RENDICONTI DEL SEMINARIO MATEMATICO

Shadowing in ordinary differential equations 95and[]N 1 =8 M 0 M 1 + 2M 1 e M 1(h max +ε 0 ) + M 2 (h max + ε 0 )e 2M 1(h max +ε 0 ),[ ()] [ ]N 2 =8 1 + 4C M 0 + e M 1(h max +ε 0 )M 1 M 1 + M 2 M 0 + 2M 2 e M 1(h max +ε 0 ),N 3 =8[1 + 4C(M 0 + e M 1(h max +ε 0 ) )] 2M1 M 2 .Now, we can state our second shadowing result for an infinite pseudo orbit interms of the associated operator L y and the constants introduced above.LEMMA 1. Infinite-time Shadowing Lemma. Let {y k } +∞k=−∞be a bounded δpseudo orbit of Eq. (1) with associated times {h k } +∞k=−∞ such that L y is invertible with‖L −1y‖ ≤ K . Then if4Cδ < ε 0 , 2M 1 Cδ ≤ 1, C 2 (N 1 δ + N 2 δ 2 + N 3 δ 3 ) < 1,the pseudo orbit {y k } +∞k=−∞ is ε-shadowed by a true orbit {x k} +∞k=−∞of Eq. (1) withassociated times {t k } +∞k=−∞and withε≤ 2 ¯Cδ.Moreover this is the unique such orbit satisfyingf (y k ) ∗ (x k − y k ) = 0f or k ∈ ZZ.Notes on infinite-time shadowing: The details of the proof of the Infinite-time ShadowingLemma are given in [22]. The idea of the proof is to set up the problem of findinga true orbit near the pseudo orbit as the solution of a nonlinear equation in a Banachspace of sequences. The invertibility of the linear operator L y implies the invertibilityof another linear operator associated with the abstract problem and this enables one toapply a Newton-Kantorovich type theorem [33] to obtain the existence and uniquenessof the true orbit.One uses the Infinite-time Shadowing Lemma to prove the Infinite-time ShadowingTheorem. The main problem is to show that hyperbolicity implies that the operatorL y : Y → Ỹ is invertible with a uniform bound on its inverse. A slightly differentproof of the Infinite-time Shadowing Theorem is in an earlier publication [16].For flows, unlike diffeomorphisms, there are various alternatives for the definitionof a pseudo orbit. Should it be a sequence of points or solution segments that arefunctions of time? Here we have elected to use sequences of points. These choices ofdefinitions have obvious advantages when we consider finite pseudo orbits that comefrom numerical computations. With such considerations in mind, the definitions of apseudo orbit and shadowing for ordinary differential equations as given here first appearedin [14] and [16]. The problem in proving the shadowing theorem for flows isthe lack of hyperbolicity in the direction of the vector field. To compensate for this, weallow a rescaling of time in our definition of shadowing.