RENDICONTI DEL SEMINARIO MATEMATICO

Shadowing in ordinary differential equations 93a δ pseudo orbit of Eq. (1) if there is an associated bounded sequence {h k } +∞k=−∞ ofpositive times with positive inf k∈ZZ h k such that‖y k+1 − ϕ h k(y k )‖ ≤ δf or k ∈ ZZ.orbit.Next, we introduce the notion of shadowing an infinite pseudo orbit by a trueDEFINITION 2. Definition of infinite-time shadowing. For a given positivenumber ε, a δ pseudo orbit {y k } +∞k=−∞ of Eq. (1) with associated times {h k} +∞said to be ε-shadowed by a true orbit of Eq. (1) if there are points {x k } +∞k=−∞true orbit and positive times {t k } +∞k=−∞ with ϕt k(x k ) = x k+1 such that‖x k − y k ‖ ≤ ε and |t k − h k | ≤ ε f or k ∈ ZZ.k=−∞ ison theIn our first Shadowing Theorem we will assume that pseudo orbits lie in a compacthyperbolic set. For completeness, we recall the definition of a hyperbolic set asgiven in, for example, [47].DEFINITION 3. Definition of hyperbolic set. A set S ⊂ U is said to be hyperbolicfor Eq. (1) if(i) f (x) ̸= 0 for all x in S;(ii) S is invariant under the flow, that is, φ t (S) = S for all t;(ii) there is a continuous splittingIR n = E 0 (x) ⊕ E s (x) ⊕ E u (x)f or x ∈ Ssuch that E 0 (x) is the one-dimensional subspace spanned by { f (x)}, and the subspacesE s (x) and E u (x) have constant dimensions; moreover, these subspaceshave the invariance propertyDφ t (x)(E s (x)) = E s (φ t (x)),Dφ t (x)(E u (x)) = E u (φ t (x))under the linearized flow and the inequalities‖Dφ t (x)ξ‖ ≤ K 1 e −α 1t ‖ξ‖‖Dφ t (x)ξ‖ ≤ K 2 e α 2t ‖ξ‖f or t ≥ 0, ξ ∈ E s (x),f or t ≤ 0, ξ ∈ E u (x)are satisfied for some positive constants K 1 , K 2 , α 1 , and α 2 .Now, we can state our first shadowing theorem for infinite pseudo orbits ofordinary differential equations.THEOREM 1. Infinite-time Shadowing Theorem. Let S be a compact hyperbolicset for Eq. (1). For a given sufficiently small ε > 0, there is a δ > 0 such that any