Computational approach for complete Lyapunov functions

Argáez, Carlos, Giesl, Peter and Hafstein, Sigurdur Freyr (2018) Computational approach for complete Lyapunov functions. In: Dynamical Systems in Theoretical Perspective. Springer International Publishing, pp. 1-11. ISBN 9783319965970

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Abstract

Ordinary differential equations arise in a variety of applications, including climate modeling, electronics, predator-prey modeling, etc., and they can exhibit highly complicated dynamical behaviour. Complete Lyapunov functions capture this behaviour by dividing the phase space into two disjoint sets: the chain-recurrent part and the transient part. If a complete Lyapunov function is known for a dynamical system the qualitative behaviour of the system’s solutions is transparent to a large degree. The computation of a complete Lyapunov function for a given system is, however, a very hard task. We present significant improvements of an algorithm recently suggested by the authors to compute complete Lyapunov functions. Previously this methodology was incapable to fully detect chain-recurrent sets in dynamical systems with high differences in speed. In the new approach we replace the system under consideration with another one having the same solution trajectories but such that they are traversed at a more uniform speed. The qualitative properties of the new system such as attractors and repellers are the same as for the original one. This approach gives a better approximation to the chain-recurrent set of the system under study.

Item Type: Book Section
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
SWORD Depositor: Mx Elements Account
Depositing User: Mx Elements Account
Date Deposited: 24 Jun 2022 10:28
Last Modified: 24 Jun 2022 10:28
URI: http://sro.sussex.ac.uk/id/eprint/106590

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