Limit cycles of a class of generalized Mathieu differential equations

Abstract

Numerous branches of science and engineering, including the design of electronic circuits and the analysis of systems with periodic forcing, use the study of limit cycles in the Mathieu equation to solve problems that have significant effects on the stability of periodic systems. In this paper, using the averaging theory of first order, we provide upper bounds for the maximum number of limit cycles bifurcating from the periodic orbits adapted to the Hamiltonian system  x=y^{2p-1}, y=-x^{2q-1} with p,q are positive integers, when perturbed in the particular class of the generalized Mathieu differential equations.