Finding positively invariant sets and proving exponential stability of limit cycles using Sum-of-Squares decompositions

Elias August; Mauricio Barahona

doi:10.3934/jcd.2022017

Article Contents

2023, Volume 10, Issue 1: 105-126. Doi: 10.3934/jcd.2022017

This issue Previous Article A novel moving orthonormal coordinate-based approach for region of attraction analysis of limit cycles Next Article Modified refinement algorithm to construct Lyapunov functions using meshless collocation

Finding positively invariant sets and proving exponential stability of limit cycles using Sum-of-Squares decompositions

Elias August^1, , and
Mauricio Barahona^2,

1.
Department of Engineering, Reykjavik University, Menntavegur 1,102 Reykjavik, Iceland
2.
Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, United Kingdom

^* Corresponding author: Elias August, eliasaugust@ru.is
^* Corresponding author: Elias August, eliasaugust@ru.is

Received: October 2021

Revised: July 2022

Early access: August 2022

Published: December 2022

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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.

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Mathematics Subject Classification: 93D05, 90-08, 34A34, 90C22.

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Figure 1. Finding positively invariant sets

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Figure 2. The Lorenz attractor of the system (13) and the classic absorbing set given by (15)

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Figure 3. The two positively invariant sets of the Lorenz system

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Figure 4. Positively invariant sets of the FitzHugh-Nagumo model (20) with $ I = 0 $

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Figure 5. Change of the positively invariant sets of the FitzHugh-Nagumo model (20) with $ I $

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Figure 6. Positively invariant set and trajectories of the van der Pol oscillator

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Figure 7. Exponential stability of the limit cycle of the van der Pol oscillator

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Figure 8. Exponential stability of the limit cycle of the system (38)

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Figure 9. Exponentially stable limit cycle of the system (41)

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References

[1]	A. A. Ahmadi and B. E. Khadir, A globally asymptotically stable polynomial vector field with rational coefficients and no local polynomial Lyapunov function, Systems Control Lett., 121 (2018), 50-53. doi: 10.1016/j.sysconle.2018.07.013.
[2]	E. August, Network Analysis of Complex Biological systems: Boundedness of Weakly Reversible Chemical Reaction Networks and Conditions for Synchronisation of Coupled Oscillators, Ph.D thesis, Imperial College London (University of London), 2007.
[3]	E. August and M. Barahona, Obtaining certificates for complete synchronisation of coupled oscillators, Physica D: Nonlinear Phenomena, 240 (2011), 795-803.
[4]	E. M. Aylward, P. A. Parrilo and J.-J. E. Slotine, Stability and robustness analysis of nonlinear systems via contraction metrics and SOS programming, Automatica, 44 (2008), 2163-2170. doi: 10.1016/j.automatica.2007.12.012.
[5]	V. N. Belykh, I. V. Belykh and M. Hasler, Connection graph stability method for synchronized coupled chaotic systems, Phy. D, 195 (2004), 159-187. doi: 10.1016/j.physd.2004.03.012.
[6]	G. Borg, A condition for the existence of orbitally stable solutions of dynamical systems, Kungl. Tekn. Högsk. Handl. Stockholm, 153 (1960).
[7]	S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, Cambridge, UK, 2004. doi: 10.1017/CBO9780511804441.
[8]	I. D. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, ACTA Scientific Publishing House, Kharkiv, Ukraine, 2002.
[9]	G. Craciun, Y. Tang and M. Feinberg, Understanding bistability in complex enzyme-driven reaction networks, PNAS, 103 (2006), 8697-8702.
[10]	R. A. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., 1 (1961), 445-466.
[11]	P. Giesl, Unbounded basins of attraction of limit cycles, Acta Math. Univ. Comenianae, 72 (2003), 81-110.
[12]	P. Giesl, Necessary conditions for a limit cycle and its basin of attraction, Nonlinear Anal., 56 (2004), 643-677. doi: 10.1016/j.na.2003.07.020.
[13]	J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, USA, 1983. doi: 10.1007/978-1-4612-1140-2.
[14]	P. Hartman and C. Olech, On global asymptotic stability of solutions of differential equations, Trans. Amer. Math. Soc., 104 (1962), 154-178. doi: 10.2307/1993939.
[15]	H. K. Khalil, Nonlinear Systems, 3$^{rd}$ edition, Prentice-Hall, Upper Saddle River, New Jersey, 2000.
[16]	C. Koch, Biophysics of Computation: Information Processing in Single Neurons, Oxford University Press, New York, USA, 1999.
[17]	D. C. Lewis, Metric properties of differential equations, Amer. J. Math., 71 (1949), 294-312. doi: 10.2307/2372245.
[18]	E. N. Lorenz, Deterministic nonperiodic flow, J. Atmospheric Sci., 20 (1963), 130-141. doi: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2.
[19]	I. R. Manchester and J.-J. E. Slotine, Transverse contraction criteria for existence, stability, and robustness of a limit cycle, Systems Control Lett., 63 (2014), 32-38. doi: 10.1016/j.sysconle.2013.10.005.
[20]	F. Meng, D. Wang, P. Yang, G. Xie and F. Guo, Application of sum-of-squares method in estimation of region of attraction for nonlinear polynomial systems, IEEE Access, 8 (2020), 14234-14243.
[21]	J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-662-08539-4.
[22]	K. G. Murty and S. N. Kabadi, Some NP-complete problems in quadratic and nonlinear programming, Math. Program., 39 (1987), 117-129. doi: 10.1007/BF02592948.
[23]	A. Papachristodoulou, Scalable Analysis of Nonlinear Systems Using Convex Optimization, Ph.D thesis, California Institute of Technology, Pasadena, California, 2005.
[24]	A. Papachristodoulou, J. Anderson, G. Valmorbida, S. Prajna, P. Seiler and P. A. Parrilo, SOSTOOLS: Sum of squares optimization toolbox for MATLAB, 2013. Available from: http://www.eng.ox.ac.uk/control/sostools, http://www.cds.caltech.edu/sostools and http://www.mit.edu/parrilo/sostools.
[25]	P. A. Parrilo, Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization, Ph.D thesis, California Institute of Technology, Pasadena, California, 2000.
[26]	P. A. Parrilo, Semidefinite programming relaxations for semialgebraic problems, Math. Program., Ser. B, 96 (2003), 293-320. doi: 10.1007/s10107-003-0387-5.
[27]	S. Prajna, A. Papachristodoulou, P. Seiler and P. Parrilo, SOSTOOLS and its control applications, In Positive Polynomials in Control, (eds. D. Henrion and G. Andrea), Springer Berlin Heidelberg, 312 (2005), 273-292. doi: 10.1007/10997703_14.
[28]	A. M. dos Santos, S. R. Lopes and R. L. Viana, Rhythm synchronization and chaotic modulation of coupled Van der Pol oscillators in a model for the heartbeat, Physica A, 338 (2004), 335-355. doi: 10.1016/j.physa.2004.02.058.
[29]	C. Schlosser and M. Korda, Converging outer approximations to global attractors using semidefinite programming, Automatica, 134 (2021), 109900. doi: 10.1016/j.automatica.2021.109900.
[30]	B. Stenström, Dynamical systems with a certain local contraction property, Math. Scand., 11 (1962), 151-155. doi: 10.7146/math.scand.a-10661.
[31]	J. F. Sturm, Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones, Optim. Methods Softw., 11/12 (1999), 625-653. doi: 10.1080/10556789908805766.
[32]	L. Vandenberghe and S. Boyd, Semidefinite programming, SIAM Rev., 38 (1996), 49-95. doi: 10.1137/1038003.
[33]	H. R. Wilson, Simplified dynamics of human and mammalian neocortical neurons, J. theor. Biol., 200 (1999), 375-388.