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Volume 1, 2014

Journal of Computational Dynamics

2014 , Volume 1 , Issue 1

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Global invariant manifolds near a Shilnikov homoclinic bifurcation
Pablo Aguirre , Bernd Krauskopf and  Hinke M. Osinga
2014, 1(1): 1-38 doi: 10.3934/jcd.2014.1.1 +[Abstract](89) +[PDF](6559.1KB)
We consider a three-dimensional vector field with a Shilnikov homoclinic orbit that converges to a saddle-focus equilibrium in both forward and backward time. The one-parameter unfolding of this global bifurcation depends on the sign of the saddle quantity. When it is negative, breaking the homoclinic orbit produces a single stable periodic orbit; this is known as the simple Shilnikov bifurcation. However, when the saddle quantity is positive, the mere existence of a Shilnikov homoclinic orbit induces complicated dynamics, and one speaks of the chaotic Shilnikov bifurcation; in particular, one finds suspended horseshoes and countably many periodic orbits of saddle type. These well-known and celebrated results on the Shilnikov homoclinic bifurcation have been obtained by the classical approach of reducing a Poincaré return map to a one-dimensional map.
    In this paper, we study the implications of the transition through a Shilnikov bifurcation for the overall organization of the three-dimensional phase space of the vector field. To this end, we focus on the role of the two-dimensional global stable manifold of the equilibrium, as well as those of bifurcating saddle periodic orbits. We compute the respective two-dimensional global manifolds, and their intersection curves with a suitable sphere, as families of orbit segments with a two-point boundary-value-problem setup. This allows us to determine how the arrangement of global manifolds changes through the bifurcation and how this influences the topological organization of phase space. For the simple Shilnikov bifurcation, we show how the stable manifold of the saddle focus forms the basin boundary of the bifurcating stable periodic orbit. For the chaotic Shilnikov bifurcation, we find that the stable manifold of the equilibrium is an accessible set of the stable manifold of a chaotic saddle that contains countably many periodic orbits of saddle type. In intersection with a suitably chosen sphere we find that this stable manifold is an indecomposable continuum consisiting of infinitely many closed curves that are locally a Cantor bundle of arcs.
The computation of convex invariant sets via Newton's method
R. Baier , M. Dellnitz , M. Hessel-von Molo , S. Sertl and  I. G. Kevrekidis
2014, 1(1): 39-69 doi: 10.3934/jcd.2014.1.39 +[Abstract](81) +[PDF](719.2KB)
In this paper we present a novel approach to the computation of convex invariant sets of dynamical systems. Employing a Banach space formalism to describe differences of convex compact subsets of $\mathbb{R}^n$ by directed sets, we are able to formulate the property of a convex, compact set to be invariant as a zero-finding problem in this Banach space. We need either the additional restrictive assumption that the image of sets from a subclass of convex compact sets under the dynamics remains convex, or we have to convexify these images. In both cases we can apply Newton's method in Banach spaces to approximate such invariant sets if an appropriate smoothness of a set-valued map holds. The theoretical foundations for realizing this approach are analyzed, and it is illustrated first by analytical and then by numerical examples.
Continuation and collapse of homoclinic tangles
Wolf-Jürgen Beyn and  Thorsten Hüls
2014, 1(1): 71-109 doi: 10.3934/jcd.2014.1.71 +[Abstract](47) +[PDF](2501.4KB)
By a classical theorem transversal homoclinic points of maps lead to shift dynamics on a maximal invariant set, also referred to as a homoclinic tangle. In this paper we study the fate of homoclinic tangles in parameterized systems from the viewpoint of numerical continuation and bifurcation theory. The new bifurcation result shows that the maximal invariant set near a homoclinic tangency, where two homoclinic tangles collide, can be characterized by a system of bifurcation equations that is indexed by a symbolic sequence. These bifurcation equations consist of a finite or infinite set of hilltop normal forms known from singularity theory. For the Hénon family we determine numerically the connected components of branches with multi-humped homoclinic orbits that pass through several tangencies. The homoclinic network found by numerical continuation is explained by combining our bifurcation result with graph-theoretical arguments.
An equation-free approach to coarse-graining the dynamics of networks
Katherine A. Bold , Karthikeyan Rajendran , Balázs Ráth and  Ioannis G. Kevrekidis
2014, 1(1): 111-134 doi: 10.3934/jcd.2014.1.111 +[Abstract](57) +[PDF](333.5KB)
We propose and illustrate an approach to coarse-graining the dynamics of evolving networks, i.e., networks whose connectivity changes dynamically. The approach is based on the equation-free framework: short bursts of detailed network evolution simulations are coupled with lifting and restriction operators that translate between actual network realizations and their appropriately chosen coarse observables. This framework is used here to accelerate temporal simulations through coarse projective integration, and to implement coarse-grained fixed point algorithms through matrix-free Newton-Krylov. The approach is illustrated through a very simple network evolution example, for which analytical approximations to the coarse-grained dynamics can be independently obtained, so as to validate the computational results. The scope and applicability of the approach, as well as the issue of selection of good coarse observables are discussed.
A closing scheme for finding almost-invariant sets in open dynamical systems
Gary Froyland , Philip K. Pollett and  Robyn M. Stuart
2014, 1(1): 135-162 doi: 10.3934/jcd.2014.1.135 +[Abstract](81) +[PDF](1584.7KB)
We explore the concept of metastability or almost-invariance in open dynamical systems. In such systems, the loss of mass through a ``hole'' occurs in the presence of metastability. We extend existing techniques for finding almost-invariant sets in closed systems to open systems by introducing a closing operation that has a small impact on the system's metastability.
Global optimal feedbacks for stochastic quantized nonlinear event systems
Stefan Jerg , Oliver Junge and  Marcus Post
2014, 1(1): 163-176 doi: 10.3934/jcd.2014.1.163 +[Abstract](41) +[PDF](911.6KB)
We consider nonlinear control systems for which only quantized and event-triggered state information is available and which are subject to random delays and losses in the transmission of the state to the controller. We present an optimization based approach for computing globally stabilizing controllers for such systems. Our method is based on recently developed set oriented techniques for transforming the problem into a shortest path problem on a weighted hypergraph. We show how to extend this approach to a system subject to a stochastic parameter and propose a corresponding model for dealing with transmission delays.
On the consistency of ensemble transform filter formulations
Sebastian Reich and  Seoleun Shin
2014, 1(1): 177-189 doi: 10.3934/jcd.2014.1.177 +[Abstract](49) +[PDF](548.7KB)
In this paper, we consider the data assimilation problem for perfect differential equation models without model error and for either continuous or intermittent observational data. The focus will be on the popular class of ensemble Kalman filters which rely on a Gaussian approximation in the data assimilation step. We discuss the impact of this approximation on the temporal evolution of the ensemble mean and covariance matrix. We also discuss options for reducing arising inconsistencies, which are found to be more severe for the intermittent data assimilation problem. Inconsistencies can, however, not be completely eliminated due to the classic moment closure problem. It is also found for the Lorenz-63 model that the proposed corrections only improve the filter performance for relatively large ensemble sizes.
Modularity revisited: A novel dynamics-based concept for decomposing complex networks
Marco Sarich , Natasa Djurdjevac Conrad , Sharon Bruckner , Tim O. F. Conrad and  Christof Schütte
2014, 1(1): 191-212 doi: 10.3934/jcd.2014.1.191 +[Abstract](69) +[PDF](1083.3KB)
Finding modules (or clusters) in large, complex networks is a challenging task, in particular if one is not interested in a full decomposition of the whole network into modules. We consider modular networks that also contain nodes that do not belong to one of modules but to several or to none at all. A new method for analyzing such networks is presented. It is based on spectral analysis of random walks on modular networks. In contrast to other spectral clustering approaches, we use different transition rules of the random walk. This leads to much more prominent gaps in the spectrum of the adapted random walk and allows for easy identification of the network's modular structure, and also identifying the nodes belonging to these modules. We also give a characterization of that set of nodes that do not belong to any module, which we call transition region. Finally, by analyzing the transition region, we describe an algorithm that identifies so called hub-nodes inside the transition region that are important connections between modules or between a module and the rest of the network. The resulting algorithms scale linearly with network size (if the network connectivity is sparse) and thus can also be applied to very large networks.



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