RENDICONTI DEL SEMINARIO MATEMATICO

Shadowing in ordinary differential equations 91restricted three-body problem, is the presence of transversal homoclinic points.He called such points “doubly asymptotic” because they are asymptotic bothin forward and backward times to a fixed point or a periodic orbit. Birkhoff[4] proved that every homoclinic point of a two-dimensional diffeomorphism isaccumulated by periodic orbits. Smale [59] confirmed Poincaré’s observation byproving that a transversal homoclinic point of a diffeomorphism in dimensiontwo and higher is contained in a hyperbolic set in which the periodic orbits areinfinitely many and dense. Sil’nikov [55] showed that a similar result holds forflows. Recently, it has been conjectured by Palis and Takens [45] that genericallychaotic orbits occur if and only if there is a transversal homoclinic orbit. This isindeed the case for continuous interval maps as shown in [6].In spite of the remarkable mathematical results above, transversal homoclinicorbits are quite difficult to exhibit in specific chaotic flows. Even the periodicorbits, to which the homoclinic orbits are to be doubly asymptotic, are hard tocome by. Recently in [22], we have formulated a practical notion of a pseudohomoclinic, more generally pseudo connecting, orbit and proved a shadowingtheorem that guarantees the existence of transversal homoclinic, or heteroclinic,orbits to periodic orbits of differential equations. The hypotheses of this theoremcan be verified for specific flows with the aid of a computer, thus enabling usto prove the existence of a multitude of periodic orbits and transversal orbitsconnecting them in, for example, yet again, the chaotic Lorenz Equations.Contents? Here is a section-by-section description of the contents of the remainder ofthis paper:• In Section 2, we first give definitions of an infinite pseudo orbit and itsshadowing by a true orbit. Then we present two infinite-time shadowingresults, one for pseudo orbits lying in hyperbolic invariant sets, and anotherfor a single pseudo orbit in terms of a certain operator.• In Section 3, shadowing definitions for finite pseudo orbits and a FinitetimeShadowing Theorem for non-uniformly hyperbolic systems are formulated.This theorem is significant in proving the existence of true orbitsnear numerically computed ones for long time intervals.• In Section 4, the shadowing of pseudo periodic orbits is considered. ThePeriodic Shadowing Theorem stated here is very effective in establishingthe existence of periodic orbits, including unstable ones in dimensions threeand higher.• In Section 5, the notion of a pseudo connecting orbit connecting two pseudoperiodic orbits is formulated. Then a Connection Orbit Shadowing Theoremthat guarantees the existence and transversality of a true connectingorbit between true periodic orbits is stated. In the particular case when thetwo periodic orbits coincide we have a Homoclinic Shadowing Theorem,with the aid of which existence of chaos, in the sense of Poincaré, can berigorously established in specific systems.