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ISSN 21582505 (online)

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JCD is focused on the intersection of computation with deterministic and stochastic dynamics. The mission of the journal is to publish papers that explore new computational methods for analyzing dynamic problems or use novel dynamical methods to improve computation. The subject matter of JCD includes both fundamental mathematical contributions and applications to problems from science and engineering. A nonexhaustive list of topics includes
1. Computation of phasespace structures and bifurcations
2. Multitimescale methods
3. Structurepreserving integration
4. Nonlinear and stochastic model reduction
5. Setvalued numerical techniques
6. Network and distributed dynamics
JCD includes both original research and survey papers that give a detailed and illuminating treatment of an important area of current interest.
The editorial board of JCD consists of worldleading researchers from mathematics, engineering, and science, all of whom are experts in both computational methods and the theory of dynamical systems.
JCD is a publication of the American Institute of Mathematical Sciences.
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TOP 10 Most Read Articles in JCD, May 2017
1 
Global invariant manifolds near a Shilnikov homoclinic bifurcation
Volume 1, Number 1, Pages: 1  38, 2014
Pablo Aguirre,
Bernd Krauskopf
and Hinke M. Osinga
Abstract
References
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We consider a threedimensional vector field with a Shilnikov homoclinic orbit that converges to a saddlefocus equilibrium in both forward and backward time. The oneparameter 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 wellknown and celebrated results on the Shilnikov homoclinic bifurcation have been obtained by the classical approach of reducing a Poincaré return map to a onedimensional map.
In this paper, we study the implications of the transition through a Shilnikov bifurcation for the overall organization of the threedimensional phase space of the vector field. To this end, we focus on the role of the twodimensional global stable manifold of the equilibrium, as well as those of bifurcating saddle periodic orbits. We compute the respective twodimensional global manifolds, and their intersection curves with a suitable sphere, as families of orbit segments with a twopoint boundaryvalueproblem 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.

2 
The computation of convex invariant sets
via Newton's method
Volume 1, Number 1, Pages: 39  69, 2014
R. Baier,
M. Dellnitz,
M. Hesselvon Molo,
S. Sertl
and I. G. Kevrekidis
Abstract
References
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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 zerofinding 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 setvalued map
holds. The theoretical foundations for realizing this
approach are analyzed, and it is illustrated first by analytical
and then by numerical examples.

3 
Continuation and collapse of homoclinic tangles
Volume 1, Number 1, Pages: 71  109, 2014
WolfJürgen Beyn
and Thorsten Hüls
Abstract
References
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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 multihumped homoclinic orbits that pass through
several tangencies.
The homoclinic network found by numerical continuation is explained
by combining our bifurcation result with graphtheoretical arguments.

4 
An equationfree approach
to coarsegraining the dynamics of networks
Volume 1, Number 1, Pages: 111  134, 2014
Katherine A. Bold,
Karthikeyan Rajendran,
Balázs Ráth
and Ioannis G. Kevrekidis
Abstract
References
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We propose and illustrate an approach to coarsegraining the dynamics of evolving networks,
i.e., networks whose connectivity changes dynamically.
The approach is based on the equationfree 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 coarsegrained fixed point algorithms through matrixfree NewtonKrylov.
The approach is illustrated through a very simple network evolution example,
for which analytical approximations to the coarsegrained 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.

5 
A closing scheme for finding almostinvariant sets in open dynamical systems
Volume 1, Number 1, Pages: 135  162, 2014
Gary Froyland,
Philip K. Pollett
and Robyn M. Stuart
Abstract
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We explore the concept of metastability or almostinvariance 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 almostinvariant sets in closed systems to open systems by introducing a closing operation that has a small impact on the system's metastability.

6 
Global optimal feedbacks for stochastic quantized nonlinear event systems
Volume 1, Number 1, Pages: 163  176, 2014
Stefan Jerg,
Oliver Junge
and Marcus Post
Abstract
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We consider nonlinear control systems for which only quantized and eventtriggered 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.

7 
On the consistency of ensemble transform filter formulations
Volume 1, Number 1, Pages: 177  189, 2014
Sebastian Reich
and Seoleun Shin
Abstract
References
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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 Lorenz63 model that the
proposed corrections only improve the filter performance for
relatively large ensemble sizes.

8 
Modularity revisited: A novel dynamicsbased concept for decomposing complex networks
Volume 1, Number 1, Pages: 191  212, 2014
Marco Sarich,
Natasa Djurdjevac Conrad,
Sharon Bruckner,
Tim O. F. Conrad
and Christof Schütte
Abstract
References
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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 hubnodes 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.

9 
Necessary and sufficient condition for the global stability of a delayed discretetime single neuron model
Volume 1, Number 2, Pages: 213  232, 2014
Ferenc A. Bartha
and Ábel Garab
Abstract
References
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We consider the global asymptotic stability of the trivial fixed point of the difference equation $x_{n+1}=m x_n\alpha \varphi(x_{n1})$, where $(\alpha,m) \in \mathbb{R}^2$ and $\varphi$ is a real function satisfying the discrete Yorke condition: $\min\{0,x\} \leq \varphi(x) \leq \max\{0,x\}$ for all $x\in \mathbb{R}$. If $\varphi$ is bounded then $(\alpha,m) \in [m1,1] \times [1,1]$, $(\alpha,m) \neq (0,1), (0,1)$ is necessary for the global stability of $0$. We prove that if $\varphi(x) \equiv \tanh(x)$, then this condition is sufficient as well.

10 
Reconstructing functions from random samples
Volume 1, Number 2, Pages: 233  248, 2014
Steve Ferry,
Konstantin Mischaikow
and Vidit Nanda
Abstract
References
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From a sufficiently large point sample lying on a compact Riemannian submanifold of Euclidean space, one can construct a simplicial complex which is homotopyequivalent to that manifold with high confidence. We describe a corresponding result for a Lipschitzcontinuous function between two such manifolds. That is, we outline the construction of a simplicial map which recovers the induced maps on homotopy and homology groups with high confidence using only finite sampled data from the domain and range, as well as knowledge of the image of every point sampled from the domain. We provide explicit bounds on the size of the point samples required for such reconstruction in terms of intrinsic properties of the domain, the codomain and the function. This reconstruction is robust to certain types of bounded sampling and evaluation noise.

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