Efficient computation of Lyapunov functions for nonlinear systems by integrating numerical solutions

  • Sigurdur Freyr HafsteinEmail author
  • Asgeir Valfells
Original Paper


A strict Lyapunov function for an equilibrium of a dynamical system asserts its asymptotic stability and gives a lower bound on its basin of attraction. For nonlinear systems, the explicit construction of a Lyapunov function taking the nonlinear dynamics into account remains a difficult problem and one often resorts to numerical methods. We improve and analyse a method that is based on a converse theorem in the Lyapunov stability theory and compare it to different methods in the literature. Our method is of low complexity, and its workload is perfectly parallel. Further, its free parameters allow it to be adapted to the problem at hand and we show that our method matches or gives a larger lower bound on the equilibrium’s basin of attraction than other approaches in the literature in most examples. Finally, we apply our method to a model of a genetic toggle switch in Escherichia coli and we demonstrate that our novel method delivers important information on the model’s dynamics for different parameters.


Nonlinear system Lyapunov function Basin of attraction Numerical method 



This work was supported by the Icelandic Research Fund in the project Algorithms to compute Lyapunov functions (No. 130677-051). Additionally, the authors would like to thank the anonymous reviewers, whose suggestions improved this manuscript considerably.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Björnsson, J., Giesl, P., Hafstein, S., Kellett, C., Li, H.: Computation of continuous and piecewise affine Lyapunov functions by numerical approximations of the Massera construction. In: Proceedings of the CDC, 53rd IEEE Conference on Decision and Control, pp. 5506–5511. Los Angeles (2014)Google Scholar
  2. 2.
    Björnsson, J., Hafstein, S.: Efficient Lyapunov function computation for systems with multiple exponentially stable equilibria. Procedia Comput. Sci. 108, 655–664 (2017). Proceedings of the International Conference on Computational Science (ICCS), Zurich, Switzerland, 2017Google Scholar
  3. 3.
    Chesi, G.: Estimating the domain of attraction for non-polynomial systems via LMI optimizations. Automatica 45, 1536–1541 (2009)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Chesi, G.: Domain of Attraction: Analysis and Control via SOS Programming. Springer, Berlin (2011)CrossRefGoogle Scholar
  5. 5.
    Doban, A.: Stability domains computation and stabilization of nonlinear systems: implications for biological systems. PhD thesis: Eindhoven University of Technology (2016)Google Scholar
  6. 6.
    Doban, A., Lazar, M.: Computation of Lyapunov functions for nonlinear differential equations via a Yoshizawa-type construction. IFAC-PapersOnLine 49(18), 29–34 (2016)CrossRefGoogle Scholar
  7. 7.
    Doban, A., Lazar, M.: Computation of Lyapunov functions for nonlinear differential equations via a Massera-type construction. IEEE Trans. Autom. Control 63(5), 1259–1272 (2018)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Gardner, T., Cantor, C., Collins, J.: Construction of a genetic toggle switch in Escherichia coli. Nature 403(6767), 339–342 (2000)CrossRefGoogle Scholar
  9. 9.
    Genesio, R., Tartaglia, M., Vicino, A.: On the estimation of asymptotic stability regions: state of the art and new proposals. IEEE Trans. Autom. Control 30(8), 747–755 (1985)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Giesl, P., Hafstein, S.: Revised CPA method to compute Lyapunov functions for nonlinear systems. J. Math. Anal. Appl. 410, 292–306 (2014)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Giesl, P., Hafstein, S.: Computation and verification of Lyapunov functions. SIAM J. Appl. Dyn. Syst. 14(4), 1663–1698 (2015)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Giesl, P., Hafstein, S.: Review of computational methods for Lyapunov functions. Discrete Contin. Dyn. Syst. Ser. B 20(8), 2291–2331 (2015)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hachicho, O., Tibken, B.: Estimating domains of attraction of a class of nonlinear dynamical systems with LMI methods based on the theory of moments. In: Proceedings of the 41th IEEE Conference on Decision and Control (CDC), pp. 3150–3155. Los Angeles (2002)Google Scholar
  14. 14.
    Hafstein, S.: A constructive converse Lyapunov theorem on exponential stability. Discrete Contin. Dyn. Syst. 10(3), 657–678 (2004)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Hafstein, S., Kellett, C., Li, H.: Computing continuous and piecewise affine Lyapunov functions for nonlinear systems. J. Comput. Dyn. 2(2), 227–246 (2015)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Hafstein, S., Valfells, A.: Study of dynamical systems by fast numerical computation of Lyapunov functions. In: Proceedings of the 14th International Conference on Dynamical Systems: Theory and Applications (DSTA), Mathematical and Numerical Aspects of Dynamical System Analysis, pp. 220–240 (2017)Google Scholar
  17. 17.
    Kellett, C.: Converse theorems in Lyapunov’s second method. Discrete Contin. Dyn. Syst. Ser. B 20(8), 2333–2360 (2015)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Khalil, H.: Nonlinear Systems, 3rd edn. Prentice-Hall, New Jersey (2002)zbMATHGoogle Scholar
  19. 19.
    Lugagne, J., Carrillo, S., Kirch, M., Köhler, A., Batt, G., Hersen, P.: Balancing a genetic toggle switch by real-time feedback control and periodic forcing. Nat. Commun. 8, 1671 (2017)CrossRefGoogle Scholar
  20. 20.
    Massera, J.: Contributions to stability theory. Ann. Math. 64, 182–206 (1956). (Erratum. Annals of Mathematics, 68:202, 1958)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Matallana, L., Blanco, A., Bandoni, J.: Estimation of domains of attraction: a global optimization approach. Math. Comput. Model. 52(3–4), 574–585 (2010)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Sastry, S.: Nonlinear Systems: Analysis, Stability, and Control. Springer, Berlin (1999)CrossRefGoogle Scholar
  23. 23.
    Vannelli, A., Vidyasagar, M.: Maximal Lyapunov functions and domains of attraction for autonomous nonlinear systems. Automatica 21(1), 69–80 (1985)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Vidyasagar, M.: Nonlinear System Analysis, Classics in Applied Mathematics, 2nd edn. SIAM, Philadelphia (2002)CrossRefGoogle Scholar
  25. 25.
    Wang, W., Ruan, S.: Bifurcations in an epidemic model with constant removal rate of infectives. J. Math. Anal. Appl. 291(1), 775–793 (2004)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Yoshizawa, T.: Stability Theory by Liapunov’s Second Method. Publications of the Mathematical Society of Japan, No. 9. The Mathematical Society of Japan, Tokyo (1966)Google Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Faculty of Physical SciencesUniversity of IcelandReykjavíkIceland

Personalised recommendations