RENDICONTI DEL SEMINARIO MATEMATICO

98 H. Koçak - K. Palmer - B. CoomesDEFINITION 6. Definition of pseudo periodic orbit. For a given positive numberδ, a sequence of points {y k }k=0 N , with f (y k) ̸= 0 for all k, is said to be a δ pseudoperiodic orbit of Eq. (1) if there is an associated sequence {h k }k=0 N of positive timessuch that‖y k+1 − ϕ h k(y k )‖ ≤ δ f or k = 0,..., N − 1,and‖y 0 − ϕ h N(y N )‖ ≤ δ.DEFINITION 7. Definition of periodic shadowing. For a given positive numberε, a δ pseudo periodic orbit {y k }k=0 N with associated times {h k}k=0 N is said to be ε-shadowed by a true periodic orbit if there are points {x k }k=0 N and positive times {t k}k=0Nwith ϕ t k(x k ) = x k+1 for k = 0,..., N − 1, and x 0 = ϕ t N(x N ) such that‖x k − y k ‖ ≤ ε and |t k − h k | ≤ ε f or k = 0,..., N.To decide if a pseudo periodic orbit is shadowed by a true periodic orbit weneed to compute certain other quantities. Let {y k }k=0 N be a δ pseudo periodic orbit ofEq. (1) with associated times {h k }k=0 N . With the subspaces Y k and the projections P kdefined as in Section 2, we define a linear operatoras follows: if v = {v k } N k=0 thenL y : Y 0 × Y 1 × ··· × Y N → Y 1 × ··· × Y N × Y 0(L y v) k = v k+1 − P k+1 Dφ h k(y k )v k , f or k = 0,..., N − 1(L y v) N = v 0 − P N Dφ h N(y N )v N .We assume the operator L y is invertible with ‖L −1y ‖ ≤ K . Also we defineconstants as before Lemma 1 with the range of k being appropriately adjusted. Now,we can state our main theorem.THEOREM 3. Periodic Shadowing Theorem. Let {y k }k=0 N be a δ pseudo periodicorbit of the autonomous system Eq. (1) such that the operator L y is invertible with‖ ≤ K . Then if‖L −1y4Cδ < ε 0 , 2M 1 Cδ ≤ 1, C 2 (N 1 δ + N 2 δ 2 + N 3 δ 3 ) < 1,the pseudo periodic orbit {y k }k=0 N is ε-shadowed by a true periodic orbit {x k}k=0 N ofEq. (1) with associated times {t k }k=0 N and withε ≤ 2 ¯Cδ.Moreover, this is the unique such orbit satisfyingf (y k ) ∗ (x k − y k ) = 0 f or 0 ≤ k ≤ N.