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

Hua Shao; Yuming Shi; Hao Zhu

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arXiv · 1603.05457

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

Hua Shao·Yuming Shi·Hao Zhu

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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.

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Hua Shao, Yuming Shi, Hao Zhu. 2016-03-17. Lyapunov exponents, sensitivity, and stability for non-autonomous discrete systems. https://arxiv.org/abs/1603.05457

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