Variational equation for discontinuous delayed systems

Abstract

For discontinuous delay differential equations, we derive and analyze the variational equation (also known as the linearization), which describes the evolution of infinitesimal perturbations to initial conditions. This variational equation incorporates delta functions that account for jumps in the right-hand side of the original equation. We establish fundamental properties of the solutions of this equation and explore its applications, which include the generalization of the theory and computational methods of Lyapunov exponents for discontinuous delayed systems, providing a powerful tool for studying stability and chaos in such systems.