Global bifurcation of homoclinic trajectories of discrete dynamical systems

We prove the existence of an unbounded connected branch of nontrivial homoclinic trajectories of a family of discrete nonautonomous asymptotically hyperbolic systems parametrized by a circle under assumptions involving the topological properties of the asymptotic stable bundles.Comment: 28 pages. arXiv admin note: text overlap with arXiv:1111.140

Topology and Homoclinic Trajectories of Discrete Dynamical Systems

We show that nontrivial homoclinic trajectories of a family of discrete, nonautonomous, asymptotically hyperbolic systems parametrized by a circle bifurcate from a stationary solution if the asymptotic stable bundles Es(+{\infty}) and Es(-{\infty}) of the linearization at the stationary branch are twisted in different ways.Comment: 19 pages, canceled the appendix (Properties of the index bundle) in order to avoid any text overlap with arXiv:1005.207

The Index Bundle and Multiparameter Bifurcation for Discrete Dynamical Systems

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

Invariant manifolds with asymptotic phase for nonautonomous difference equations

AbstractFor autonomous difference equations with an invariant manifold, conditions are known which guarantee that a solution approaching this manifold eventually behaves like a solution on this manifold. In this paper, we extend the fundamental result in this context to difference equations which are nonautonomous and whose solutions are guaranteed only in forward time

Bet-hedging in stochastically switching environments.

We investigate the evolution of bet-hedging in a population that experiences a stochastically switching environment by means of adaptive dynamics. The aim is to extend known results to the situation at hand, and to deepen the understanding of the range of validity of these results. We find three different types of evolutionarily stable strategies (ESSs) depending on the frequency at which the environment changes: for a rapid change, a monomorphic phenotype adapted to the mean environment; for an intermediate range, a bimorphic bet-hedging phenotype; for slowly changing environments, a monomorphic phenotype adapted to the current environment. While the last result is only obtained by means of heuristic arguments and simulations, the first two results are based on the analysis of Lyapunov exponents for stochastically switching systems