Journal of Dynamics and Differential Equations

, Volume 31, Issue 4, pp 1873–1920 | Cite as

Stability of Periodic Solutions for Hysteresis-Delay Differential Equations

  • Pavel GurevichEmail author
  • Eyal Ron


We study an interplay between delay and discontinuous hysteresis in dynamical systems. After having established existence and uniqueness of solutions, we focus on the analysis of stability of periodic solutions. The main object we study is a Poincaré map that is infinite-dimensional due to delay and non-differentiable due to hysteresis. We propose a general functional framework based on the fractional order Sobolev–Slobodeckij spaces and explicitly obtain a formal linearization of the Poincaré map in these spaces. Furthermore, we prove that the spectrum of this formal linearization determines the stability of the periodic solution and then reduce the spectral analysis to an equivalent finite-dimensional problem.


Hysteresis Delay Periodic orbits Stability 



The research of the first author was supported by the DFG Heisenberg Programme (Grant No. GU 1482/1-2) and the RUDN University Program 5-100. Both authors acknowledge the support of the DFG project SFB 910.


  1. 1.
    Akian, M., Bliman, P.-A., Sorine, M.: Control of delay systems with relay. Special issue on analysis and design of delay and propagation systems. IMA J. Math. Control Inf. 19(1–2), 133–155 (2002)zbMATHCrossRefGoogle Scholar
  2. 2.
    Barton, D.A.W., Krauskopf, B., Wilson, R.E.: Periodic solutions and their bifurcations in a non-smooth second-order delay differential equation. Dyn. Syst. 21(3), 289–311 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Besov, O.V., Il\(^{\prime }\)in, V.P., Nikol\(^{\prime }\)skiĭ, S.M.: Integral representations of functions and imbedding theorems. In: Taibleson, M.H. (eds.) Scripta Series in Mathematics, vol. II. V. H. Winston & Sons, Halsted Press [Wiley], Washington, New York-Toronto, Ont.-London (1979)Google Scholar
  4. 4.
    Bliman, P.-A., Krasnosel\(^{\prime }\)skii, A.M.: Periodic solutions of linear systems coupled with relay. In: Proceedings of the Second World Congress of Nonlinear Analysts, Part 2 (Athens, 1996), vol. 30, pp. 687–696 (1997)Google Scholar
  5. 5.
    Brokate, M., Sprekels, J.: Hysteresis and Phase Transitions. Applied Mathematical Sciences, vol. 121. Springer, New York (1996)zbMATHCrossRefGoogle Scholar
  6. 6.
    Burenkov, V.I.: Additivity of the classes of \(W_{p}^{(r)}\,(\Omega )\). Trudy Mat. Inst. Steklov. 89, 31–55 (1967)MathSciNetGoogle Scholar
  7. 7.
    Cahlon, B., Schmidt, D., Shillor, M., Zou, X.: Analysis of thermostat models. Eur. J. Appl. Math. 8(5), 437–455 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Co, T.: Relay-stabilization and bifurcations of unstable SISO processes with time delay. IEEE Trans. Autom. Control 55(5), 1131–1141 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Colombo, A., di Bernardo, M., Hogan, S.J., Kowalczyk, P.: Complex dynamics in a hysteretic relay feedback system with delay. J. Nonlinear Sci. 17(2), 85–108 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Diekmann, O., van Gils, S.A., Verduyn Lunel, S.M., Walther, H.-O.: Delay equations. In: Diekmann, O. (ed.) Functional, Complex, and Nonlinear Analysis. Applied Mathematical Sciences, vol. 110. Springer, New York (1995)zbMATHGoogle Scholar
  11. 11.
    Dunford, N., Schwartz, J.T.: Linear Operators. Part I. General Theory, With the Assistance of William G. Bade and Robert G. Bartle, Reprint of the 1958 Original. Wiley, New York (1988)zbMATHGoogle Scholar
  12. 12.
    Fridman, L., Strygin, V., Polyakov, A.: Stabilization of amplitude of oscillations via relay delay control. Int. J. Control 76(8), 770–780 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Friedman, A., Jiang, L.S.: Periodic solutions for a thermostat control problem. Commun. Partial Differ. Equ. 13(5), 515–550 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Glashoff, K., Sprekels, J.: An application of Glicksberg’s theorem to set-valued integral equations arising in the theory of thermostats. SIAM J. Math. Anal. 12(3), 477–486 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Glashoff, K., Sprekels, J.: The regulation of temperature by thermostats and set-valued integral equations. J. Integral Equ. 4(2), 95–112 (1982)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Götz, I.G., Hoffmann, K.-H., Meirmanov, A.M.: Periodic solutions of the stefan problem with hysteresis-type boundary conditions. Manuscripta Mathematica 78, 179–199 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Gurevich, P.: Periodic solutions of parabolic problems with hysteresis on the boundary. Discrete Contin. Dyn. Syst. 29, 1041–1083 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Gurevich, P., Jäger, W., Skubachevskii, A.: On periodicity of solutions for thermocontrol problems with hysteresis-type switches. SIAM J. Math. Anal. 41(2), 733–752 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Gurevich, P., Tikhomirov, S.: Symmetric periodic solutions of parabolic problems with discontinuous hysteresis. J. Dyn. Differ. Equ. 23(4), 923–960 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Hale, J.K., Verduyn Lunel, S.M.: Introduction to Functional-Differential Equations. Applied Mathematical Sciences, vol. 99. Springer, New York (1993)zbMATHCrossRefGoogle Scholar
  21. 21.
    Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics, vol. 840. Springer, Berlin-New York (1981)zbMATHCrossRefGoogle Scholar
  22. 22.
    Kamachkin, A.M., Stepanov, A.V.: Stable periodic solutions of time delay systems containing hysteresis nonlinearities. In: Michiels, W., et al. (eds.) Topics in Time Delay Systems. Lecture Notes in Control and Inform. Sci., vol. 388, pp. 121–132. Springer, Berlin (2009)CrossRefGoogle Scholar
  23. 23.
    Kopfová, J., Kopf, T.: Differential equations, hysteresis, and time delay. Z. Angew. Math. Phys. 53(4), 676–691 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Krasnosel\(^{\prime }\)skiĭ, M.A., Pokrovskiĭ, A.V.: Systems with hysteresis. Springer, Berlin (1989). (Translated from the Russian by Marek Niezgódka)Google Scholar
  25. 25.
    Krejčí, P.: Hysteresis, Convexity and Dissipation in Hyperbolic Equations. GAKUTO International Series. Gattötoscho, Tokyo (1996)zbMATHGoogle Scholar
  26. 26.
    Lin, C., Wang, Q.-G., Lee, T.H., Lam, J.: Local stability of limit cycles for time-delay relay-feedback systems. IEEE Trans. Circuits Systems I Fund. Theory Appl. 49(12), 1870–1875 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Logemann, H., Ryan, E.P., Shvartsman, I.: A class of differential-delay systems with hysteresis: asymptotic behaviour of solutions. Nonlinear Anal. 69(1), 363–391 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Mayergoyz, I.: Mathematical Models of Hysteresis and Their Applications. Electromagnetism Series. Elsevier, Amsterdam (2003)Google Scholar
  29. 29.
    Mikhailov, V.P.: Partial differential equations. “Mir”, Moscow; distributed by Imported Publications, Inc., Chicago, Ill., (1978)Google Scholar
  30. 30.
    Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, vol. 44. Springer, New York (1983)zbMATHCrossRefGoogle Scholar
  31. 31.
    Prüss, J.: Periodic solutions of the thermostat problem. In: Angelo, F., Enrico, O. (eds.) Differential Equations in Banach Spaces (Bologna, 1985). Lecture Notes in Math., vol. 1223, pp. 216–226. Springer, Berlin (1986)CrossRefGoogle Scholar
  32. 32.
    Pyragas, K.: Continuous control of chaos by self-controlling feedback. Phys. Lett. A 170, 421–428 (1992)CrossRefGoogle Scholar
  33. 33.
    Pyragas, K.: A twenty-year review of time-delay feedback control and recent developments. In: International Symposium on Nonlinear Theory and its Applications (2012)Google Scholar
  34. 34.
    Seidman, T.I.: Nonlinear Semigroups, Partial Differential Equations and Attractors (Washington, DC, 1987). In: Gill, T.L., Zachary, W.W. (eds.) Switching Systems and Periodicity. Lecture Notes in Math., vol. 1394, pp. 199–210. Springer, Berlin (1989)Google Scholar
  35. 35.
    Sieber, J., Kowalczyk, P., Hogan, S.J., di Bernardo, M.: Dynamics of symmetric dynamical systems with delayed switching. J. Vib. Control 16(7–8), 1111–1140 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Slobodeckij, L.N.: Generalized Sobolev spaces and their application to boundary problems for partial differential equations; English translation: American mathematical society translations. series 2, vol. 57. Leningrad. Gos. Ped. Inst. Učen. Zap. 197, 54–112 (1958)MathSciNetGoogle Scholar
  37. 37.
    Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. VEB Deutscher Verlag der Wissenschaften, Berlin (1978)zbMATHGoogle Scholar
  38. 38.
    Visintin, A.: Differential Models of Hysteresis. Applied Mathematical Sciences, vol. 111. Springer, Berlin (1994)zbMATHCrossRefGoogle Scholar
  39. 39.
    Zhang, X., Lin, Y.: Adaptive control for a class of nonlinear time-delay systems preceded by unknown hysteresis. Int. J. Syst. Sci. 44(8), 1468–1482 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Zou, X., Jordan, J.A., Shillor, M.: A dynamic model for a thermostat. J. Eng. Math. 36(4), 291–310 (1999)MathSciNetzbMATHCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Free University of BerlinBerlinGermany
  2. 2.RUDN UniversityMoscowRussia
  3. 3.Cryptom TechnologiesBerlinGermany

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