Skip to main content
SpringerLink
Log in

Morse Decompositions for Periodic General Dynamical Systems and Differential Inclusions

  • Published:
Set-Valued and Variational Analysis Aims and scope Submit manuscript

Abstract

We first establish the Morse decomposition theory of periodic invariant sets for non-autonomous periodic general dynamical systems (set-valued dynamical systems). Then we discuss the stability of Morse decompositions of periodic uniform forward attractors. We also apply the abstract results to non-autonomous periodic differential inclusions with only upper semi-continuous right-hand side. We show that Morse decompositions are robust with respect to both internal and external perturbations (upper semi-continuity of Morse sets). Finally as an application we study the effect of small time delays to asymptotic behavior of control systems from the point of view of Morse decompositions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Akin, E.: The general topology of dynamical systems. In: Graduate Studies in Mathematics 1. American Mathematical Society, Providence (1993)

    Google Scholar 

  2. Aubin, J.P., Cellina, A.: Differential Inclusions. Springer, Berlin (1984)

    Book  MATH  Google Scholar 

  3. Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Birkhäuser (1990)

  4. Caraballo, T., Langa, J.A.: Global attractors for multivalued random dynamical systems generated by random differential inclusions with multiplicative noise. J. Math. Anal. Appl. 260, 602–622 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  5. Caraballo, T., Marin-Rubio, P., Robinson, J.C.: A comparison between two theories for multivalued semiflows and their asymptotic behaviour. Set-Valued Anal. 11, 297–322 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  6. Caraballo, T., Kloeden, P.E., Marin-Rubio, P.: Weak pullback attractors of setvalued processes. J. Math. Anal. Appl. 288, 692–707 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cherene Jr., L.J.: Set valued dynamical system and economic flow. In: Lecture Notes in Economics and Mathematical Systems 158. Springer, Berlin (1978)

    Google Scholar 

  8. Clarke, F.H., Ledyaev, Yu.S., Stern, R.J.: Asymptotic stability and smooth Lyapunov functions. J. Differ. Equ. 149, 69–114 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  9. Conley, C.: Isolated invariant sets and the Morse index. In: Regional Conference Series in Mathematics 38. American Matematical Society, Providence (1978)

    Google Scholar 

  10. Crauel, H., Duc, L.H., Siegmund, S.: Towards a Morse theory for random dynamical systems. Stoch. Dyn. 4, 277–296 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  11. Deimling, K.: Multivalued Differential Equations. De Gruyter (1992)

  12. Filippov, A.F.: Differential Equations with Discontinuous Righthand Side. Kluwer Academic, Dordrecht (1998)

    Google Scholar 

  13. Gedeon, T., Hines, G.: Upper semicontinuity of Morse sets of a discretization of a delayed-differential equation: an improvement. J. Differ. Equ. 151, 36–78 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gedeon, T., Hines, G.: Upper semicontinuity of Morse sets of a discretization of a delayed-differential equation: an improvement. J. Differ. Equ. 179, 369–383 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hale, J.K., Verduyn Lunel, S.M.: Effects of small delays on stability and control. In: Operator Theory and Analysis (Amsterdam, 1997), Operator Theory and Advanced Applications, vol. 122, pp. 275–301. Birkhäuser, Basel (2001)

    Chapter  Google Scholar 

  16. Kapustyan, A.V., Mrozek, M.: Conley index for discrete multivalued systems. Topol. its Appl. 65, 83–96 (1995)

    Article  Google Scholar 

  17. Kapustyan, A.V., Valero, J.: Attractors of multivalued semiflows generated by differential inclusions and their approximations. Abstr. Appl. Anal. 5, 33–46 (2000)

    Article  MathSciNet  Google Scholar 

  18. Kloeden, P.E.: Asymptotic invariance and limit sets of general control systems. J. Differ. Equ. 19, 91–105 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kloeden, P.E.: General control systems. In: Coppel, W.A. (ed.) Mathematical Control Theory, Lecture Notes in Mathematics, vol. 680, pp. 119–138. Springer (1978)

  20. Kloeden, P.E., Valero, J.: Attractors of weakly asymptotically compact setvalued dynamical systems. Set-Valued Anal. 13, 381–304 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  21. Li, D.S.: On dynamical stability in general dynamical systems. J. Math. Anal. Appl. 263, 455–478 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  22. Li, D.S.: Morse decompositions for general dynamical systems and differential inclusions with applications to control systems. SIAM J. Control Optim. 46, 35–60 (2007)

    Article  MathSciNet  Google Scholar 

  23. Li, D.S., Kloeden, P.E.: Robustness of asymptotic stability to small time delays. Discrete Contin. Dyn. Syst. 13, 1007–1034 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  24. Li, D.S., Kloeden, P.E.: On the dynamics of nonautonomous periodic general dynamical systems and differential inclusions. J. Differ. Equ. 224, 1–38 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  25. Li, X.J., Liu, K.S.: The effects of small delays in the feedbacks on boundary stabilization. Sci. China 36, 1443–1453 (1993)

    Google Scholar 

  26. Li, D.S., Zhang, X.X., On the stability in general dynamical systems and differential inclusions. J. Math. Anal. Appl. 274, 705–724 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  27. Li, D.S., Wang, Y.J., Wang, S.Y.: On the dynamics of non-autonomous general dynamical systems and differential inclusions. Set-Valued Anal. 16, 651–671 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  28. Mallet-Paret, J.: Morse decompositions for differential delay equations. J. Differ. Equ. 72, 270–315 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  29. Mrozek, M.: A cohomological index of conley type for multi-valued addmissible flows. J. Differ. Equ. 84, 15–51 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  30. Rasmussen, M.: Morse decompositions of nonautonomous dynamical systems. Trans. Am. Math. Soc. 359, 5091–5115 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  31. Roxin, E.: Stability in general control systems. J. Differ. Equ. 1, 115–150 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  32. Roxin, E.: On generelized dynamical systems defined by contingent equations. J. Differ. Equ. 1, 188–205 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  33. Rybakowski, K.P.: The Homotopy Index and Partial Differential Equations. Springer, Berlin (1987)

    Book  MATH  Google Scholar 

  34. Wang, Y.J., Zhou, S.F.: Kernel sections and uniform attractors of multi-valued semiprocesses. J. Differ. Equ. 232, 573–622 (2007)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, People’s Republic of China

    Yejuan Wang

  2. Department of Mathematics, Tianjin University, Tianjin, 300072, People’s Republic of China

    Desheng Li

Authors
  1. Yejuan Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Desheng Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yejuan Wang.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wang, Y., Li, D. Morse Decompositions for Periodic General Dynamical Systems and Differential Inclusions. Set-Valued Var. Anal 20, 519–549 (2012). https://doi.org/10.1007/s11228-012-0212-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11228-012-0212-5

Keywords

Mathematics Subject Classifications (2000)

Search

Navigation