References
N. D. Alikakos & P. Hess, On stabilization of discrete monotone dynamical systems, Israel J. Math. 59 (1987), 185–194.
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.
E. N. Dancer & P. Hess, Stability of fixed points for order-preserving discrete-time dynamical systems, J. reine angew. Math, to appear.
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.
P. Hess, Periodic Parabolic Boundary-Value Problems and Positivity, Pitman Research Notes in Mathematics, Vol. 247, 1991.
M. W. Hirsch, Stability and convergence in strongly monotone dynamical Systems, J. reine angew. Math. 383 (1988), 1–58.
M. W. Hirsch, Differential equations and convergence almost everywhere in strongly monotone flows, Contemp. Math. 17, Amer. Math. Soc. 1983, 267–285.
T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, 1966.
I. P. Kornfeld, S. V. Fomin & Y. G. Sinai, Ergodic Theory, Springer-Verlag, 1982.
R. Mañé, Ergodic Theory and Differentiable Dynamics, Springer-Verlag, 1987.
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.
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.
J. Mierczyński, On a generic behavior in strongly cooperative differential equations, Colloquia Mathematica Societatis János Bolyai Vol. 53, North-Holland, 1990, 402–406.
J. Mierczyński, Flows on ordered bundles, preprint.
X. Mora, Semilinear problems define semiflows on C kspaces, Trans. Amer. Math. Soc. 278 (1983), 1–55.
V. I. Oseledec, A multiplicative ergodic theorem, Trans. Moscow Math. Soc. 19 (1968), 197–231.
P. Poláčik, Convergence in smooth strongly monotone flows defined by semilinear parabolic equations, J. Diff. Eq. 79 (1989), 89–110.
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.
P. Poláčik, Imbedding of any vector field in scalar semilinear parabolic equation, Proc. Amer. Math. Soc., to appear.
P. Poláčik & I. Tereščák, in preparation.
D. Ruelle, Analyticity properties of the characteristic exponents of random matrix products, Advances Math. 32 (1979), 68–80.
H. L. Smith & H. R. Thieme, Quasi convergence and stability for order-preserving semiflows, SIAM J. Math. Anal. 21 (1990), 673–692.
H. L. Smith & H. R. Thieme, Convergence for strongly order-preserving semiflows, SIAM J. Math. Anal., 22 (1991), 1081–1101.
P. Takáč, Convergence to equilibrium on invariant d-hypersurfaces for strongly increasing discrete-time dynamical systems, J. Math. Anal. Appl. 148 (1990), 223–244.
P. Takáč, Domains of attraction of generic ω-limit set for strongly monotone semiflows, Z. Anal. Anwendungen, to appear.
P. Takáč, Asymptotic behavior of strongly monotone time-periodic dynamical processes with symmetry, J. Diff. Eq., to appear.
P. Takáč, Linearly stable subharmonic orbits in strongly monotone time-periodic dynamical systems, Proc. Amer. Math. Soc., to appear.
P. Takáč, Domains of attraction of generic ω-limit sets for strongly monotone discrete-time semigroups, J. reine angew. Math., to appear.
Author information
Authors and Affiliations
Additional information
Communicated by M. Golubitsky
Rights and permissions
About this article
Cite this article
Poláčik, P., Tereščák, I. Convergence to cycles as a typical asymptotic behavior in smooth strongly monotone discrete-time dynamical systems. Arch. Rational Mech. Anal. 116, 339–360 (1992). https://doi.org/10.1007/BF00375672
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00375672