An analytical method for Mathieu oscillator based on method of variation of parameter

Periodic solutions of a generalized Van der Pol-Mathieu differential equation

Abstract This paper addresses the steady-state periodic and quasi-periodic responses of van der Pol–Mathieu system subject to three excitations (i.e., self, parametric and external excitations). Method of multiple scales and double perturbation technique are employed to study the original system. The cases of van der Pol–Mathieu oscillator with and without external excitation are considered, and periodic and quasi-periodic solutions are obtained and discussed. In the parametric study, the effects of various parameters and self, parametric and external excitations on the system behaviors are studied. Results from method of multiple scales well agree with those from numerical method.

Periodic and Quasi-periodic Responses of Van der Pol–Mathieu System Subject to Various Excitations

The Lienard–Mathieu equation with a small parameter is
examined. The existence of periodic and quasiperiodic solutions
is proved using the averaging method, the method of
complexification and the phase space analysis of an associated
autonomous equation. The results extend and generalize those of
Kalas and Kadeřabek (2014).

Periodic solutions of Liénard–Mathieu differential equation with a small parameter

The modified generalized Van der Pol-Mathieu equation is generalization of the equation that is investigated by authors Momeni et al. (2007), Veerman and Verhulst (2009) and Kadeřabek (2012). In this article the Bautin bifurcation of the autonomous system associated with the modified generalized Van der Pol-Mathieu equation has been proved. The existence of limit cycles is studied and the Lyapunov quantities of the autonomous system associated with the modified Van der Pol-Mathieu equation are computed.

Bautin bifurgation of a modified generalized Van der Pol-Mathieu equation

This work is devoted to the autonomous system derived from Van
der Pol-Mathieu equation which was applied to the study the
dynamics of dusty plasmas. In this work we shall investigate
mathematically this autonomous system and shall find how large
region of the plane will be attracted to the equilibrium point.

Zdeněk Kadeřábek

Papers

The autonomous system derived from Van der Pol-Mathieu equation