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### Open Access Journals

DCDS-B

With a view to stabilization issues of hybrid systems exhibiting a regular structure in terms of symmetry, we introduce the concept of

*symmetry switching*and relate symmetry-induced switching strategies to the asymptotic stability of switched linear systems. To this end, a general notion of*hybrid symmetries*for switched systems is established whereupon*orbital switching*is treated which builds on the existence of hybrid symmetries. In the main part, we formulate and prove sufficient conditions for asymptotic stability under slow symmetry switching. As an example of both theoretical and practical interest, we examine time-varying networks of dynamical systems and perform stabilization by means of orbital switching. Behind all that, this work is meant to provide the groundwork for the treatment of equivariant bifurcation phenomena of hybrid systems.
DCDS

The aim of this paper is the construction of numerical tools for
the efficient approximation of transport phenomena in non-autonomous dynamical systems.
We focus on

*transfer operator*methods which have been developed in the last years for the treatment of non-autonomous dynamical systems. For instance Froyland et al. [11] proposed a method for the approximation of so-called*coherent pairs*-- these pairs of sets represent time-dependent slowly mixing structures -- by thresholding singular vectors of a normalized transfer operator over a fixed time-interval. In principle such transfer operator methods involve long term simulations of trajectories on the whole state space. In our main result we show that transport phenomena over a fixed (long) time horizon imply the existence of*almost invariant sets*over shorter time intervals if the transport process is slow enough. This fact is used to formulate an algorithm that preselects part of state space as a candidate for containing one of the sets of a coherent pair. By this we significantly reduce the related numerical effort.
DCDS

We present a technique for the rigorous computation of periodic
orbits in certain ordinary differential equations. The method
combines set oriented numerical techniques for the computation of
invariant sets in dynamical systems with topological index
arguments. It not only allows for the proof of existence of
periodic orbits but also for a precise (and rigorous) approximation
of these. As an example we compute a periodic orbit for a
differential equation introduced in [2].

JCD

In this work we present a novel framework for the computation of
finite dimensional invariant sets of infinite dimensional dynamical
systems. It extends a classical subdivision technique [7] for the computation of
such objects of finite dimensional systems to the infinite
dimensional case by utilizing results on embedding techniques for
infinite dimensional systems. We show how to implement this approach for
the analysis of delay differential equations and illustrate the
feasibility of our implementation by computing invariant sets for
three different delay differential equations.

DCDS-B

We present a new technique for the numerical detection and
localization of connecting orbits between hyperbolic invariant
sets in parameter
dependent dynamical systems. This method is based on set-oriented multilevel
methods for the computation of invariant manifolds and it can be applied
to systems of moderate dimension. The main idea of
the algorithm is to detect intersections of coverings of the stable and
unstable manifolds of the invariant sets on different levels of the
approximation. We demonstrate the applicability of the new method by
three examples.

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