Time-preserving Structural Stability of Hyperbolic Differential Dynamics with Noncompact Phase Spaces

Let S : E → R where TwE = R for all w ∈ E, be a C-differential system on an n-dimensional Euclidean w-space E, which naturally gives rise to a flow φ : (t, w) 7→ t·w on E, and let Λ be a φ-invariant closed subset containing no any singularities of S. If Λ is compact and hyperbolic, then Anosov’s theorem asserts that S is structurally stable on Λ in the sense of topological equivalence; that is, for any C-perturbation V close to S, there is an ε-homeomorphism H : Λ → ΛV sending orbits φ(R, w) of S into orbits φV (R, H(w)) of V for all w in Λ. In this paper, using Liao theory Anosov’s result is generalized as follows: Let ψV : R×Σ → Σ be the crosssection flow of V relative to S locally defined on the Poincaré cross-section bundle Σ = S w∈Λ Σw of S, where Σw = {w ∈ E | 〈S(w), w − w〉 = 0}. If S is hyperbolic on Λ and V is C-close to S, then there is an ε-homeomorphism w 7→ H(w) ∈ Σw from Λ onto a closed set ΛV such that ψV (t, H(w)) = H(t·w) for all w ∈ Λ, where Λ need not be compact. Finally, an example is provided to illustrate our theoretical outcome.