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Discrete & Continuous Dynamical Systems - A

Open Access Articles

degenerate with respect to parameters fold-Hopf bifurcations
Gheorghe Tigan
2017, 37(4): 2115-2140 doi: 10.3934/dcds.2017091 +[Abstract](132) +[HTML](203) +[PDF](813.0KB)
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.

Haldane linearisation done right: Solving the nonlinear recombination equation the easy way
Ellen Baake and  Michael Baake
2016, 36(12): 6645-6656 doi: 10.3934/dcds.2016088 +[Abstract](56) +[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.
Low-dimensional Galerkin approximations of nonlinear delay differential equations
Mickaël D. Chekroun , Michael Ghil , Honghu Liu and  Shouhong Wang
2016, 36(8): 4133-4177 doi: 10.3934/dcds.2016.36.4133 +[Abstract](70) +[PDF](2260.8KB)
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.
Preface
Alexandre J. Chorin and  Jeffrey Rauch
2016, 36(8): i-ii doi: 10.3934/dcds.2016.36.8i +[Abstract](38) +[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.

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The unsteady transonic small disturbance equation: Data on oblique curves
Mary Chern and  Barbara Lee Keyfitz
2016, 36(8): 4213-4225 doi: 10.3934/dcds.2016.36.4213 +[Abstract](48) +[PDF](360.7KB)
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.
On some two phase problem for compressible and compressible viscous fluid flow separated by sharp interface
Takayuki Kubo , Yoshihiro Shibata and  Kohei Soga
2016, 36(7): 3741-3774 doi: 10.3934/dcds.2016.36.3741 +[Abstract](40) +[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.
Stability analysis of reaction-diffusion models on evolving domains: The effects of cross-diffusion
Anotida Madzvamuse , Hussaini Ndakwo and  Raquel Barreira
2016, 36(4): 2133-2170 doi: 10.3934/dcds.2016.36.2133 +[Abstract](41) +[PDF](5124.7KB)
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.
Preface
Kung-Ching Chang , Minyou Chi , Wei-Ming Ni and  Zhuoqun Wu
2016, 36(2): i-iii doi: 10.3934/dcds.2016.36.2i +[Abstract](20) +[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.

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Polynomial and linearized normal forms for almost periodic differential systems
Weigu Li , Jaume Llibre and  Hao Wu
2016, 36(1): 345-360 doi: 10.3934/dcds.2016.36.345 +[Abstract](158) +[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.
The general recombination equation in continuous time and its solution
Ellen Baake , Michael Baake and  Majid Salamat
2016, 36(1): 63-95 doi: 10.3934/dcds.2016.36.63 +[Abstract](67) +[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.
Preface
Baojun Bian , Shanjian Tang and  Qi Zhang
2015, 35(11): i-iv doi: 10.3934/dcds.2015.35.11i +[Abstract](45) +[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.

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Preface
Jean-Baptiste Caillau , Maria do Rosário de Pinho , Lars Grüne , Emmanuel Trélat and  Hasnaa Zidani
2015, 35(9): i-iv doi: 10.3934/dcds.2015.35.9i +[Abstract](48) +[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.

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Preface
Alessio Figalli and  Filippo Santambrogio
2014, 34(4): i-ii doi: 10.3934/dcds.2014.34.4i +[Abstract](35) +[PDF](73.8KB)
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.

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Preface
Elena Celledoni , Jesus M. Sanz-Serna and  Antonella Zanna Munthe-Kaas
2014, 34(3): i-ii doi: 10.3934/dcds.2014.34.3i +[Abstract](27) +[PDF](95.7KB)
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.

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Preface
Yuncheng You
2014, 34(1): i-iii doi: 10.3934/dcds.2014.34.1i +[Abstract](26) +[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.

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Preface
Gisèle Ruiz Goldstein and  Alain Miranville
2013, 33(11/12): i-ii doi: 10.3934/dcds.2013.33.11i +[Abstract](40) +[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.

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Persistence and global stability for a class of discrete time structured population models
Hal L. Smith and  Horst R. Thieme
2013, 33(10): 4627-4646 doi: 10.3934/dcds.2013.33.4627 +[Abstract](42) +[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.
Sequences of compatible periodic hybrid orbits of prefractal Koch snowflake billiards
Michel L. Lapidus and  Robert G. Niemeyer
2013, 33(8): 3719-3740 doi: 10.3934/dcds.2013.33.3719 +[Abstract](28) +[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.
Preface
Amadeu Delshams , Marian Gidea and  Ernesto Pérez-Chavela
2013, 33(3): i-i doi: 10.3934/dcds.2013.33.3i +[Abstract](23) +[PDF](87.2KB)
Abstract:
The material of this special issue of DCDS-A was originally dedicated in honor of the 65-th birthday of Prof. Ernesto A. Lacomba. Some of the papers in this issue reflect the joyful spirit surrounding this celebration. Sadly, shortly after the preparation of this volume was completed, Prof. Ernesto A. Lacomba passed away on June 26, 2012. Therefore this special issue is also paying a tribute to his long standing mathematical legacy.
    The work of Prof. Lacomba comprised research on geometric theory of ordinary differential equations, dynamical systems, and symplectic geometry, with applications to celestial mechanics, classical mechanics, vortex theory, thermodynamics and electrical circuits. Prof.~Lacomba was the leader of a strong research group working in these areas. In 1991 he started organizing, jointly with some members of his group and with other collaborators, the International Symposium on Hamiltonian Systems and Celestial Mechanics (HAMSYS), which became a great success over the next several years. These symposia brought together top researches from several countries, working in the aforementioned topics, as well as many graduate students who had the opportunity to learn from and connect with the experts in the field, and often get inspiration and motivation to improve and finalize their doctoral theses.
    The framework for the celebration of the 65-th birthday of Prof. Lacomba was the VI-th edition of HAMSYS, which was held in México D.F. between November 29 -- December 3, 2010.
    This symposium assembled an impressive number of highly respected researches who generated important discussions among the participants, presented new problems, and identified future research directions. The emphasis of the talks was on Hamiltonian dynamics and its relationship to several aspects of mechanics, geometric mechanics, and dynamical systems in general. The papers in this volume are an outgrowth of the themes of the symposium. All papers that were submitted to this special issue underwent a through refereeing process typical to any top mathematical journal. The accepted papers form the present issue of DCDS-A.
    The symposium received generous support from CONACYT México and UAM-I. Special thanks are due to Universidad Autónoma Metropolitana for hosting the symposium in the beautiful colonial building Casa de la primera imprenta de América. Last but not least, we thank all participants for contributing to a week-long intense and highly productive mathematical experience.
Orbit structure and (reversing) symmetries of toral endomorphisms on rational lattices
Michael Baake , Natascha Neumärker and  John A. G. Roberts
2013, 33(2): 527-553 doi: 10.3934/dcds.2013.33.527 +[Abstract](51) +[PDF](542.5KB)
Abstract:
We study various aspects of the dynamics induced by integer matrices on the invariant rational lattices of the torus in dimension $2$ and greater. Firstly, we investigate the orbit structure when the toral endomorphism is not invertible on the lattice, characterising the pretails of eventually periodic orbits. Next we study the nature of the symmetries and reversing symmetries of toral automorphisms on a given lattice, which has particular relevance to (quantum) cat maps.

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