Stabilization via symmetry switching in hybrid dynamical systems
Sebastian Hage-Packhäuser Michael Dellnitz
Discrete & Continuous Dynamical Systems - B 2011, 16(1): 239-263 doi: 10.3934/dcdsb.2011.16.239
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.
keywords: symmetries stability. Hybrid systems
The efficient approximation of coherent pairs in non-autonomous dynamical systems
Michael Dellnitz Christian Horenkamp
Discrete & Continuous Dynamical Systems - A 2012, 32(9): 3029-3042 doi: 10.3934/dcds.2012.32.3029
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.
keywords: Non-autonomous dynamical systems transfer operators. almost invariant sets coherent structures
Topological method for rigorously computing periodic orbits using Fourier modes
Anthony W. Baker Michael Dellnitz Oliver Junge
Discrete & Continuous Dynamical Systems - A 2005, 13(4): 901-920 doi: 10.3934/dcds.2005.13.901
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].
keywords: periodic orbit set oriented methods Dynamical system computer assisted proof.
On the computation of attractors for delay differential equations
Michael Dellnitz Mirko Hessel-Von Molo Adrian Ziessler
Journal of Computational Dynamics 2016, 3(1): 93-112 doi: 10.3934/jcd.2016005
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.
keywords: set oriented numerical methods. delay differential equations Attractors invariant manifolds infinite dimensional dynamical systems embedding theory
The numerical detection of connecting orbits
Michael Dellnitz O. Junge B Thiere
Discrete & Continuous Dynamical Systems - B 2001, 1(1): 125-135 doi: 10.3934/dcdsb.2001.1.125
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.
keywords: Heteroclinic orbit detection of connecting orbits.

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