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.

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The effects of plasma inhomogeneities on the propagation of a cloud of hot electrons through a cold background plasma and generation of Langmuir waves are investigated using numerical simulations of the quasilinear equations. It is found that in a plasma with decreasing density the quasilinear relaxation of the electron distribution in velocity space is accelerated and the levels of the generated Langmuir waves are enhanced. The magnitude of the induced emission rate is increased and its maximum value moves to lower velocities. Due to density gradient the height of plateau shows an increase at small distances and a corresponding decrease at large distances. It is also found that in a plasma with decreasing temperature, the relaxation of the beam is retarded, the spectral density of Langmuir waves is broadened, and the height of the plateau decreases below its value in a uniform plasma. In the presence of both density and temperature gradients, at given position, the height and upper boundary of the plateau and the level of Langmuir waves are all increased at small velocities. The spatial expansion of the beam is increased by the plasma inhomogeneities, but its average velocity of propagation decreases. Initially, at a given position, the velocity at the upper boundary of the plateau is smaller in the presence of the density gradient than in the uniform plasma but the reverse is true at longer times. Due to temperature gradient, at large times and small distances, the upper boundary of the plateau is increased above its value in the uniform plasma. Because of fast relaxation, the value of the lower boundary of the plateau in the plasma with decreasing density is always less than its value in the uniform plasma. It is found that the local velocity of the beam decreases when the density gradient is present. The local velocity spread of the beam remains unchanged during the propagation of the beam in the uniform plasma, but increases in the presence of inhomogeneities.

The dynamics of a beam of hot electrons traveling through a cold plasma and the generation of Langmuir waves are investigated in the presence of a nonthermal tail of electrons in the background distribution function. Using quasilinear simulations, it is shown that in the presence of the nonthermal electrons, the relaxation of the beam distribution function in velocity space is retarded and the Langmuir waves are strongly damped at low velocities. The average velocity of beam propagation is almost constant but its magnitude is larger in the presence of nonthermal electrons than their absence. It is found that the self-similarity of the system is preserved in the presence of nonthermal electrons. The effects of nonthermal electrons on the evolution of gas-dynamical parameters of the beam, including the height of plateau in the beam distribution function, its upper and lower velocity boundaries, and beam velocity width, are also studied. It is found that initially the values of the upper and lower velocity boundaries are almost unaltered, but at large times the lower (upper) boundary velocity is larger (smaller) in the presence of nonthermal electrons than without the nonthermal electrons.

Gas-dynamic theory is generalized to incorporate the effects of beam-driven Langmuir waves scattering off ambient density fluctuations, and the consequent effects on the propagation of a cloud of hot electrons in an inhomogeneous plasma. Assuming Langmuir scattering as the limit of nonlinear three-wave interactions with fluctuations that are weak, low-frequency, long-wavelength ion-sound waves, the net effect of scattering is equivalent to effective damping of the Langmuir waves. Under the assumption of self-similarity in the evolution of the beam and Langmuir wave distribution functions, gas-dynamic theory shows that the effects of Langmuir scattering on the beam distribution are equivalent to a perturbation in the injection profile of the beam. Analytical expressions are obtained for the height of the plateau of the beam distribution function, wave spectral number density, total wave and particle energy density, and the beam number density. The main results of gas-dynamic theory are then compared with simulation results from numerical solutions of quasilinear equations. The relaxation of the beam in velocity space is retarded in the presence of density fluctuations and the magnitude of the upper velocity boundary is less than that in the absence of fluctuations. There are four different regimes for the height of the plateau, corresponding to different stages of relaxation of the beam in velocity space. Moreover, Langmuir scattering results in transfer of electrons from moderate velocity to low velocity; this effect produces an enhancement in the beam number density at small distances near the injection site and a corresponding decrease at large distances. There are sharp decreases in the profiles of the beam and total wave energy densities, which are related to dissipation of energy at large phase velocities. Due to a slower velocity space diffusion of the beam distribution in the presence of scattering effects, the spatial width of the beam is reduced while its mean velocity of propagation increases slightly.