Set-valued discrete chaos. (English) Zbl 1079.37024

Peris, Alfredo

Chaos Solitons Fractals 26, No. 1, 19-23 (2005).

When studying the chaotic dynamics of individual members, the following natural questions arise: How does individual chaos affect the chaotic behaviour of the ecosystem dynamics as a whole? And vice versa? Here, the author gives a characterization of topological transitivity for collective dynamics in terms of the weakly mixing property for individual dynamics.

Reviewer: Messoud A. Efendiev (Berlin)

MSC:

37D45	Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37A25	Ergodicity, mixing, rates of mixing
37N25	Dynamical systems in biology

Keywords:

ecosystem dynamics; chaotic dynamics; mixing; topological transitivity

References:

[1]	Auslander, J.; Yorke, J. A., Interval maps, factors of maps, and chaos, Tohoku Math J, 32, 177-188 (1980) · Zbl 0448.54040
[2]	Banks, J., Topological mapping properties defined by digraphs, Discr Cont Dyn Syst, 5, 83-92 (1999) · Zbl 0957.54020
[3]	Barnsley, M. F., Fractals everywhere (1988), Academic Press · Zbl 0691.58001
[5]	Bès, J.; Peris, A., Hereditarily hypercyclic operators, J Funct Anal, 167, 94-112 (1999) · Zbl 0941.47002
[6]	Birkhoff, G. D., Démonstration d’un théoréme elémentaire sur les fonctions entiéres, C R Acad Sci Paris, 189, 473-475 (1929) · JFM 55.0192.07
[7]	Bonet, J.; Martı́nez-Giménez, F.; Peris, A., Linear chaos on Fréchet spaces, Int J Bifurcat Chaos, 13, 1649-1655 (2003) · Zbl 1079.47008
[8]	Devaney, R. L., An introduction to chaotic dynamical systems (1989), Addison-Wesley · Zbl 0695.58002
[9]	Fedeli, A., On chaotic set-valued discrete dynamical systems, Chaos, Solitons & Fractals, 23, 1381-1384 (2005) · Zbl 1079.37021
[10]	Furstenberg, H., Disjointness in ergodic theory, minimal sets and a problem in diophantine approximation, Syst Theory, 1, 1-49 (1967) · Zbl 0146.28502
[11]	Gethner, R. M.; Shapiro, J. H., Universal vectors for operators on spaces of holomorphic functions, Proc Am Math Soc, 100, 281-288 (1987) · Zbl 0618.30031
[12]	Godefroy, G.; Shapiro, J. H., Operators with dense, invariant, cyclic vector manifolds, J Funct Anal, 98, 229-269 (1991) · Zbl 0732.47016
[13]	Grosse-Erdmann, K. G., Universal families and hypercyclic operators, Bull Am Math Soc, 36, 345-381 (1999) · Zbl 0933.47003
[14]	Grosse-Erdmann, K. G., Recent developments in hypercyclicity, Rev R Acad Cienc Ser A Mat., 97, 273-286 (2003) · Zbl 1076.47005
[15]	Herzog, G.; Lemmert, R., On universal subsets of Banach spaces, Math Z, 229, 615-619 (1998) · Zbl 0928.47003
[17]	Román-Flores, H., A note on transitivity in set-valued discrete systems, Chaos, Solitons & Fractals, 17, 99-104 (2003) · Zbl 1098.37008