Bridging the Gap between Reactivity, Contraction and Finite-Time Lyapunov Exponents

This paper investigates the concept of reactivity for a nonlinear discrete-time system and generalizes this concept to the case of the p-iteration system, p > 1. We introduce a definition of reactivity for nonlinear discrete-time systems based on a general weighted norm. Stability conditions of the first iteration system based on the reactivity of p-iteration system, p > 1 are provided for both linear-time varying systems and nonlinear maps. We provide examples of stability analysis of linear time-varying, and synchronization of chaotic maps using reactivity. We discuss the connection between reactivity and well-established stability criteria, contraction, and finite-time Lyapunov exponents. For the case of a linearized system about a given trajectory and the study of the local stability of the synchronous solution for networks of coupled maps, the reactivity of the p-iteration system coincides with the finite-time Lyapunov exponent, with this time being equal to p. In the limit of infinite p, the reactivity becomes the maximum Lyapunov exponent, which allows us to bridge the gap between the master stability function approach and the contraction theory approach to study the stability of the synchronous solution for networks.