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-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 dense Gδ-set Sϵu(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.