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DCDS, series A includes peerreviewed original papers and invited expository papers on the theory and methods of analysis, differential equations and dynamical systems. This journal is committed to recording important new results in its field and maintains the highest standards of innovation and quality. To be published in this journal, an original paper must be correct, new, nontrivial and of interest to a substantial number of readers.
DCDS is published monthly in 2016 and is a publication of the American Institute of Mathematical Sciences. All rights reserved.
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TOP 10 Most Read Articles in DCDSA, October 2016
1 
$H^1$ Solutions of a class of fourth order nonlinear equations for image processing
Volume 10, Number 1/2, Pages: 349  366, 2003
John B. Greer
and Andrea L. Bertozzi
Abstract
Full Text
Related Articles
Recently fourth order equations of the form
$u_t = \nabla\cdot((\mathcal G(J_\sigma u)) \nabla \Delta u)$ have been proposed
for noise reduction and simplification of two dimensional images.
The operator $\mathcal G$ is a nonlinear functional involving
the gradient or Hessian of its argument, with decay in the far field.
The operator $J_\sigma$ is a standard mollifier.
Using ODE methods on Sobolev spaces,
we prove existence and uniqueness of solutions of this problem
for $H^1$ initial data.

2 
A biharmonicmodified forward time stepping
method for fourth order nonlinear diffusion equations
Volume 29, Number 4, Pages: 1367  1391, 2010
Andrea L. Bertozzi,
Ning Ju
and HsiangWei Lu
Abstract
References
Full Text
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We consider a class of splitting schemes for fourth order nonlinear
diffusion equations. Standard backwardtime differencing requires
the solution of a higher order elliptic problem, which can be both
computationally expensive and workintensive to code, in higher space
dimensions.
Recent papers in the literature provide computational evidence that
a biharmonicmodified, forward timestepping method, can provide good
results for these problems.
We provide a theoretical explanation of the results.
For a basic nonlinear 'thin film' type equation we prove $H^1$
stability of the method given very simple boundedness constraints
of the numerical solution. For a more general class of longwave
unstable problems, we prove stability and convergence, using
only constraints on the smooth solution.
Computational examples include both the model of 'thin film' type
problems and a quantitative model for electrowetting in a HeleShaw
cell (Lu et al J. Fluid Mech. 2007).
The methods considered here are related to 'convexity splitting'
methods for gradient flows with nonconvex energies.

3 
Adaptation of an ecological territorial model to street gang spatial patterns in Los Angeles
Volume 32, Number 9, Pages: 3223  3244, 2012
Laura M. Smith,
Andrea L. Bertozzi,
P. Jeffrey Brantingham,
George E. Tita
and Matthew Valasik
Abstract
References
Full Text
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Territorial behavior is often found in nature. Coyotes and wolves organize themselves around a den site and mark their territory to distinguish their claimed region.
Moorcroft et al. model the formation of territories and spatial distributions of coyote packs and their markings in [31].
We modify this ecological approach to simulate spatial gang dynamics in the Hollenbeck policing division of eastern Los Angeles. We incorporate important geographical features from the region that would inhibit movement, such as rivers and freeways. From the gang and marking densities created by this method, we create a rivalry network from overlapping territories and compare the graph to both the observed network and those constructed through other methods. Data on the locations of where gang members have been observed is then used to analyze the densities created by the model.

4 
Pointwise asymptotic convergence of solutions for a phase separation model
Volume 16, Number 1, Pages: 1  18, 2006
Pavel Krejčí
and Songmu Zheng
Abstract
Full Text
Related Articles
A new technique, combining the global energy and entropy balance equations
with the local stability theory for dynamical systems, is used for proving
that every solution to a nonsmooth temperaturedriven phase separation model
with conserved energy converges pointwise in space to an equilibrium as time
tends to infinity. Three main features are observed: the limit
temperature is uniform in space, there exists a partition of the physical body
into at most three constant limit phases, and the phase separation process has
a hysteresislike character.

5 
An interface problem: The twolayer shallow water equations
Volume 33, Number 11/12, Pages: 5327  5345, 2013
Madalina Petcu
and Roger Temam
Abstract
References
Full Text
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The aim of this article is to study a model of two superposed layers of fluid governed by the shallow water equations in space dimension one. Under some suitable hypotheses the governing equations are hyperbolic. We introduce suitable boundary conditions and establish a result of existence and uniqueness of smooth solutions for a limited time for this model.

6 
Persistence and global stability for a class of discrete time structured population models
Volume 33, Number 10, Pages: 4627  4646, 2013
Hal L. Smith
and Horst R. Thieme
Abstract
References
Full Text
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We obtain sharp conditions distinguishing extinction from persistence and provide sufficient conditions for
global stability of a positive fixed point for a class of discrete time
dynamical systems on the positive cone of an ordered Banach space generated by a map which is, roughly speaking,
a nonlinear, rank one perturbation of a linear contraction. Such maps were considered by
Rebarber, Tenhumberg, and Towney (Theor. Pop. Biol. 81, 2012) as abstractions of a restricted class of density dependent integral population projection models
modeling plant population dynamics.
Significant improvements of their results are provided.

7 
Waves in random neural media
Volume 32, Number 8, Pages: 2951  2970, 2012
Stephen Coombes,
Helmut Schmidt,
Carlo R. Laing,
Nils Svanstedt
and John A. Wyller
Abstract
References
Full Text
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Translationally invariant integrodifferential equations are a common choice of model in neuroscience for describing the coarsegrained dynamics of cortical tissue. Here we analyse the propagation of travelling wavefronts in models of neural media that incorporate some form of modulation or randomness such that translational invariance is broken. We begin with a study of neural architectures in which there is a periodic modulation of the neuronal connections. Recent techniques from twoscale convergence analysis are used to construct a homogenized model in the limit that the spatial modulation is rapid compared with the scale of the long range connections.
For the special case that the neuronal firing rate is a Heaviside we calculate the speed of a travelling homogenized front exactly and find how the wave speed changes as a function of the amplitude of the modulation. For this special case we further show how to obtain more accurate results about wave speed and the conditions for propagation failure by using an interface dynamics approach that circumvents the requirement of fast modulation. Next we turn our attention to forms of disorder that arise via the variation of firing rate properties across the tissue. To model this we draw parameters of the firing rate function from a distribution with prescribed spatial correlations and analyse the corresponding fluctuations in the wave speed. Finally we consider generalisations of the model to incorporate adaptation and stochastic forcing and show how recent numerical techniques developed for stochastic partial differential equations can be used to determine the wave speed by minimising the $L^2$ norm of a travelling disordered activity profile against a fixed test function.

8 
Sequences of compatible periodic hybrid orbits of prefractal Koch snowflake billiards
Volume 33, Number 8, Pages: 3719  3740, 2013
Michel L. Lapidus
and Robert G. Niemeyer
Abstract
References
Full Text
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The Koch snowflake $KS$ is a nowhere differentiable curve. The billiard table $Ω (KS)$ with boundary $KS$ is, a priori, not well defined. That is, one cannot a priori determine the minimal path traversed by a billiard ball subject to a collision in the boundary of the table. It is this problem which makes $Ω (KS)$ such an interesting, yet difficult, table to analyze.
In this paper, we approach this problem by approximating (from the inside) $Ω (KS)$ by welldefined (prefractal) rational polygonal billiard tables $Ω (KS_{n})$. We first show that the flat surface $S(KS_{n})$ determined from the rational billiard $Ω (KS_{n})$ is a branched cover of the singly punctured hexagonal torus. Such a result, when combined with the results of [6], allows us to define a sequence of compatible orbits of prefractal billiards $Ω (KS_{n})$. Using a particular addressing system, we define a hybrid orbit of a prefractal billiard $Ω (KS_{n})$ and show that every dense orbit of a prefractal billiard $Ω (KS_{n})$ is a dense hybrid orbit of $Ω (KS_{n})$. This result is key in obtaining a topological dichotomy for a sequence of compatible orbits. Furthermore, we determine a sufficient condition for a sequence of compatible orbits to be a sequence of compatible periodic hybrid orbits.
We then examine the limiting behavior of a sequence of compatible periodic hybrid orbits. We show that the trivial limit of particular (eventually) constant sequences of compatible hybrid orbits constitutes an orbit of $Ω(KS)$. In addition, we show that the union of two suitably chosen nontrivial polygonal paths connects two elusive limit points of the Koch snowflake. We conjecture that such a path is indeed the subset of what will eventually be an orbit of the Koch snowflake fractal billiard, once an appropriate `fractal law of reflection' is determined.
Finally, we close with a discussion of several open problems and potential directions for further research. We discuss how it may be possible for our results to be generalized to other fractal billiard tables and how understanding the structures of the Veech groups of the prefractal billiards may help in determining 'fractal flat surfaces' naturally associated with the billiard flows.

9 
Recovering damping and potential coefficients for an inverse nonhomogeneous secondorder hyperbolic problem via a localized Neumann boundary trace
Volume 33, Number 11/12, Pages: 5217  5252, 2013
Shitao Liu
and Roberto Triggiani
Abstract
References
Full Text
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We consider a secondorder hyperbolic equation defined on an open bounded domain $\Omega$ in $\mathbb{R}^n$ for $n \geq 2$, with $C^2$boundary $\Gamma = \partial \Omega = \overline{\Gamma_0 \cup \Gamma_1}$, $\Gamma_0 \cap \Gamma_1 = \emptyset$, subject to nonhomogeneous Dirichlet boundary conditions for the entire boundary $\Gamma$. We then study the inverse problem of determining both the damping and the potential (source) coefficients simultaneously, in one shot, by means of an additional measurement of the Neumann boundary trace of the solution, in a suitable, explicit subportion $\Gamma_1$ of the boundary $\Gamma$, and over a computable time interval $T > 0$. Under sharp conditions on the complementary part $\Gamma_0 = \Gamma \backslash \Gamma_1$, $T > 0$, and under sharp regularity requirements on the data, we establish the two canonical results in inverse problems: (i) uniqueness and (ii) Lipschitzstability. The latter (ii) is the main result of the paper. Our proof relies on a few main ingredients: (a) sharp Carleman estimates at the $H^1(\Omega) \times L^2(\Omega)$level for secondorder hyperbolic equations [23], originally introduced for control theory issues; (b) a correspondingly implied continuous observability inequality at the same energy level [23]; (c) sharp interior and boundary regularity theory for secondorder hyperbolic equations with Dirichlet boundary data [14,15,16]. The proof of the linear uniqueness result (Section 3) also takes advantage of a convenient tactical route ``postCarleman estimates" proposed by V. Isakov in [8, Thm. 8.2.2, p. 231]. Expressing the final results for the nonlinear inverse problem directly in terms of the data offers an additional challenge.

10 
Feedforward networks, center manifolds, and forcing
Volume 32, Number 8, Pages: 2913  2935, 2012
Martin Golubitsky
and Claire Postlethwaite
Abstract
References
Full Text
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This paper discusses feedforward chains near points of synchronybreaking Hopf bifurcation. We show that at synchronybreaking bifurcations the center manifold inherits a feedforward structure and use this structure to provide a simplified proof of the theorem of Elmhirst and Golubitsky that there is a branch of periodic solutions in such bifurcations whose amplitudes grow at the rate of $\lambda^{\frac{1}{6}}$. We also use this center manifold structure to provide a method for classifying the bifurcation diagrams of the forced feedforward chain where the amplitudes of the periodic responses are plotted as a function of the forcing frequency. The bifurcation diagrams depend on the amplitude of the forcing, the deviation of the system from Hopf bifurcation, and the ratio $\gamma$ of the imaginary part of the cubic term in the normal form of Hopf bifurcation to the real part. These calculations generalize the results of Zhang on the forcing of systems near Hopf bifurcations to threecell feedforward chains.

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