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 ${\dot{x}} = A_0 x + \varepsilon (A x + \phi (x_1) b)$, where $\phi$ is a continuous or discontinuous piecewise linear function, $A_0$ is a $2n\times 2n$ matrix with only purely imaginary eigenvalues, $\varepsilon$ is a small parameter, A is an arbitrary $2n\times 2n$ matrix, and b is an arbitrary vector of ${\mathbb {R}}^n$.