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Complete Lyapunov Functions: Computation and Applications

Conference paper
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Part of the Advances in Intelligent Systems and Computing book series (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.

Keywords

Dynamical system Complete Lyapunov Function Quasi-potential Mesh-free collocation Radial basis functions 

Notes

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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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Science InstituteUniversity of IcelandReykjavíkIceland
  2. 2.Department of MathematicsUniversity of SussexFalmerUK

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