0 (1+hcosgt) (a time dependent oscillation frequency). Since
parametric resonance is stronger for a frequency w(t) nearly twice the eigenfrequency w0 (see e.g. in Ref. , §27),
we shall consider the parametric excitation frequency to be g = 2w0+e , where e ¿1 is a (small) real parameter.
We assume a solution given by the ansatz NUMERICAL RESULTS AND DISCUSSION
The VdPM-type nonlinear ODE (4) possesses an oscillatory (periodic) solution, which is a periodic attractor. Every
nontrivial solution tend to this periodic solution. The periodic solution may be sought by varying a and b parameters.
In order to investigate the dynamical profile of Eq. (4) numerically, we have chosen a set of fixed values: w = 1:0,
w0 = 1:0 and h = 0:01, in addition to the initial conditions fx0;y0; t0g = f1:0;1:0; 0:0g. Employing a fourth-order
Runge-Kutta method, we have solved Eq. (4). The system was found to possess various stable and unstable limit
cycles. The phase diagram in (x;y) and (t;x) planes, for different values of a and b , is depicted in the Figures.
Periodic states occur when we choose a11+a22 = 0, i.e. a =b ; see in Fig. 1. For a <b the system exhibits a stable
limit cycle: large amplitude initial states are attracted to the limit cycle; cf. Fig. 2. In Fig. 3 (for a = 100b = 0:1, the
system’s behavior is initially unstable, and a typical chaotic limit cycle picture is obtained; the solution later tends to
a limit cycle from inside. As a increases, the system attains a stable state in a deformed limit cycle; cf. Fig. 4.
The Van der-Pol oscillator (with no external force) is known to converge to a limit cycle. Here, we see that lower
values of a lead to a limit cycle similar to that of the (stable) Mathieu equation, while for higher values of a, a
profile similar to the limit cycle of the Van der-Pol equation is recovered. We conclude that this system features a
balance among an instability region, where it behaves according to the Mathieu equation, and a stability region, where
it follows the Van der-Pol equation profile (recall that the VdP equation always possesses a periodic solution).
These results complement previous findings on dust charging instabilities occurring in dusty plasmas.
Acknowledgements M.M. would like to thank Prof. J. Mahmodi for helpful discussions. His research was partially
supported by the University of Tabriz (Iran). I.K. acknowledges partial support by the Deutsche Forschungsgemeinschaft
(Bonn, Germany) through the Sonderforschungsbereich (SFB) 591 Programme.
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4. M.W. Hirsch and S.Smale, Differential Equations, Dynamical Systems and Linear Algebra (Academic Press, N.Y., 1974).
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8. A. Maccari, Nonlinear Dynamics, 15, 329-343 (1998).
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