Nonlinear dynamics of a charged dust grain in a plasma

For an arbitrary monotonic charging function, the dynamics of a dust grain is dissipative and energy is a Liapunov function. In an arbitrary external potential two types of equilibria exist. The first type, with uncharged grain, is always unstable. The second type of equilibrium, admitting states of both positive and negative charge, can be marginally stable; stability depends on the local potential. Under spatially uniform (constant or time-dependent) potentials, motion is free while the charge adapts to the potential. For a spatially oscillating potential, the phase space is that of the simple pendulum with one additional degree of freedom, the charge. Dissipation in the charging process forbids periodic behavior and ensures the existence of attractors: A grain is at stable equilibrium only when charged positively and trapped in a potential well, or when charged negatively on top of a hill. The small oscillations near a stable equilibrium decay weakly, and the grain charge oscillates at twice the oscillation frequency.