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.

翻译：本文研究了非线性离散时间系统的反应性概念，并将该概念推广至p次迭代系统（p > 1）的情形。我们基于一般加权范数提出了非线性离散时间系统反应性的定义。针对线性时变系统和非线性映射，给出了基于p次迭代系统（p > 1）反应性的首次迭代系统稳定性条件。我们通过反应性概念展示了线性时变系统稳定性分析及混沌映射同步的实例。文中探讨了反应性与经典稳定性判据、收缩性及有限时间李雅普诺夫指数之间的关联。对于沿给定轨迹线性化的系统及耦合映射网络同步解的局部稳定性研究，p次迭代系统的反应性与有限时间李雅普诺夫指数相吻合，此时时间参数等于p。当p趋于无穷时，反应性转化为最大李雅普诺夫指数，这使我们能够弥合主稳定函数方法与收缩理论方法在研究网络同步解稳定性时的理论间隙。