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Efficient computation of Lyapunov functions for nonlinear systems by integrating numerical solutions

  • Sigurdur Freyr Hafstein
  • Asgeir Valfells
Original Paper
  • 53 Downloads

Abstract

A strict Lyapunov function for an equilibrium of a dynamical system asserts its asymptotic stability and gives a lower bound on its basin of attraction. For nonlinear systems, the explicit construction of a Lyapunov function taking the nonlinear dynamics into account remains a difficult problem and one often resorts to numerical methods. We improve and analyse a method that is based on a converse theorem in the Lyapunov stability theory and compare it to different methods in the literature. Our method is of low complexity, and its workload is perfectly parallel. Further, its free parameters allow it to be adapted to the problem at hand and we show that our method matches or gives a larger lower bound on the equilibrium’s basin of attraction than other approaches in the literature in most examples. Finally, we apply our method to a model of a genetic toggle switch in Escherichia coli and we demonstrate that our novel method delivers important information on the model’s dynamics for different parameters.

Keywords

Nonlinear system Lyapunov function Basin of attraction Numerical method 

Notes

Acknowledgements

This work was supported by the Icelandic Research Fund in the project Algorithms to compute Lyapunov functions (No. 130677-051). Additionally, the authors would like to thank the anonymous reviewers, whose suggestions improved this manuscript considerably.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

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Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Faculty of Physical SciencesUniversity of IcelandReykjavíkIceland

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