Chaos and stability in some random dynamical systems

Janková, Katarína

doi:10.2478/v10127-012-0008-x

Abstract

Nonchaotic behavior in the sense of Li and Yorke chaos in discrete dynamical systems generated by a continuous selfmapping of a real compact inter- val means that every trajectory can be approximated by a periodic one. Stability of this behavior was analyzed also for dynamical systems with small random perturbations. In this paper we study similar properties for nonautonomous periodic dynamical systems with random perturbations and for random dynamical systems generated by two continuous maps and their perturbations.

References

[1] BHATTACHARYA,R.-MAJUMDAR,M.: Random Dynamical Systems. Theory and Applications , Cambridge University Press, Cambridge, 2007.10.1017/CBO9780511618628
Search in Google Scholar
[2] BHATTACHARYA, R.-WAYMIRE E. C.: An approach to the existence of unique in- variant probabilities for Markov processes, in: Limit Theorems in Probability and Statis- tics, Vol. 1, 4th Hungarian Colloquium on Limit Theorems in Probability and Statistics (I. Berkes et al., eds.), Balatonlelle, Hungary, 1999, J´anos Bolyai Math. Soc., Budapest, 2002, pp. 181-200.
Search in Google Scholar
[3] C´ANOVAS, J. S.: Recent results on nonautonomous discrete systems, Bol. Soc. Esp. Mat. Apl. 31 (2010), 33-41.
Search in Google Scholar
[4] FRANKE, J. E.-SELGRADE, J. F.: Attractors for discrete periodic dynamical systems, J. Math. Anal. Appl. 286 (2003), 64-79.10.1016/S0022-247X(03)00417-7
Search in Google Scholar
[5] JANKOV´A , K.-SM´I TAL, J.: Maps with random perturbations are generically nonchaotic, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 5 (1995), 1375-1378.10.1142/S0218127495001058
Search in Google Scholar
[6] JANKOV´A , K.-SM´ITAL, J.: A characterization of chaos, Bull. Austral. Math. Soc. 34 (1986), 283-292.10.1017/S0004972700010157
Search in Google Scholar
[7] JANKOV´A , K.: Chaos in dynamical systems with randomly perturbed trajectories, in: Pro- ceedings of European Conference on Iteration Theory, Lisbon, 1991, pp. 146-156.
Search in Google Scholar
[8] KOLYADA, S.-SNOHA, L’.: Topological entropy of nonautonomous dynamical sysytems, Random Comput. Dynam. 4 (1996), 205-233.
Search in Google Scholar
[9] PREISS, D.-SM´ITAL, J.: Characterization of nonchaotic continuous maps of the interval stable under small perturbations, Trans. Amer. Math. Soc. 313 (1989), 687-697.10.1090/S0002-9947-1989-0997677-9
Search in Google Scholar
[10] SHI, Y.-CHEN, G.: Chaos of time varying discrete dynamical systems, J. Difference Equ. Appl. 15 (2009), 429-449.10.1080/10236190802020879
Search in Google Scholar

Chaos and stability in some random dynamical systems

Abstract

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