Computing Sacker–Sell spectra in Discrete Time Dynamical Systems

Abstract

In this paper we develop two boundary value methods for detecting SackerSell 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 P P − P . If this residual is large, then the difference equation has no exponential dichotomy. A second criterion for detecting Sacker-Sell spectral intervals is the norm of end points 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 Hénon's map.

Qualitative Analysis of a Nonautonomous Beverton--Holt Ricker Model

“…Changes in them are caused by multiple nonautonomous bifurcation scenarios (see [23,31,12,27]), among which we particularly address counterparts to transcritical and flip (period doubling) bifurcations. The associated stability transitions are described using Bohl exponents, which are boundary points of the dichotomy (also known as Sacker-Sell) spectrum (see [33] and [2,1,14]). Gaps in the dichotomy spectrum in turn give rise to invariant manifolds, and we particularly perform a nonautonomous center manifold reduction (cf.…”

“…[30]). In this endeavor it turns out at an early stage that up-to-date numerical techniques are indispensable when quantitative information on the dichotomy spectrum [13,14] or the continuation of bounded entire solutions [11,15] is required. Also, explicit perturbation bounds for the persistence of hyperbolicity under nonautonomous forcing are given in a representative special case.…”

“…Hence, the appropriate tools for stability investigations are the dichotomy spectrum and Bohl exponents (cf. [1,14]); for readers unfamiliar with this concept, we have summarized some essential aspects in Appendix B. When dealing with Bohl exponents let us implicitly assume that the associated sequences have bounded inverses (cf.…”

Fine Structure of the Dichotomy Spectrum

“…This indicates that numerical methods to approximate Σ(A) (cf. [23]) remain reliable despite the upper-semicontinuity of Σ.…”

“…[23]). However, for scalar, autonomous, periodic and asymptotically autonomous equations on Z, explicit formulas for Σ ED (A) have been derived in [7, Sect.…”