Limit cycles for two classes of control piecewise linear differential systems

TL;DR

This paper investigates the emergence of limit cycles from linear centers in high-dimensional piecewise linear control systems, analyzing bifurcations under small perturbations with both continuous and discontinuous nonlinearities.

Contribution

It introduces a framework for studying bifurcations of limit cycles in high-dimensional piecewise linear systems with control features, extending classical results to more complex control-theoretic models.

Findings

Identifies conditions for bifurcation of limit cycles in piecewise linear systems.

Provides analytical methods for detecting limit cycles in high-dimensional control systems.

Extends bifurcation theory to systems with discontinuous nonlinearities.

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 $\dot{x} = A_0 x + \varepsilon \big(A x + \phi(x_1) b\big)$, 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 $\mathbb{R}^n$.