RENDICONTI DEL SEMINARIO MATEMATICO

Shadowing in ordinary differential equations 97The operator L y has right inverses and we choose one such right inverse L −1y‖L −1y ‖ ≤ K.withAlso we define constants as before Lemma 1 with the range of k being appropriatelyadjusted. Now we can state our theorem.THEOREM 2. Finite-time Shadowing Theorem. Let {y k }k=0 N be a δ pseudoorbit of Eq. (1) with associated times {h k } N−1k=0and let L−1 y be a right inverse of theoperator L y 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 } N k=0 is ε-shadowed by a true orbit {x k} N k=0 withε ≤ 2 ¯Cδ.Notes on finite-time shadowing: The proof of the Finite-time Shadowing Theoremabove is quite similar to that of Lemma 1 except that here we use Brouwer’s fixedpoint theorem rather than the contraction mapping principle.It was first observed in [28], [29] that pseudo orbits of certain chaotic mapscould be shadowed for long times by true orbits, despite the lack of uniform hyperbolicity.Others, [9], [10], and [52] realized that these observations could be generalizedusing shadowing techniques. Here the key idea is the construction of a right inverseof small norm for a linear operator similar to the one used for infinite-time shadowing.The choice of this right inverse is guided by the infinite-time case—one takes the formulafor the inverse in the infinite-time case and truncates it appropriately (see [18]).However, the ordinary differential equation case is somewhat more complicated. It isnot simply a matter of looking at the time-one map and applying the theory for the mapcase. One must somehow “quotient-out” the direction of the vector field and allow forrescaling of time as is done in Theorem 2; this needs to be done because of lack ofhyperbolicity in the direction of the vector field. That this leads to much better shadowingresults is shown in [17]. For other finite-time shadowing theorems in the contextof autonomous ordinary differential equations, see [11], [12], [32], [62].4. Periodic shadowingIn simulations of differential equations apparent periodic orbits, usually asymptoticallystable, are often calculated. In this section we show how shadowing can be used toverify that there do indeed exist true periodic orbits near the computed orbits. Ourmethod can be applied even to unstable periodic orbits which are ubiquitous in chaoticsystems. We first recall the notions of pseudo periodic orbit and periodic shadowingfor autonomous ordinary differential equations. Then, we state a Periodic ShadowingTheorem which guarantees the existence of a true periodic orbit near a pseudo periodicorbit.