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Journal of Computational Dynamics

2015 , Volume 2 , Issue 1

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Preface: Special issue on the occasion of the 4th International Workshop on Set-Oriented Numerics (SON 13, Dresden, 2013)
Gary Froyland, Oliver Junge and Kathrin Padberg-Gehle
2015, 2(1): i-ii doi: 10.3934/jcd.2015.2.1i +[Abstract](314) +[PDF](86.8KB)
Abstract:
This issue comprises manuscripts collected on the occasion of the 4th International Workshop on Set-Oriented Numerics which took place at the Technische Universität Dresden in September 2013. The contributions cover a broad spectrum of different subjects in computational dynamics ranging from purely discrete problems on graphs to computer assisted proofs of bifurcations in dissipative PDEs. In many cases, ideas related to set-oriented paradigms turn out to be useful in the computations, for example by quantizing the state space, or by using interval arithmetic to perform rigorous computations.

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Modularity of directed networks: Cycle decomposition approach
Nataša Djurdjevac Conrad, Ralf Banisch and Christof Schütte
2015, 2(1): 1-24 doi: 10.3934/jcd.2015.2.1 +[Abstract](464) +[PDF](1434.0KB)
Abstract:
The problem of decomposing networks into modules (or clusters) has gained much attention in recent years, as it can account for a coarse-grained description of complex systems, often revealing functional subunits of these systems. A variety of module detection algorithms have been proposed, mostly oriented towards finding hard partitionings of undirected networks. Despite the increasing number of fuzzy clustering methods for directed networks, many of these approaches tend to neglect important directional information. In this paper, we present a novel random walk based approach for finding fuzzy partitions of directed, weighted networks, where edge directions play a crucial role in defining how well nodes in a module are interconnected. We will show that cycle decomposition of a random walk process connects the notion of network modules and information transport in a network, leading to a new, symmetric measure of node communication. Finally, we will use this measure to introduce a communication graph, for which we will show that although being undirected it inherits important directional information of modular structures from the original network.
Symmetry exploiting control of hybrid mechanical systems
Kathrin Flasskamp, Sebastian Hage-Packhäuser and Sina Ober-Blöbaum
2015, 2(1): 25-50 doi: 10.3934/jcd.2015.2.25 +[Abstract](346) +[PDF](896.4KB)
Abstract:
Symmetry properties such as invariances of mechanical systems can be beneficially exploited in solution methods for control problems. A recently developed approach is based on quantization by so called motion primitives. A library of these motion primitives forms an artificial hybrid system. In this contribution, we study the symmetry properties of motion primitive libraries of mechanical systems in the context of hybrid symmetries. Furthermore, the classical concept of symmetry in mechanics is extended to hybrid mechanical systems and an extended motion planning approach is presented.
An elementary way to rigorously estimate convergence to equilibrium and escape rates
Stefano Galatolo, Isaia Nisoli and Benoît Saussol
2015, 2(1): 51-64 doi: 10.3934/jcd.2015.2.51 +[Abstract](332) +[PDF](505.8KB)
Abstract:
We show an elementary method to obtain (finite time and asymptotic) computer assisted explicit upper bounds on convergence to equilibrium (decay of correlations) and escape rates for systems satisfying a Lasota Yorke inequality. The bounds are deduced from the ones of suitable approximations of the system's transfer operator. We also present some rigorous experiments on some nontrivial example.
Numerical event-based ISS controller design via a dynamic game approach
Lars Grüne and Manuela Sigurani
2015, 2(1): 65-81 doi: 10.3934/jcd.2015.2.65 +[Abstract](267) +[PDF](621.2KB)
Abstract:
We present an event-based numerical design method for an input-to-state practically stabilizing (ISpS) state feedback controller for perturbed nonlinear discrete time systems. The controllers are designed to be constant on quantization regions which are not assumed to be small. A transition of the state from one quantization region to another triggers an event upon which the control value changes.
    The controller construction relies on the conversion of the ISpS design problem into a robust controller design problem which is solved by a set oriented discretization technique followed by the solution of a dynamic game on a hypergraph. We present and analyze this approach with a particular focus on keeping track of the quantitative dependence of the resulting gain and the size of the exceptional region for practical stability from the design parameters of our event-based controller.
Attraction-based computation of hyperbolic Lagrangian coherent structures
Daniel Karrasch, Mohammad Farazmand and George Haller
2015, 2(1): 83-93 doi: 10.3934/jcd.2015.2.83 +[Abstract](394) +[PDF](2835.1KB)
Abstract:
Recent advances enable the simultaneous computation of both attracting and repelling families of Lagrangian Coherent Structures (LCS) at the same initial or final time of interest. Obtaining LCS positions at intermediate times, however, has been problematic, because either the repelling or the attracting family is unstable with respect to numerical advection in a given time direction. Here we develop a new approach to compute arbitrary positions of hyperbolic LCS in a numerically robust fashion. Our approach only involves the advection of attracting material surfaces, thereby providing accurate LCS tracking at low computational cost. We illustrate the advantages of this approach on a simple model and on a turbulent velocity data set.
Steady state bifurcations for the Kuramoto-Sivashinsky equation: A computer assisted proof
Piotr Zgliczyński
2015, 2(1): 95-142 doi: 10.3934/jcd.2015.2.95 +[Abstract](418) +[PDF](623.1KB)
Abstract:
We apply the method of self-consistent bounds to prove the existence of multiple steady state bifurcations for Kuramoto-Sivashinski PDE on the line with odd and periodic boundary conditions.

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