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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, February 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 
Continuation and collapse of homoclinic tangles
Volume 1, Number 1, Pages: 71  109, 2014
WolfJürgen Beyn
and Thorsten Hüls
Abstract
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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.

3 
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.

4 
On dynamic mode decomposition: Theory and applications
Volume 1, Number 2, Pages: 391  421, 2014
Jonathan H. Tu,
Clarence W. Rowley,
Dirk M. Luchtenburg,
Steven L. Brunton
and J. Nathan Kutz
Abstract
References
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Originally introduced in the fluid mechanics community, dynamic mode decomposition (DMD) has emerged as a powerful tool for analyzing the dynamics of nonlinear systems.
However, existing DMD theory deals primarily with sequential time series for which the measurement dimension is much larger than the number of measurements taken.
We present a theoretical framework in which we define DMD as the eigendecomposition of an approximating linear operator.
This generalizes DMD to a larger class of datasets, including nonsequential time series.
We demonstrate the utility of this approach by presenting novel sampling strategies that increase computational efficiency and mitigate the effects of noise, respectively.
We also introduce the concept of linear consistency, which helps explain the potential pitfalls of applying DMD to rankdeficient datasets, illustrating with examples.
Such computations are not considered in the existing literature but can be understood using our more general framework.
In addition, we show that our theory strengthens the connections between DMD and Koopman operator theory.
It also establishes connections between DMD and other techniques, including the eigensystem realization algorithm (ERA), a system identification method, and linear inverse modeling (LIM), a method from climate science.
We show that under certain conditions, DMD is equivalent to LIM.

5 
Steady state bifurcations for the
KuramotoSivashinsky equation: A computer assisted proof
Volume 2, Number 1, Pages: 95  142, 2015
Piotr Zgliczyński
Abstract
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We apply the method of selfconsistent bounds to prove the
existence of multiple steady state bifurcations for
KuramotoSivashinski PDE on the line with odd and periodic
boundary conditions.

6 
Attractionbased computation of hyperbolic Lagrangian coherent structures
Volume 2, Number 1, Pages: 83  93, 2015
Daniel Karrasch,
Mohammad Farazmand
and George Haller
Abstract
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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.

7 
Preface:
Special issue on the occasion of the 4th International Workshop on SetOriented Numerics (SON 13, Dresden, 2013)
Volume 2, Number 1, Pages: i  ii, 2015
Gary Froyland,
Oliver Junge
and Kathrin PadbergGehle
Abstract
Full Text
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This issue comprises manuscripts collected on the occasion of the 4th International Workshop on SetOriented 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 setoriented 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.
For more information please click the “Full Text” above.

8 
Modularity of directed networks: Cycle decomposition approach
Volume 2, Number 1, Pages: 1  24, 2015
Nataša Djurdjevac Conrad,
Ralf Banisch
and Christof Schütte
Abstract
References
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The problem of decomposing networks into modules (or clusters) has gained much attention in recent years, as it can account for a coarsegrained 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.

9 
Symmetry exploiting control of hybrid mechanical systems
Volume 2, Number 1, Pages: 25  50, 2015
Kathrin Flasskamp,
Sebastian HagePackhäuser
and Sina OberBlöbaum
Abstract
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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.

10 
An elementary way to rigorously estimate convergence to equilibrium
and escape rates
Volume 2, Number 1, Pages: 51  64, 2015
Stefano Galatolo,
Isaia Nisoli
and Benoît Saussol
Abstract
References
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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.

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