Wolf-Juergen Beyn

Wolf-Juergen Beyn, Yuri Latushkin, Jens Rottmann-Matthes

Investigating the stability of nonlinear waves often leads to linear or nonlinear eigenvalue problems for differential operators on unbounded domains. In this paper we propose to detect and approximate the point spectra of such operators (and the associated eigenfunctions) via contour integrals of solutions to resolvent equations. The approach is based on Keldysh' theorem and extends a recent method for matrices depending analytically on the eigenvalue parameter. We show that errors are well-controlled under very general assumptions when the resolvent equations are solved via boundary value problems on finite domains. Two applications are presented: an analytical study of Schrödinger operators on the real line as well as on bounded intervals and a numerical study of the FitzHugh-Nagumo system. We also relate the contour method to the well-known Evans function and show that our approach provides an alternative to evaluating and computing its zeroes.

Wolf-Juergen Beyn, Thorsten Huels

By a classical theorem transversal homoclinic points of maps lead to shift dynamics on a maximal invariant set, also referred to as a homoclinic tangle. In this paper we study the fate of homoclinic tangles in parameterized systems from the viewpoint of numerical continuation and bifurcation theory. The bifurcation result shows that the maximal invariant set near a homoclinic tangency, where two homoclinic tangles collide, can be characterized by a system of bifurcation equations that is indexed by a symbolic sequence. For the Hénon family we investigate in detail the bifurcation structure of multi-humped orbits originating from several tangencies. The homoclinic network found by numerical continuation is explained by combining our bifurcation result with graph-theoretical arguments.