A Poincaré–Bendixson theorem for hybrid dynamical systems on directed graphs

  • William ClarkEmail author
  • Anthony Bloch
Original Article


The purpose of this work is to obtain restrictions on the asymptotic structure of two-dimensional hybrid dynamical systems. Previous results have been achieved by the authors concerning hybrid dynamical systems with a single impact surface and a single state space. Here, this work is extended to hybrid dynamical systems defined on a directed graph; each vertex corresponds to a state space and each directed edge corresponds to an impact.


Poincaré–Bendixson theorem Hybrid systems Periodic orbits Directed graphs 

Mathematics Subject Classification

37E25 37C27 34A38 37C10 34A34 



  1. 1.
    Barreiro A, Baños A, Dormido S, González-Prieto JA (2014) Reset control systems with reset band: well-posedness, limit cycles and stability analysis. Syst Control Lett 63:1–11., MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bendixson I (1901) Sur les courbes definies par des equations differentielles. Acta Mathematica 21:1–88MathSciNetCrossRefGoogle Scholar
  3. 3.
    Chartrand G, Lesniak L, Zhang P (2010) Graphs & digraphs, 5th edn. A Chapman & Hall book, LondonzbMATHGoogle Scholar
  4. 4.
    Clark W, Bloch A, Colombo L (2019) A Poincaré–Bendixson theorem for hybrid systems. Math Control Relat Fields. CrossRefGoogle Scholar
  5. 5.
    Crampin M, Heal B (1994) On the chaotic behaviour of the tent map. Teach Math Appl Int J IMA 13(2):83–89Google Scholar
  6. 6.
    Dulac H (1923) Sur les cycles limites. Bulletin de la Société Mathématique de France 51:45–188MathSciNetCrossRefGoogle Scholar
  7. 7.
    Efimov D, Perruquetti W, Shiriaev A (2014) On existence of oscillations in hybrid systems. Nonlinear Anal Hybrid Syst 12:104–116., MathSciNetzbMATHGoogle Scholar
  8. 8.
    Goebel R, Sanfelice RG, Teel AR (2012) Hybrid dynamical systems. Princeton University Press, PrincetonCrossRefGoogle Scholar
  9. 9.
    Lamperski A, Ames AD (2007) Lyapunov-like conditions for the existence of Zeno behavior in hybrid and Lagrangian hybrid systems. In: 2007 46th IEEE conference on decision and control, IEEE, pp 115–120Google Scholar
  10. 10.
    Lou X, Li Y, Sanfelice RG (2017) Existence of hybrid limit cycles and Zhukovskii stability in hybrid systems. In: 2017 American control conference (ACC), pp 1187–1192,
  11. 11.
    Morris B, Grizzle JW (2009) Hybrid invariant manifolds in systems with impulse effects with application to periodic locomotion in bipedal robots. IEEE Trans Autom Control 54(8):1751–1764. MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Perko L (1991) Differential equations and dynamical systems. Texts in applied mathematics, Springer. CrossRefGoogle Scholar
  13. 13.
    Poincaré H (1886) Sur les courbes definies par les equations differentielles. J Math Pures Appl 2:151–217zbMATHGoogle Scholar
  14. 14.
    Rudin W (1976) Principles of mathematical analysis. International series in pure and applied mathematics, McGraw-Hill, New York.
  15. 15.
    Saglam CO, Teel AR, Byl K (2014) Lyapunov-based versus Poincaré map analysis of the rimless wheel. In: 2014 IEEE 53rd annual conference on decision and control (CDC), IEEE, pp 1514–1520Google Scholar
  16. 16.
    Sanfelice RG, Goebel R, Teel AR (2007) Invariance principles for hybrid systems with connections to detectability and asymptotic stability. IEEE Trans Autom Control 52(12):2282–2297. MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Simic SN, Sastry S, Johansson KH, Lygeros J (2002) Hybrid limit cycles and hybrid Poincaré-Bendixson. IFAC Proc Vol 35(1):197–202.,, 15th IFAC World CongressCrossRefGoogle Scholar
  18. 18.
    Strogatz S (2014) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering. Studies in Nonlinearity, Avalon Publishing.
  19. 19.
    Tang JZ, Manchester IR (2014) Transverse contraction criteria for stability of nonlinear hybrid limit cycles. In: 53rd IEEE Conference on decision and control, pp 31–36.
  20. 20.
    Tu L (2010) An introduction to manifolds. Universitext. Springer, New YorkGoogle Scholar
  21. 21.
    Zhang Z (1981) On the existence of exactly two limit cycles for the Liénard equation. Acta Math Sinica 24:710–716MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MichiganAnn ArborUSA

Personalised recommendations