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Complete Lyapunov Functions: Computation and Applications

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Simulation and Modeling Methodologies, Technologies and Applications (SIMULTECH 2017)

Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 873))

Abstract

Many phenomena in disciplines such as engineering, physics and biology can be represented as dynamical systems given by ordinary differential equations (ODEs). For their analysis as well as for modelling purposes it is desirable to obtain a complete description of a dynamical system. Complete Lyapunov functions, or quasi-potentials, describe the dynamical behaviour without solving the ODE for many initial conditions. In this paper, we use mesh-free numerical approximation to compute a complete Lyapunov function and to determine the chain-recurrent set, containing the attractors and repellers of the system. We use a homogeneous evaluation grid for the iterative construction, and thus improve a previous method. Finally, we apply our methodology to several examples, including one to compute an epigenetic landscape, modelling a bistable network of two genes. This illustrates the capability of our method to solve interdisciplinary problems.

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Acknowledgements

First author in this paper is supported by the Icelandic Research Fund (Rannís) grant number 163074-052, Complete Lyapunov functions: Efficient numerical computation. Special thanks to Dr. Jean-Claude Berthet for all his good comments and advices on C++.

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Authors and Affiliations

  1. Science Institute, University of Iceland, 107, Reykjavík, Iceland

    Carlos Argáez & Sigurdur Hafstein

  2. Department of Mathematics, University of Sussex, Falmer, BN1 9QH, UK

    Peter Giesl

Authors
  1. Carlos Argáez
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  2. Peter Giesl
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  3. Sigurdur Hafstein
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Corresponding author

Correspondence to Carlos Argáez .

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Editors and Affiliations

  1. King Abdullah II School of Information Technology, The University of Jordan, Amman, Jordan

    Mohammad S. Obaidat

  2. School of Electrical Engineering and Computer Science, University of Ottawa, Ottawa, ON, Canada

    Tuncer Ören

  3. Department of Informatics, Modeling, Electronics and System Engineering, University of Calabria, Rende, Italy

    Floriano De Rango

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Argáez, C., Giesl, P., Hafstein, S. (2019). Complete Lyapunov Functions: Computation and Applications. In: Obaidat, M., Ören, T., Rango, F. (eds) Simulation and Modeling Methodologies, Technologies and Applications . SIMULTECH 2017. Advances in Intelligent Systems and Computing, vol 873. Springer, Cham. https://doi.org/10.1007/978-3-030-01470-4_11

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