Lyapunov exponents, sensitivity, and stability for non-autonomous discrete systems

TL;DR

This paper explores how Lyapunov exponents relate to sensitivity and stability in non-autonomous discrete systems, introducing new concepts and generalizing existing results from autonomous systems.

Contribution

It introduces new concepts for non-autonomous systems and extends existing autonomous system results to this broader context, weakening previous conditions.

Findings

Positive Lyapunov exponent implies strong sensitivity.

Negative Lyapunov exponent implies exponential asymptotic stability.

Results are illustrated with an example.

Abstract

This paper is concerned with relationships of Lyapunov exponents with sensitivity and stability for non-autonomous discrete systems. Some new concepts are introduced for non-autonomous discrete systems, including Lyapunov exponents, strong sensitivity at a point and in a set, Lyapunov stability, and exponential asymptotical stability. It is shown that the positive Lyapunov exponent at a point implies strong sensitivity for a class of non-autonomous discrete systems. Furthermore, the uniformly positive Lyapunov exponents in a totally invariant set imply strong sensitivity in this set under certain conditions. It is also shown that the negative Lyapunov exponent at a point implies exponential asymptotical stability for a class of non-autonomous discrete systems. The related existing results for autonomous discrete systems are generalized to non-autonomous discrete systems and their conditions are weaken. One example is provided for illustration.