RENDICONTI DEL SEMINARIO MATEMATICO

96 H. Koçak - K. Palmer - B. CoomesA shadowing theorem for flows, with a somewhat different notion of shadowing,was first proved in [26]. Different versions of the shadowing theorem have been provedin [42], [34], [35], [16], and [49]. For a “continuous” shadowing theorem, see [47] andalso [49]. In the latter book the author also proves a shadowing theorem for structurallystable systems, which includes that for hyperbolic systems.3. Finite-time shadowingSince chaotic systems exhibit sensitive dependence on initial conditions, a numericallygenerated orbit will diverge quickly from the true orbit with the same initial condition.However, we observe that a computed orbit is a pseudo orbit and so, according to ourInfinite-time Shadowing Lemma, would be shadowed by a true orbit in the presenceof hyperbolicity, albeit with slightly different initial condition. It turns out chaoticsystems are seldom uniformly hyperbolic but still exhibit enough hyperbolicity thatpseudo orbits can still be shadowed for long times. To this end, we formulate a finitetimeshadowing theorem in this section. As in the Infinite-time Shadowing Lemma,our condition involves a linear operator associated with the pseudo orbit but now theaim is to choose a right inverse with small norm. First we recall a precise notion ofshadowing of a finite pseudo orbit by an associated nearby true orbit. Then we presentthe Finite-time Shadowing Theorem.DEFINITION 4. Definition of finite pseudo orbit. For a given positive numberδ, a sequence of points {y k }k=0 N is said to be a δ pseudo orbit of Eq. (1) if f (y k) ̸= 0and there is an associated sequence {h k } N−1k=0of positive times such that‖y k+1 − ϕ h k(y k )‖ ≤ δ f or k = 0,..., N − 1.DEFINITION 5. Definition of finite-time shadowing. For a given positivenumber ε, an orbit of Eq. (1) is said to ε-shadow a δ pseudo orbit {y k }k=0 N with associatedtimes {h k } N−1k=0 if there are points {x k}k=0 N on the true orbit and times {t k} N−1k=0with ϕ t k(x k ) = x k+1 such that‖x k − y k ‖ ≤ ε f or k = 0,..., N and |t k − h k | ≤ ε f or k = 0,..., N − 1.To state our theorem we need to develop a bit of notation and introduce certainrelevant mathematical constructs. Let {y k }k=0 N be a δ pseudo orbit of Eq. (1) withassociated times {h k } N−1k=0 . With the subspaces Y k and the projections P k defined as inSection 2, we define a linear operatorL y : Y 0 × ··· × Y N → Y 1 × ··· × Y Nin the following way: If v = {v k }k=0 N is in Y 0 × ··· × Y N , then we take L y v ={[L y v] k } N−1k=0to be[L y v] k = v k+1 − P k+1 Dφ h k(y k )v k f or k = 0,..., N − 1.