Abstract
In this paper, we study some chaotic properties ofs-dimensional dynamical system of the formΨa1,a2,…,as=gsas,g1a1,…,gs−1as−1,whereak∈Hkfor anyk∈1,2,…,s, s≥2is an integer, andHkis a compact subinterval of the real lineℝ=−∞,+∞for anyk∈1,2,…,s. Particularly, a necessary and sufficient condition for a cyclic permutation mapΨa1,a2,…,as=gsas,g1a1,…,gs−1as−1to be LY-chaotic or h-chaotic or RT-chaotic or D-chaotic is obtained. Moreover, the LY-chaoticity, h-chaoticity, RT-chaoticity, and D-chaoticity of such a cyclic permutation map is explored. Also, we proved that the topological entropyhΨof such a cyclic permutation map is the same as the topological entropy of each of the following maps:gj∘gj−1∘⋯∘g1l∘gs∘gs−1∘⋯∘gj+1,ifj=1,…,s−1andgs∘gs−1∘⋯∘g1, and thatΨis sensitive if and only if at least one of the coordinates maps ofΨsis sensitive.
Highlights
A very important generalization is distributional chaos, proposed by Schweizer and Smıtal [5], mainly because it is equivalent to positive topological entropy and some other concepts of chaos when restricted to some spaces
It is clear that a continuous cyclic permutation map defined by
From [31], we know that one can find continuous cyclic permutation maps in age-structured population models, as in [24], where it is analyzed as the Leslie model: Complexity y1(n + 1) yNg yN, y2(n + 1) y1(n), . . . , yN(n + 1)
Summary
We proved that the topological entropy h(Ψ) of such a cyclic permutation map is the same as the topological entropy of each of the following maps: gj ∘ gj−1 ∘ · · · ∘ g1l ∘ gs ∘ gs−1 ∘ · · · ∘ gj+1, if j 1, . There exist LY-chaotic interval maps with zero topological entropy (see [21]). They extended well-known properties of transitivity from interval maps to cyclically permuted direct product maps.
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