RENDICONTI DEL SEMINARIO MATEMATICO

Shadowing in ordinary differential equations 103tolerance control to generate a good pseudo orbit. However, in order to claim theexistence of a true orbit near the computed approximate orbit, we need a rigorousbound on the local discretization error δ. We have found a high-order Taylor Method tobe the most effective numerical integration method for this purpose. To get a rigorous δ,one must also account for the floating point errors in the calculation of φ h k(y k ), whichwe handle using the techniques of Wilkinson [65]. Details of the implementation of theTaylor Method with floating error estimates for the Lorenz Equations are given in [17].In the case of periodic or homoclinic shadowing, it is very difficult to find apseudo periodic orbit or pseudo homoclinic orbit with δ small enough by a routine useof a numerical integrator using simple shooting, that is, various initial conditions aretried until one is found with a small δ. Usually the δ found in this way is not smallenough to apply our theorems. To get a pseudo orbit with a smaller δ, we refine the“crude” pseudo orbit with a suitable global Newton’s method. “Global” means we workwith the whole pseudo orbit, not just its initial point since it turns out that working withjust the initial point is not effective.Now we describe what we do in the periodic case. Let {y k }k=0 N be a δ pseudoperiodic orbit of Eq. (1), found perhaps by simple shooting, or by concatenating segmentsof several orbits. In general, the δ associated with such a crude pseudo orbit willnot be sufficiently small to apply our Periodic Orbit Shadowing Theorem. We want toreplace this pseudo periodic orbit by a nearby one with a smaller δ. Ideally there wouldbe a nearby sequence of points {x k }k=0 N and a sequence of times {t k}k=0 N such thatx k+1 = φ t k(x k ) for k = 0, ..., N − 1x 0 = ϕ t N(x N ).We write x k = y k + z k , where z k is orthogonal to f (y k ), and t k = h k + s k . Sowe need to solve the equationsz k+1 = φ h k+s k(y k + z k ) − y k+1 for k = 0, ..., N − 1z 0 = ϕ h N+s N(y N + z N ) − y 0 .As in Newton’s method, we linearize:φ h k+s k(y k + z k ) − y k+1 ≈ f (φ h k(y k ))s k + Dφ h k(y k )z k + φ h k(y k ) − y k+1 .Next we writez k = S k u k ,where u k ∈ IR n−1 and {S k }k=0 N is a sequence of n × (n − 1) matrices chosen so that[ f (y k )/‖ f (y k )‖ S k ] is orthogonal. So now we solve the linear equationsS k+1 u k+1 = f (y k+1 )s k + Dφ h k(y k )S k u k + g k for k = 0, ..., N − 1S 0 u 0 = f (y N )s N + Dφ h N(y N )S N u N + g Nfor s k and u k , where g k = φ h k(y k ) − y k+1 . Multiplying each equation in the first setby S ∗ k+1 and f (y k+1) ∗ and multiplying the last equation by S ∗ 0 and f (y 0), under the