Chaotic dynamics of continuous-time topological semi-flows on Polish spaces

Differently from Lyapunov exponents, Li–Yorke, Devaney and others that appeared in the literature, we introduce the concept, chaos, for a continuous semi-flow f:R+×X→X on a Polish space X with a metric d, which is useful in the theory of ODE and is invariant under topological equivalence of semi-flo...

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Bibliographic Details
Published in:Journal of Differential Equations 2015-04, Vol.258 (8), p.2794-2805
Main Author: Dai, Xiongping
Format: Article
Language:English
Subjects:
Chaos
Maximal chaotic subsystem
Quasi-transitivity
Semi-flow
Sensitivity
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Summary:Differently from Lyapunov exponents, Li–Yorke, Devaney and others that appeared in the literature, we introduce the concept, chaos, for a continuous semi-flow f:R+×X→X on a Polish space X with a metric d, which is useful in the theory of ODE and is invariant under topological equivalence of semi-flows. Our definition is weaker than Devaney's one since here f may have neither fixed nor periodic elements; but it implies repeatedly observable sensitive dependence on initial data: there is an ϵ>0 such that for any x∈X, there corresponds a denseGδ-setSϵu(x) in X satisfyinglimsupt→+∞d(ft(x),ft(y))≥ϵ∀y∈Sϵu(x). This sensitivity is obviously stronger than Guckenheimer's one that requires only d(ft(x),ft(y))≥ϵ for some moment t>0 and some y arbitrarily close to x.
ISSN:0022-0396
1090-2732
DOI:10.1016/j.jde.2014.12.027