The index bundle and multiparameter bifurcation for discrete dynamical systems

Abstract

We develop a K-theoretic approach to multiparameter bifurcation theory of homoclinic solutions of discrete non-autonomous dynamical systems from a branch of stationary solutions. As a byproduct we obtain a family index theorem for asymptotically hyperbolic linear dynamical systems which is of independent interest. In the special case of a single parameter, our bifurcation theorem weakens the assumptions in previous work by Pejsachowicz and the first author.

Fredholm theory of families of discrete dynamical systems and its applications to bifurcation theory

“…In the final part of this work, we apply the obtained index theorem to bifurcation theory along the lines of our previous work [SW17], which substantially widens its applicability. Indeed, the only example that we could give in [SW17] was a two dimensional system parametrized by a torus that was adapted from an example of [Pe08a]. Here we use our new approach to construct a whole class of examples for general parameter spaces by perturbing simple systems.…”

“…of linear discrete dynamical systems, where A n (λ) = D 2 f n (λ, 0). In our previous work [SW17] we imposed assumptions from [PS13] on f that allow to study the bifurcation problem of (1) by topological methods in the Banach space ℓ 0 (R d ) of all sequences converging to 0 for n → ±∞. A fundamental assumption in [SW17] is that the matrices A n (λ) in (2) are asymptotically hyperbolic, i.e., they converge for n → ±∞ uniformly to families of hyperbolic matrices.…”

“…In our previous work [SW17] we imposed assumptions from [PS13] on f that allow to study the bifurcation problem of (1) by topological methods in the Banach space ℓ 0 (R d ) of all sequences converging to 0 for n → ±∞. A fundamental assumption in [SW17] is that the matrices A n (λ) in (2) are asymptotically hyperbolic, i.e., they converge for n → ±∞ uniformly to families of hyperbolic matrices. The most powerful concept for studying homoclinic trajectories of dynamical systems are socalled exponential dichotomies (ED for short), which generalize the concept of hyperbolicity from autonomous to nonautonomous linear equations (c.f., e.g., [Pa84,Pa88,Po10,Po11a,Po11b]).…”

“…As it can be shown that every asymptotically hyperbolic system (2) has an ED, it is a natural question whether our assumptions in [SW17] can be weakened to this far more common setting. The aim of this article is to give an affirmative answer.…”

“…The main theorem of this article is a formula that computes the index bundle for discrete dynamical systems (2) having an ED in terms of the associated projections. In the special case that (2) is asymptotically hyperbolic, we reobtain the family index theorem that we proved in [SW17], where we need the perturbation theory for EDs that we previously established.…”