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Attractors for discrete periodic dynamical systems. (English) Zbl 1035.37020
The paper introduces a notion of an attractor for a discrete time periodic dynamical system, given by $$p$$ maps $$f_0,\ldots,f_{p-1}$$ on a metric space $$X$$, by extending the system to $$\{0,\ldots,p-1\}\times X$$. It is shown that such an attractor is the union of attractors for the $$p$$ autonomous maps $$F_i=f_{p-1+i}\circ\cdots\circ f_i$$, $$0\leq i\leq p$$. Conditions are given under which the constituents of an attractor of a periodic perturbation of an autonomous map are homeomorphic to the attractor of this map. Several examples, some of them originating from population dynamics, are discussed. The results fit nicely into the theory of attractors for nonautonomous and for random systems, which has been developed during the last decade.

##### MSC:
 37C70 Attractors and repellers of smooth dynamical systems and their topological structure 92D25 Population dynamics (general) 37N25 Dynamical systems in biology
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##### References:
 [1] Costantino, R.F.; Cushing, J.M.; Dennis, B.; Desharnais, R.A., Resonant population cycles in temporally fluctuating habitats, Bull. math. biol., 60, 247-273, (1998) · Zbl 0973.92034 [2] Cushing, J.M., Periodic time-dependent predator – prey systems, SIAM J. appl. math., 32, 82-95, (1977) · Zbl 0348.34031 [3] Cushing, J.M., Two species competition in a periodic environment, J. math. biol., 10, 385-400, (1980) · Zbl 0455.92012 [4] Güémez, J.; Matı́as, M.A., Control of chaos in unidimensional maps, Phys. lett. A, 181, 29-32, (1993) [5] Henson, S.M., Multiple attractors and resonance in periodically forced population models, Phys. D, 140, 33-49, (2000) · Zbl 0957.37018 [6] Henson, S.M.; Costantino, R.F.; Cushing, J.M.; Dennis, B.; Desharnais, R.A., Multiple attractors, saddles, and population dynamics in periodic habitats, Bull. math. biol., 61, 1121-1149, (1999) · Zbl 1323.92169 [7] Henson, S.M.; Cushing, J.M., The effect of periodic habitat fluctuations on a nonlinear insect population model, J. math. biol., 36, 201-226, (1997) · Zbl 0890.92023 [8] Jillson, D., Insect populations respond to fluctuating environments, Nature, 288, 699-700, (1980) [9] Marsden, J.E.; Hoffman, M.J., Elementary classical analysis, (1993), Freeman New York · Zbl 0777.26001 [10] Palis, J.; Takens, F., Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcatioins, (1993), Cambridge Univ. Press Cambridge [11] Roberds, J.H.; Selgrade, J.F., Dynamical analysis of density-dependent selection in a discrete one-island migration model, Math. biosci., 164, 1-15, (2000) · Zbl 0947.92023 [12] Robinson, C., Dynamical systems: stability, symbolic dynamics, and chaos, (1995), CRC Press Boca Raton, FL · Zbl 0853.58001 [13] Selgrade, J.F.; Roberds, J.H., On the structure of attractors for discrete, periodically forced systems with applications to population models, Phys. D, 158, 69-82, (2001) · Zbl 1018.37047 [14] Thieme, H.R., Uniform weak implies uniform strong persistence for non-autonomous semiflows, Proc. amer. math. soc., 127, 2395-2403, (1999) · Zbl 0918.34053 [15] Thieme, H.R., Uniform persistence and permanence for non-autonomous semiflows in population biology, Math. biosci., 166, 173-201, (2000) · Zbl 0970.37061
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