Preprint

Constructing pseudo-orbits on invariant manifolds of maps as seeds for boundary value problems

Dana C'Julio, Sanaz Amani, Sam Doak, Bernd Krauskopf, and Hinke M. Osinga

Abstract

An important tool for understanding complicated dynamics in the iteration of a map is the ability to find certain orbits of interest. In particular, connecting orbits between saddle fixed points and/or saddle periodic points are special orbits that are closely associated with intersections of stable or unstable manifolds. They can be found numerically and then be continued in parameters as solutions of suitably formulated boundary value problems (BVPs). We address the key issue that one needs to find a seed solution when solving a BVP with Newton's method. In our setting, a seed solution takes the form of a finite sequence of points that almost satisfies the required conditions of the BVP. To construct such seeds, we modify a numerical method that finds and represents a one-dimensional stable or unstable manifold of a diffeomorphism in such a way that a so-called pseudo-orbit for any computed point is readily available from the data. We explain how pseudo-orbits of relevant points on a computed invariant manifold can successfully be used as seeds for solving a variety of BVPs in a systematic and efficient way. We illustrate our findings with two three-dimensional diffeomorphism with complicated dynamics, by computing many orbits on an invariant manifold that end in a chosen section, as well as numerous homoclinic and heteroclinic connectiong orbits.

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