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1078-0947

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## Discrete & Continuous Dynamical Systems - A

Open Access Articles

*+*[Abstract](481)

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*+*[PDF](12373.99KB)

**Abstract:**

Motivated by the question whether higher-order nonlinear model equations, which go beyond the Camassa-Holm regime of moderate amplitude waves, could point us to new types of waves profiles, we study the traveling wave solutions of a quasilinear evolution equation which models the propagation of shallow water waves of large amplitude. The aim of this paper is a complete classification of its traveling wave solutions. Apart from symmetric smooth, peaked and cusped solitary and periodic traveling waves, whose existence is well-known for moderate amplitude equations like Camassa-Holm, we obtain entirely new types of singular traveling waves: periodic waves which exhibit singularities on both crests and troughs simultaneously, waves with asymmetric peaks, as well as multi-crested smooth and multi-peaked waves with decay. Our approach uses qualitative tools for dynamical systems and methods for integrable planar systems.

*+*[Abstract](307)

*+*[HTML](367)

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**Abstract:**

In this work we study degenerate with respect to parameters fold-Hopfbifurcations in three-dimensional differential systems. Such degeneraciesarise when the transformations between parameters leading to a normal formare not regular at some points in the parametric space. We obtain newgeneric results for the dynamics of the systems in such a degenerateframework. The bifurcation diagrams we obtained show that in a degeneratecontext the dynamics may be completely different than in a non-degenerateframework.

*+*[Abstract](75)

*+*[HTML](12)

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**Abstract:**

Professor Paul Chase Fife was born in Cedar City, Utah, on February 14, 1930. After undergraduate studies at the University of Chicago, he obtained a Master's degree in physics from the University of California Berkeley where he also received a Phi Beta Kappa Award. He then completed a PhD in Applied Mathematics at New York University, Courant Institute, in June 1959. While at NYU he met Jayne Winters, and they married on December 22, 1959. They then moved to Palo Alto, California, where Paul joined the Department of Mathematics at Stanford University.

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*+*[Abstract](129)

*+*[PDF](429.7KB)

**Abstract:**

The nonlinear recombination equation from population genetics has a long history and is notoriously difficult to solve, both in continuous and in discrete time. This is particularly so if one aims at full generality, thus also including degenerate parameter cases. Due to recent progress for the continuous time case via the identification of an underlying stochastic fragmentation process, it became clear that a direct general solution at the level of the corresponding ODE itself should also be possible. This paper shows how to do it, and how to extend the approach to the discrete-time case as well.

*+*[Abstract](116)

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**Abstract:**

This article revisits the approximation problem of systems of nonlinear delay differential equations (DDEs) by a set of ordinary differential equations (ODEs). We work in Hilbert spaces endowed with a natural inner product including a point mass, and introduce polynomials orthogonal with respect to such an inner product that live in the domain of the linear operator associated with the underlying DDE. These polynomials are then used to design a general Galerkin scheme for which we derive rigorous convergence results and show that it can be numerically implemented via simple analytic formulas. The scheme so obtained is applied to three nonlinear DDEs, two autonomous and one forced: (i) a simple DDE with distributed delays whose solutions recall Brownian motion; (ii) a DDE with a discrete delay that exhibits bimodal and chaotic dynamics; and (iii) a periodically forced DDE with two discrete delays arising in climate dynamics. In all three cases, the Galerkin scheme introduced in this article provides a good approximation by low-dimensional ODE systems of the DDE's strange attractor, as well as of the statistical features that characterize its nonlinear dynamics.

*+*[Abstract](79)

*+*[PDF](2472.6KB)

**Abstract:**

The papers in this special issue of Discrete and Continuous Dynamical Systems are dedicated to Professor Peter D. Lax, of the Courant Institute, on the occasion of his ninetieth birthday, by some of his friends, associates, and students.

For more information please click the “Full Text” above.

*+*[Abstract](94)

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**Abstract:**

We propose and solve a new problem for the unsteady transonic small disturbance equation. Data are given for the self-similar equation in a fixed, bounded region of similarity space, where on a part of the boundary the equation has degenerate type (a `sonic line') and on the remainder it is elliptic. Previous results on this problem have chosen data so that the solution is constant on the sonic line, but we set up a situation where the solution is not constant on the sonic part of the boundary. The solution we find is Lipschitz up to the boundary. Our solution sets the stage for resolution of some interesting Riemann problems for this equation and for other multidimensional conservation laws.

*+*[Abstract](84)

*+*[PDF](553.2KB)

**Abstract:**

In this paper, we prove a local in time unique existence theorem for some two phase problem of compressible and compressible barotropic viscous fluid flow without surface tension in the $L_p$ in time and the $L_q$ in space framework with $2< p <\infty$ and $N< q <\infty$ under the assumption that the initial domain is a uniform $W^{2-1/q}_q$ domain in $\mathbb{R}^N (N\ge 2)$. After transforming a unknown time dependent domain to the initial domain by the Lagrangian transformation, we solve the problem by the contraction mapping principle with the maximal $L_p$-$L_q$ regularity of the generalized Stokes operator for the compressible viscous fluid flow with free boundary condition. The key step of our method is to prove the existence of $\mathcal{R}$-bounded solution operator to resolvent problem corresponding to linearized problem. The $\mathcal{R}$-boundedness combined with Weis's operator valued Fourier multiplier theorem implies the generation of analytic semigroup and the maximal $L_p$-$L_q$ regularity theorem.

*+*[Abstract](92)

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**Abstract:**

This article presents stability analytical results of a two component reaction-diffusion system with linear cross-diffusion posed on continuously evolving domains. First the model system is mapped from a continuously evolving domain to a reference stationary frame resulting in a system of partial differential equations with time-dependent coefficients. Second, by employing appropriately asymptotic theory, we derive and prove cross-diffusion-driven instability conditions for the model system for the case of slow, isotropic domain growth. Our analytical results reveal that unlike the restrictive diffusion-driven instability conditions on stationary domains, in the presence of cross-diffusion coupled with domain evolution, it is no longer necessary to enforce cross nor pure kinetic conditions. The restriction to

*activator-inhibitor*kinetics to induce pattern formation on a growing biological system is no longer a requirement. Reaction-cross-diffusion models with equal diffusion coefficients in the principal components as well as those of the

*short-range inhibition, long-range activation*and

*activator-activator*form can generate patterns only in the presence of cross-diffusion coupled with domain evolution. To confirm our theoretical findings, detailed parameter spaces are exhibited for the special cases of isotropic exponential, linear and logistic growth profiles. In support of our theoretical predictions, we present evolving or moving finite element solutions exhibiting patterns generated by a

*short-range inhibition, long-range activation*reaction-diffusion model with linear cross-diffusion in the presence of domain evolution.

*+*[Abstract](52)

*+*[PDF](1914.3KB)

**Abstract:**

Professor Rou-Huai Wang (October 30, 1924 - November 5, 2001) was a mathematician who proved fundamental results for partial differential equations, helped to introduce modern PDE theory to the Chinese mathematics community since early 50's, and played a leading role in revitalizing PDE research in China after the disastrous "Cultural Revolution" (1966-1976). He was regarded by many Chinese mathematicians of younger generations as a visionary, generous and caring mentor.

For more information please click the “Full Text” above.

*+*[Abstract](227)

*+*[PDF](418.1KB)

**Abstract:**

For almost periodic differential systems $\dot x= \varepsilon f(x,t,\varepsilon)$ with $x\in \mathbb{C}^n$, $t\in \mathbb{R}$ and $\varepsilon>0$ small enough, we get a polynomial normal form in a neighborhood of a hyperbolic singular point of the system $\dot x= \varepsilon \lim_{T \to \infty} \frac {1} {T} \int_0^T f(x,t,0) dt$, if its eigenvalues are in the Poincaré domain. The normal form linearizes if the real part of the eigenvalues are non--resonant.

*+*[Abstract](145)

*+*[PDF](581.9KB)

**Abstract:**

The process of recombination in population genetics, in its deterministic limit, leads to a nonlinear ODE in the Banach space of finite measures on a locally compact product space. It has an embedding into a larger family of nonlinear ODEs that permits a systematic analysis with lattice-theoretic methods for general partitions of finite sets. We discuss this type of system, reduce it to an equivalent finite-dimensional nonlinear problem, and establish a connection with an ancestral partitioning process, backward in time. We solve the finite-dimensional problem recursively for generic sets of parameters and briefly discuss the singular cases, and how to extend the solution to this situation.

*+*[Abstract](92)

*+*[PDF](132.3KB)

**Abstract:**

The workshop on ``Analysis and Control of Stochastic Partial Differential Equations" was held in Fudan University on December 3--6, 2012, which was jointly organized and financially supported by Fudan University and Tongji University. Many of the contributions in the special issue were reported in the workshop, and there are also some few others which are solicited from renowned researchers in the fields of stochastic partial differential equations (SPDEs). The contents of the special issue are divided into the following three parts.

For more information please click the “Full Text” above.

*+*[Abstract](103)

*+*[PDF](131.6KB)

**Abstract:**

This special volume gathers a number of new contributions addressing various topics related to the field of optimal control theory and sensitivity analysis. The field has a rich and varied mathematical theory, with a long tradition and a vibrant body of applications. It has attracted a growing interest across the last decades, with the introduction of new ideas and techniques, and thanks to various new applications.

For more information please click the “Full Text” above.

*+*[Abstract](75)

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**Abstract:**

Optimal mass transportation can be traced back to Gaspard Monge's paper in 1781. There, for engineering/military reasons, he was studying how to minimize the cost of transporting a given distribution of mass from one location to another, giving rise to a challenging mathematical problem. This problem, an optimization problem in a certain class of maps, had to wait for almost two centuries before seeing significant progress (starting with Leonid Kantorovich in 1942), even on the very fundamental question of the existence of an optimal map. Due to these connections with several other areas of pure and applied mathematics, optimal transportation has received much renewed attention in the last twenty years. Indeed, it has become an increasingly common and powerful tool at the interface between partial differential equations, fluid mechanics, geometry, probability theory, and functional analysis. At the same time, it has led to significant developments in applied mathematics, with applications ranging from economics, biology, meteorology, design, to image processing. Because of the success and impact that this subject is still receiving, we decided to create a special issue collecting selected papers from leading experts in the area.

For more information please click the “Full Text” above.

*+*[Abstract](68)

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**Abstract:**

Arieh Iserles was born in Poland, on September 2, 1947. He was educated in Israel, where he received BSc and MSc degrees from the Hebrew University and obtained his PhD degree under the supervision of Giacomo Della Riccia at Ben Gurion University with the dissertation

*Numerical Solution of Stiff Differential Equations*(1978). He first arrived in Cambridge, in 1978 and has remained there ever since. He has successively been Junior and Senior Research Fellow at King's College, and Lecturer (1987) and Professor (1999) at Cambridge University where he holds a chair in Numerical Analysis and Differential Equations. Arieh has received many honours, in particuluar the Lars Onsager Medal (1999) from the Nowegian University of Science and Technology and the David G. Crighton Medal (2012) from the London Mathematical Society and the Institute of Mathematics and its Applications. He holds Honorary Professorships at Huazhong University of Science and Technology, Wuhan, since 2002 and Jilin University, Changchun, since 2004.

For more information please click the “Full Text” above.

*+*[Abstract](63)

*+*[PDF](139.2KB)

**Abstract:**

The theory of infinite dimensional and stochastic dynamical systems is a rapidly expanding and vibrant field of mathematics. In the recent three decades it has been highlighted as a core knowledge and an advancing thrust in the qualitative study of complex systems and processes described by evolutionary partial differential equations in many different settings, stochastic differential equations, functional differential equations and lattice differential equations. The central research topics include the invariant and attracting sets, stability and bifurcation of patterns and waves, asymptotic theory of dissipative systems and reduction of dimensions, and more and more problems of nonlocal systems, ill-posed systems, multicomponent and network dynamics, random dynamics and chaotic dynamics.

For more information please click the “Full Text” above

*+*[Abstract](86)

*+*[PDF](83.8KB)

**Abstract:**

Jerome Arthur Goldstein (Jerry) was born on August 5, 1941 in Pittsburgh, PA. He attended Carnegie Mellon University, then called Carnegie Institute of Technology, where he earned his Bachelors of Science degree (1963), Masters of Science degree (1964) and Ph.D. (1967) in Mathematics. Jerry took a postdoctoral position at the Institute for Advanced Study in Princeton, followed by an Assistant Professorship at Tulane University. He became a Full Professor at Tulane University in 1975. After twenty-four years there, Jerry moved ``upriver" to join the faculty (and his wife, Gisèle) at Louisiana State University, where he was Professor of Mathematics from 1991 to 1996. In 1996 Jerry and Gisèle moved to the University of Memphis.

For more information please click the “Full Text” above.

*+*[Abstract](83)

*+*[PDF](425.0KB)

**Abstract:**

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.

*+*[Abstract](64)

*+*[PDF](1019.7KB)

**Abstract:**

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 well-defined (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.

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