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Journal of Computational Dynamics

January 2016 , Volume 3 , Issue 1

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Discretization strategies for computing Conley indices and Morse decompositions of flows
Konstantin Mischaikow, Marian Mrozek and Frank Weilandt
2016, 3(1): 1-16 doi: 10.3934/jcd.2016001 +[Abstract](3248) +[PDF](1053.2KB)
Conley indices and Morse decompositions of flows can be found by using algorithms which rigorously analyze discrete dynamical systems. This usually involves integrating a time discretization of the flow using interval arithmetic. We compare the old idea of fixing a time step as a parameter to a time step continuously varying in phase space. We present an example where this second strategy necessarily yields better numerical outputs and prove that our outputs yield a valid Morse decomposition of the given flow.
Towards a formal tie between combinatorial and classical vector field dynamics
Tomasz Kaczynski, Marian Mrozek and Thomas Wanner
2016, 3(1): 17-50 doi: 10.3934/jcd.2016002 +[Abstract](4438) +[PDF](562.5KB)
Forman's combinatorial vector fields on simplicial complexes are a discrete analogue of classical flows generated by dynamical systems. Over the last decade, many notions from dynamical systems theory have found analogues in this combinatorial setting, such as for example discrete gradient flows and Forman's discrete Morse theory. So far, however, there is no formal tie between the two theories, and it is not immediately clear what the precise relation between the combinatorial and the classical setting is. The goal of the present paper is to establish such a formal tie on the level of the induced dynamics. Following Forman's paper [6], we work with possibly non-gradient combinatorial vector fields on finite simplicial complexes, and construct a flow-like upper semi-continuous acyclic-valued mapping on the underlying topological space whose dynamics is equivalent to the dynamics of Forman's combinatorial vector field on the level of isolated invariant sets and isolating blocks.
On the numerical approximation of the Perron-Frobenius and Koopman operator
Stefan Klus, Péter Koltai and Christof Schütte
2016, 3(1): 51-79 doi: 10.3934/jcd.2016003 +[Abstract](9564) +[PDF](2004.7KB)
Information about the behavior of dynamical systems can often be obtained by analyzing the eigenvalues and corresponding eigenfunctions of linear operators associated with a dynamical system. Examples of such operators are the Perron-Frobenius and the Koopman operator. In this paper, we will review di erent methods that have been developed over the last decades to compute nite-dimensional approximations of these in nite-dimensional operators - in particular Ulam's method and Extended Dynamic Mode Decomposition (EDMD) - and highlight the similarities and di erences between these approaches. The results will be illustrated using simple stochastic di erential equations and molecular dynamics examples.
Rigorous enclosures of rotation numbers by interval methods
Anna Belova
2016, 3(1): 81-91 doi: 10.3934/jcd.2016004 +[Abstract](3349) +[PDF](515.1KB)
We apply set-valued numerical methods to compute an accurate enclosure of the rotation number. The described algorithm is supplemented with a method of proving the existence of periodic points, which is used to check the rationality of the rotation number. A few numerical experiments are presented to show that the implementation of interval methods produces a good enclosure of the rotation number of a circle map.
On the computation of attractors for delay differential equations
Michael Dellnitz, Mirko Hessel-Von Molo and Adrian Ziessler
2016, 3(1): 93-112 doi: 10.3934/jcd.2016005 +[Abstract](4433) +[PDF](2548.8KB)
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.

2021 CiteScore: 1.7




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