Thorsten Hüls

Homoclinic trajectories of non-autonomous maps

For non-autonomous difference equations of the form we consider homoclinic trajectories. These are pairs of trajectories that converge in both time directions towards each other. Assuming hyperbolicity, we derive a numerical method to compute homoclinic trajectories in two steps. In the first step, one trajectory is approximated by the solution of a boundary value problem and precise error estimates are given. In particular, influences of parameters with |n| large are discussed in detail. A second trajectory that is homoclinic to the first one is computed in a subsequent step as follows. We transform the original system into a topologically equivalent form having 0 as an n-independent fixed point. Applying the boundary value ansatz to the transformed system, we obtain a non-autonomous homoclinic orbit, converging towards the origin (T. Hüls, J. Difference Equ. Appl. 12(11) (2006), pp. 1103–1126). Transforming back to the original coordinates leads to the desired homoclinic trajectories. The numerical method and the validity of the error estimates are illustrated by examples.

Computing Sacker-Sell spectra in Discrete Time Dynamical Systems

In this paper we develop boundary value methods for detecting Sacker-Sell spectra in discrete time dynamical systems. The algorithms are advancements of earlier methods for computing projectors of exponential dichotomies. The first method is based on the projector residual

NUMERICAL ANALYSIS OF DEGENERATE CONNECTING ORBITS FOR MAPS

P^2-P

A model function for polynomial rates in discrete dynamical systems

. If this residual is large, then the difference equation has no exponential dichotomy. Further criterions for detecting Sacker-Sell spectral intervals are the norm of end points and midpoints of the solution of a specific boundary value problem. Refined error estimates for the underlying approximation process are given, and the resulting algorithms are applied to an example with known continuous Sacker-Sell spectrum, as well as to the variational equation along orbits of Henons map.

Bifurcation of Connecting Orbits with One Nonhyperbolic Fixed Point for Maps

This paper contains a survey of numerical methods for connecting orbits in discrete dynamical systems. Special emphasis is put on degenerate cases where either the orbit loses transversality or one of its endpoints loses hyperbolicity. Numerical methods that approximate the connecting orbits by finite orbit sequences are described in detail and theoretical results on the error analysis are provided. For most of the degenerate cases we present examples and numerical results that illustrate the applicability of the methods and the validity of the error estimates.

Homoclinic orbits of non-autonomous maps and their approximation

In this paper, we construct a one-dimensional map with a nonhyperbolic fixed point at zero for which the orbits converging to zero and the solution of the associated variational equation can be determined explicitly. We extend the construction to parameterized systems where the fixed point undergoes bifurcations. Applications are indicated to heteroclinic orbits that connect a hyperbolic to a nonhyperbolic fixed point with one-dimensional center manifold.

Error estimates for approximating non-hyperbolic heteroclinic orbits of maps

In this paper we consider the bifurcation of transversal heteroclinic orbits in discrete time dynamical systems. We assume that a nonhyperbolic transversal heteroclinic orbit exists at some critical parameter value. This situation appears, for example, when one end point undergoes a fold or flip bifurcation. In these two cases the bifurcation analysis of the orbit is performed in detail. In particular, we prove, using implicit function techniques, that the orbit can be continued beyond the bifurcation point. Finally, we show numerical computations for the fold and for the flip bifurcations.

A Contour Algorithm for Computing Stable Fiber Bundles of Nonautonomous, Noninvertible Maps

We consider homoclinic orbits in non-autonomous discrete time dynamical systems of the form where it is assumed that an n independent fixed point exists. A numerical method for computing finite approximations of transversal homoclinic orbits is introduced and a detailed error analysis is presented. The non-autonomous setup requires special tools. We prove that the analytic condition of transversality of the orbit corresponds to a transversal intersection of the corresponding invariant fiber bundles. The approximation method and the validity of the error estimate is illustrated by an example.

On r-periodic orbits of k-periodic maps

Summary.In this paper we consider heteroclinic orbits in discrete time dynamical systems that connect a hyperbolic fixed point to a non-hyperbolic fixed point with a one-dimensional center direction. A numerical method for approximating the heteroclinic orbit by a finite orbit sequence is introduced and a detailed error analysis is presented. The loss of hyperbolicity requires special tools for proving the error estimate – the polynomial dichotomy of linear difference equations and a (partial) normal form transformation near the non-hyperbolic fixed point. This situation appears, for example, when one fixed point undergoes a flip bifurcation. For this case, the approximation method and the validity of the error estimate is illustrated by an example.

Qualitative Analysis of a Nonautonomous Beverton--Holt Ricker Model

Stable fiber bundles are the nonautonomous analogue of stable manifolds, and these objects provide valuable information on the underlying dynamics. We propose an algorithm for their computation that applies to a wide class of models, including noninvertible and nonautonomous discrete time systems. Precise error estimates are provided, and fiber bundles are computed for several examples.