Limit cycles for two classes of control piecewise linear differential systems

Abstract

We study the bifurcation of limit cycles from the periodic orbits of $2n$--dimensional linear centers $\dot{x}=A_{0}x$ when they are perturbed inside classes of continuous and discontinuous piecewise linear differential systems of control theory of the form $\dot{x}=A_{0}x+\epsilon (Ax+\varphi (x_1)b)$, where $\varphi$ is a continuous or discontinuous piecewise linear function, $A_0$ is a $2n\times 2n$ matrix with only purely imaginary eigenvalues, $\epsilon$ is a small parameter, $A$ is an arbitrary $2n\times 2n$ matrix, and $b$ is an arbitrary vector of ${\mathbb{R}}^n$.