Finding positively invariant sets and proving exponential stability of limit cycles using sum-of-squares decompositions
File(s)
Accepted version
Author(s)
August, Elias
Barahona, Mauricio
Type
Journal Article
Abstract
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 each
positively 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.
positively 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.
Date Issued
2022-12
Date Acceptance
2022-07-23
Citation
Journal of Computational Dynamics, 2022, 10, pp.105-126
ISSN
2158-2505
Publisher
American Institute of Mathematical Sciences
Start Page
105
End Page
126
Journal / Book Title
Journal of Computational Dynamics
Volume
10
Identifier
https://www.aimsciences.org/article/doi/10.3934/jcd.2022017
Publication Status
Published
Date Publish Online
2022-08