Imperial College London
Alternate

ProfessorMauricioBarahona

Faculty of Natural Sciences, Department of Mathematics

Director of Research, Chair in Biomathematics
 
 
 
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Contact

 

m.barahona Website

 
 
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Location

 

6M31Huxley BuildingSouth Kensington Campus

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Summary

 

Publications

Citation

BibTex format

@article{August:2022:10.3934/jcd.2022017,
author = {August, E and Barahona, M},
doi = {10.3934/jcd.2022017},
journal = {Journal of Computational Dynamics},
pages = {105--126},
title = {Finding positively invariant sets and proving exponential stability of limit cycles using sum-of-squares decompositions},
url = {http://dx.doi.org/10.3934/jcd.2022017},
volume = {10},
year = {2022}
}
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RIS format (EndNote, RefMan)

TY  - JOUR
AB  - The dynamics of many systems from physics, economics, chemistry, and biology can be modelled through polynomial functions. In this paper, we provide a computational means to find positively invariant sets of polynomial dynamical systems by using semidefinite programming to solve sum-of-squares (SOS) programmes. With the emergence of SOS programmes, it is possible to efficiently search for Lyapunov functions that guarantee stability of polynomial systems. Yet, SOS computations often fail to find functions, such that the conditions hold in the entire state space. We show here that restricting the SOS optimisation to specific domains enables us to obtain positively invariant sets, thus facilitating the analysis of the dynamics by considering separately eachpositively invariant set. In addition, we go beyond classical Lyapunov stability analysis and use SOS decompositions to computationally implement sufficient positivity conditions that guarantee existence, uniqueness, and exponential stability of a limit cycle. Importantly, this approach is applicable to systems of any dimension and, thus, goes beyond classical methods that are restricted to two dimensional phase space. We illustrate our different results with applications to classical systems, such as the van der Pol oscillator, the Fitzhugh-Nagumo neuronal equation, and the Lorenz system.
AU  - August,E
AU  - Barahona,M
DO  - 10.3934/jcd.2022017
EP  - 126
PY  - 2022///
SN  - 2158-2505
SP  - 105
TI  - Finding positively invariant sets and proving exponential stability of limit cycles using sum-of-squares decompositions
T2  - Journal of Computational Dynamics
UR  - http://dx.doi.org/10.3934/jcd.2022017
UR  - https://www.aimsciences.org/article/doi/10.3934/jcd.2022017
UR  - http://hdl.handle.net/10044/1/98529
VL  - 10
ER  -
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