Bifurcation analysis of Van der Pol–Mathieu equation for the dust grain charge with regularized (κ)-distributed electron-ion.

This paper presents a novel viewpoint on nonlinear dust-acoustic waves (DAWs) in an unmagnetized collisionless plasma containing regularized κ distributed electrons and ions (ei) and negative dust grain. The nonlinear oscillatory system based on hybridization of the Van der Pol–Mathieu equation (VdPME) is derived by a new technique. By bifurcation analysis of the planar dynamical system (DS), the effects of parameters with the assistance of phase planes and time series of VdPME are studied. After analyzing the equation to identify the resonance region, a fourth-order Runge–Kutta method is used to solve it numerically. We explained the behavior of DA periodic, stable limit cycle, and chaotic limit cycle wave solutions with different parameters. These types of numerical solutions are illustrated in two-dimensional and three-dimensional graphics by changing the rate at which charged dust grain is produced α, as well as waste β and comparing the results with those of earlier research. A novel bifurcation analysis of VdPE and VdPME is obtained with the effects of the cut-off factor δ of regularized κ-distribution (RKD) distributed ei, the superthermality of ei particles κe,i, the ratio of ion to electron temperature σ, and the ratio of dust to electron density ρ illustrated.  It is noticed that the DAW shows promotion in width and amplitude as the frequency γ increases and it becomes rarefaction as the cut-off parameter δ increases. On the contrary, it becomes compressive under the impact of superthermality κe and κi. The obtained conclusions may help explain and comprehend a variety of applications in experimental plasma containing highly energy regularized κ dispersed ei, as well as in the interstellar medium. [ABSTRACT FROM AUTHOR]