Archive for Rational Mechanics and Analysis

, Volume 116, Issue 4, pp 339–360 | Cite as

Convergence to cycles as a typical asymptotic behavior in smooth strongly monotone discrete-time dynamical systems

  • P. Poláčik
  • I. Tereščák
Article

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [AH]
    N. D. Alikakos & P. Hess, On stabilization of discrete monotone dynamical systems, Israel J. Math. 59 (1987), 185–194.Google Scholar
  2. [AHM]
    N. D. Alikakos, P. Hess & H. Matano, Discrete order preserving semigroups and stability for periodic parabolic differential equations, J. Diff. Eq. 82 (1989), 322–341.Google Scholar
  3. [DH]
    E. N. Dancer & P. Hess, Stability of fixed points for order-preserving discrete-time dynamical systems, J. reine angew. Math, to appear.Google Scholar
  4. [He1]
    P. Hess, On stabilization of discrete strongly order-preserving semigroups and dynamical processes, in Semigroup Theory and Applications, P. Clément (ed.), Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, 1989.Google Scholar
  5. [He2]
    P. Hess, Periodic Parabolic Boundary-Value Problems and Positivity, Pitman Research Notes in Mathematics, Vol. 247, 1991.Google Scholar
  6. [Hi1]
    M. W. Hirsch, Stability and convergence in strongly monotone dynamical Systems, J. reine angew. Math. 383 (1988), 1–58.Google Scholar
  7. [Hi2]
    M. W. Hirsch, Differential equations and convergence almost everywhere in strongly monotone flows, Contemp. Math. 17, Amer. Math. Soc. 1983, 267–285.Google Scholar
  8. [K]
    T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, 1966.Google Scholar
  9. [KFS]
    I. P. Kornfeld, S. V. Fomin & Y. G. Sinai, Ergodic Theory, Springer-Verlag, 1982.Google Scholar
  10. [M1]
    R. Mañé, Ergodic Theory and Differentiable Dynamics, Springer-Verlag, 1987.Google Scholar
  11. [M2]
    R. Mañé, Lyapunov exponents and stable manifolds for compact transformations, in: Geometric Dynamics, J. Palis (ed.), Lecture Notes in Mathematics vol. 1007, Springer-Verlag, 1983, 522–577.Google Scholar
  12. [Ma]
    H. Matano, Strong comparison principle in nonlinear parabolic equations, in Nonlinear Parabolic Equations: Qualitative Properties of Solutions, L. Boccardo, A. Tesei (eds.), Pitman, 1987, 148–155.Google Scholar
  13. [Mi1]
    J. Mierczyński, On a generic behavior in strongly cooperative differential equations, Colloquia Mathematica Societatis János Bolyai Vol. 53, North-Holland, 1990, 402–406.Google Scholar
  14. [Mi2]
    J. Mierczyński, Flows on ordered bundles, preprint.Google Scholar
  15. [Mo]
    X. Mora, Semilinear problems define semiflows on C kspaces, Trans. Amer. Math. Soc. 278 (1983), 1–55.Google Scholar
  16. [Os]
    V. I. Oseledec, A multiplicative ergodic theorem, Trans. Moscow Math. Soc. 19 (1968), 197–231.Google Scholar
  17. [P1]
    P. Poláčik, Convergence in smooth strongly monotone flows defined by semilinear parabolic equations, J. Diff. Eq. 79 (1989), 89–110.Google Scholar
  18. [P2]
    P. Poláčik, Generic properties of strongly monotone semiflows defined by ordinary and parabolic differential equations, Colloquia Mathematica Societatis János Bolyai Vol. 53, North-Holland, 1990, 402–406.Google Scholar
  19. [P3]
    P. Poláčik, Imbedding of any vector field in scalar semilinear parabolic equation, Proc. Amer. Math. Soc., to appear.Google Scholar
  20. [PT]
    P. Poláčik & I. Tereščák, in preparation.Google Scholar
  21. [R]
    D. Ruelle, Analyticity properties of the characteristic exponents of random matrix products, Advances Math. 32 (1979), 68–80.Google Scholar
  22. [ST1]
    H. L. Smith & H. R. Thieme, Quasi convergence and stability for order-preserving semiflows, SIAM J. Math. Anal. 21 (1990), 673–692.Google Scholar
  23. [ST2]
    H. L. Smith & H. R. Thieme, Convergence for strongly order-preserving semiflows, SIAM J. Math. Anal., 22 (1991), 1081–1101.Google Scholar
  24. [T1]
    P. Takáč, Convergence to equilibrium on invariant d-hypersurfaces for strongly increasing discrete-time dynamical systems, J. Math. Anal. Appl. 148 (1990), 223–244.Google Scholar
  25. [T2]
    P. Takáč, Domains of attraction of generic ω-limit set for strongly monotone semiflows, Z. Anal. Anwendungen, to appear.Google Scholar
  26. [T3]
    P. Takáč, Asymptotic behavior of strongly monotone time-periodic dynamical processes with symmetry, J. Diff. Eq., to appear.Google Scholar
  27. [T4]
    P. Takáč, Linearly stable subharmonic orbits in strongly monotone time-periodic dynamical systems, Proc. Amer. Math. Soc., to appear.Google Scholar
  28. [T5]
    P. Takáč, Domains of attraction of generic ω-limit sets for strongly monotone discrete-time semigroups, J. reine angew. Math., to appear.Google Scholar

Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • P. Poláčik
    • 1
    • 2
  • I. Tereščák
    • 1
    • 2
  1. 1.Institute of Applied MathematicsComenius UniversityBratislavaCzechoslovakia
  2. 2.Department of Mathematical AnalysisComenius UniversityBratislavaCzechoslovakia

Personalised recommendations