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Discrete & Continuous Dynamical Systems - A

2013 , Volume 33 , Issue 10

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An introduction to migration-selection PDE models
Yuan Lou, Thomas Nagylaki and Wei-Ming Ni
2013, 33(10): 4349-4373 doi: 10.3934/dcds.2013.33.4349 +[Abstract](77) +[PDF](457.9KB)
This expository article concerns a system of semilinear parabolic partial differential equations that describes the evolution of the gene frequencies at a single locus under the joint action of migration and selection. We shall review mathematical techniques suited for the models under investigation; discuss some of the main mathematical results, including most recent developments; and also propose some open problems.
Pentagonal domain exchange
Shigeki Akiyama and Edmund Harriss
2013, 33(10): 4375-4400 doi: 10.3934/dcds.2013.33.4375 +[Abstract](62) +[PDF](819.1KB)
Self-inducing structure of pentagonal piecewise isometry is applied to show detailed description of periodic and aperiodic orbits, and further dynamical properties. A Pisot number appears as a scaling constant and plays a crucial role in the proof. Further generalization is discussed in the last section.
On the large deviation rates of non-entropy-approachable measures
Masayuki Asaoka and Kenichiro Yamamoto
2013, 33(10): 4401-4410 doi: 10.3934/dcds.2013.33.4401 +[Abstract](92) +[PDF](370.4KB)
We construct a non-ergodic maximal entropy measure of a $C^{\infty}$ diffeomorphism with a positive entropy such that neither the entropy nor the large deviation rate of the measure is influenced by that of ergodic measures near it.
Construction of response functions in forced strongly dissipative systems
Renato C. Calleja, Alessandra Celletti and Rafael de la Llave
2013, 33(10): 4411-4433 doi: 10.3934/dcds.2013.33.4411 +[Abstract](60) +[PDF](494.0KB)
We study the existence of quasi--periodic solutions of the equation \[ ε \ddot x + \dot x + ε g(x) = ε f(\omega t)\ , \] where $x: \mathbb{R} \rightarrow \mathbb{R}$ is the unknown and we are given $g:\mathbb{R} \rightarrow \mathbb{R}$, $f: \mathbb{T}^d \rightarrow \mathbb{R}$, $\omega \in \mathbb{R}^d$ (without loss of generality we can assume that $\omega\cdot k\not=0$ for any $k \in \mathbb{Z}^d\backslash\{0\}$). We assume that there is a $c_0\in \mathbb{R}$ such that $g(c_0) = \hat f_0$ (where $\hat f_0$ denotes the average of $f$) and $g'(c_0) \ne 0$. Special cases of this equation, for example when $g(x)=x^2$, are called the ``varactor problem'' in the literature.
    We show that if $f$, $g$ are analytic, and $\omega$ satisfies some very mild irrationality conditions, there are families of quasi--periodic solutions with frequency $\omega$. These families depend analytically on $ε$, when $ε$ ranges over a complex domain that includes cones or parabolic domains based at the origin.
    The irrationality conditions required in this paper are very weak. They allow that the small denominators $|\omega \cdot k|^{-1}$ grow exponentially with $k$. In the case that $f$ is a trigonometric polynomial, we do not need any condition on $|\omega \cdot k|$. This answers a delicate question raised in [8].
    We also consider the periodic case, when $\omega$ is just a number ($d = 1$). We obtain that there are solutions that depend analytically in a domain which is a disk removing countably many disjoint disks. This shows that in this case there is no Stokes phenomenon (different resummations on different sectors) for the asymptotic series.
    The approach we use is to reduce the problem to a fixed point theorem. This approach also yields results in the case that $g$ is a finitely differentiable function; it provides also very effective numerical algorithms and we discuss how they can be implemented.
About the unfolding of a Hopf-zero singularity
Freddy Dumortier, Santiago Ibáñez, Hiroshi Kokubu and Carles Simó
2013, 33(10): 4435-4471 doi: 10.3934/dcds.2013.33.4435 +[Abstract](140) +[PDF](3065.5KB)
We study arbitrary generic unfoldings of a Hopf-zero singularity of codimension two. They can be written in the following normal form: \begin{eqnarray*} \left\{ \begin{array}{l} x'=-y+\mu x-axz+A(x,y,z,\lambda,\mu) \\ y'=x+\mu y-ayz+B(x,y,z,\lambda,\mu) \\ z'=z^2+\lambda+b(x^2+y^2)+C(x,y,z,\lambda,\mu), \end{array} \right. \end{eqnarray*} with $a>0$, $b>0$ and where $A$, $B$, $C$ are $C^\infty$ or $C^\omega$ functions of order $O(\|(x,y,z,\lambda,\mu)\|^3)$.
    Despite that the existence of Shilnikov homoclinic orbits in unfoldings of Hopf-zero singularities has been discussed previously in the literature, no result valid for arbitrary generic unfoldings is available. In this paper we present new techniques to study global bifurcations from Hopf-zero singularities. They allow us to obtain a general criterion for the existence of Shilnikov homoclinic bifurcations and also provide a detailed description of the bifurcation set. Criteria for the existence of Bykov cycles are also provided. Main tools are a blow-up method, including a related invariant theory, and a novel approach to the splitting functions of the invariant manifolds. Theoretical results are applied to the Michelson system and also to the so called extended Michelson system. Paper includes thorough numerical explorations of dynamics for both systems.
Dispersive estimates for matrix Schrödinger operators in dimension two
M. Burak Erdoǧan and William R. Green
2013, 33(10): 4473-4495 doi: 10.3934/dcds.2013.33.4473 +[Abstract](89) +[PDF](546.2KB)
We consider the non-selfadjoint operator \[ H = \left[\begin{array}{cc} -\Delta + \mu-V_1 & -V_2\\ V_2 & \Delta - \mu + V_1 \end{array} \right] \] where $\mu>0$ and $V_1,V_2$ are real-valued decaying potentials. Such operators arise when linearizing a focusing NLS equation around a standing wave. Under natural spectral assumptions we obtain $L^1(\mathbb{R}^2)\times L^1(\mathbb{R}^2)\to L^\infty(\mathbb{R}^2)\times L^\infty(\mathbb{R}^2)$ dispersive decay estimates for the evolution $e^{it H}P_{ac}$. We also obtain the following weighted estimate $$ \|w^{-1} e^{it\mathcal H}P_{ac}f\|_{L^\infty(\mathbb R^2)\times L^\infty(\mathbb R^2)} ≲ \frac{1}{|t|\log^2(|t|)} \|w f\|_{L^1(\mathbb R^2)\times L^1(\mathbb R^2)},\,\,\,\,\,\,\,\, |t| >2, $$with $w(x)=\log^2(2+|x|)$.
Weak solutions of a gas-liquid drift-flux model with general slip law for wellbore operations
Steinar Evje and Huanyao Wen
2013, 33(10): 4497-4530 doi: 10.3934/dcds.2013.33.4497 +[Abstract](78) +[PDF](540.3KB)
In this work we study a compressible gas-liquid models highly relevant for wellbore operations like drilling. The model is a drift-flux model and is composed of two continuity equations together with a mixture momentum equation. The model allows unequal gas and liquid velocities, dictated by a so-called slip law, which is important for modeling of flow scenarios involving for example counter-current flow. The model is considered in Lagrangian coordinates. The difference in fluid velocities gives rise to new terms in the mixture momentum equation that are challenging to deal with. First, a local (in time) existence result is obtained under suitable assumptions on initial data for a general slip relation. Second, a global in time existence result is proved for small initial data subject to a more specialized slip relation.
Polynomial and rational first integrals for planar quasi--homogeneous polynomial differential systems
Jaume Giné, Maite Grau and Jaume Llibre
2013, 33(10): 4531-4547 doi: 10.3934/dcds.2013.33.4531 +[Abstract](98) +[PDF](409.4KB)
In this paper we find necessary and sufficient conditions in order that a planar quasi--homogeneous polynomial differential system has a polynomial or a rational first integral. We also prove that any planar quasi--homogeneous polynomial differential system can be transformed into a differential system of the form $\dot{u} \, = \, u f(v)$, $\dot{v} \, = \, g(v)$ with $f(v)$ and $g(v)$ polynomials, and vice versa.
A Liouville theorem of degenerate elliptic equation and its application
Genggeng Huang
2013, 33(10): 4549-4566 doi: 10.3934/dcds.2013.33.4549 +[Abstract](61) +[PDF](436.9KB)
In this paper, we apply the moving plane method to the following degenerate elliptic equation arising from isometric embedding,\begin{equation*} yu_{yy}+au_y+\Delta_x u+u^\alpha=0\text{ in } \mathbb R^{n+1}_+,n\geq 1. \end{equation*} We get a Liouville theorem for subcritical case and classify the solutions for critical case. As an application, we derive the a priori bounds for positive solutions of some semi-linear degenerate elliptic equations.
An equivalent characterization of the summability condition for rational maps
Huaibin Li
2013, 33(10): 4567-4578 doi: 10.3934/dcds.2013.33.4567 +[Abstract](69) +[PDF](347.0KB)
We give an equivalent characterization of the summability condition in terms of the backward contracting property defined by Juan Rivera-Letelier, for rational maps of degree at least two which are expanding away from critical points.
Statistical stability for multi-substitution tiling spaces
Rui Pacheco and Helder Vilarinho
2013, 33(10): 4579-4594 doi: 10.3934/dcds.2013.33.4579 +[Abstract](97) +[PDF](437.1KB)
Given a finite set $\{S_1\dots,S_k \}$ of substitution maps acting on a certain finite number (up to translations) of tiles in $\mathbb{R}^d$, we consider the multi-substitution tiling space associated to each sequence $\bar a\in \{1,\ldots,k\}^{\mathbb{N}}$. The action by translations on such spaces gives rise to uniquely ergodic dynamical systems. In this paper we investigate the rate of convergence for ergodic limits of patches frequencies and prove that these limits vary continuously with $\bar a$.
Canard trajectories in 3D piecewise linear systems
Rafel Prohens and Antonio E. Teruel
2013, 33(10): 4595-4611 doi: 10.3934/dcds.2013.33.4595 +[Abstract](68) +[PDF](752.0KB)
We present some results on singularly perturbed piecewise linear systems, similar to those obtained by the Geometric Singular Perturbation Theory. Unlike the differentiable case, in the piecewise linear case we obtain the global expression of the slow manifold ${\mathcal S}_{\varepsilon}$. As a result, we characterize the existence of canard orbits in such systems. Finally, we apply the above theory to a specific case where we show numerical evidences of the existence of a canard cycle.
Existence and uniqueness of singular solution to stationary Schrödinger equation with supercritical nonlinearity
Fouad Hadj Selem, Hiroaki Kikuchi and Juncheng Wei
2013, 33(10): 4613-4626 doi: 10.3934/dcds.2013.33.4613 +[Abstract](98) +[PDF](414.3KB)
In this paper, we study a singular solution to the following elliptic equations: \begin{equation*} \left\{\begin{array}{ll} - \Delta u + |x|^{2}u - \lambda u - |u|^{p-1}u = 0, \quad x \in \mathbb{R}^{d}, & \\ u(x) > 0, \quad x \in \mathbb{R}^{d}, & \\ u(x) \to 0 \quad \text{as}\; |x| \to \infty, & \end{array}\right. \end{equation*} where $d \geq 3, \lambda >0$ and $p > 1$. In the spirit of Merle and Peletier [9], we shall show that in case of $p>(d+2)/(d-2)$, there is a unique value $\lambda = \lambda_{*}$ such that the equation with $\lambda = \lambda_{*}$ has a unique radial singular solution.
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](77) +[PDF](425.0KB)
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.
Continuity of Hausdorff measure for conformal dynamical systems
Tomasz Szarek, Mariusz Urbański and Anna Zdunik
2013, 33(10): 4647-4692 doi: 10.3934/dcds.2013.33.4647 +[Abstract](72) +[PDF](641.0KB)
Developing the pioneering work of Lars Olsen [14], we deal with the question of continuity of the numerical value of Hausdorff measures of some natural families of conformal dynamical systems endowed with an appropriate natural topology. In particular, we prove such continuity for hyperbolic polynomials from the Mandelbrot set, and more generally for the space of hyperbolic rational functions of a fixed degree. We go beyond hyperbolicity by proving continuity for maps including parabolic rational functions, for example that the parameter $1/4$ is such a continuity point for quadratic polynomials $z\mapsto z^2+c$ for $c\in [0,1/4]$. We prove the continuity of the numerical value of Hausdorff measures also for the spaces of conformal expanding repellers and parabolic ones, more generally for parabolic Walters conformal maps. We also prove some partial continuity results for all conformal Walters maps; these are in general of infinite degree. In order to do this, as one of our tools, we provide a detailed local analysis, uniform with respect to the parameter, of the behavior of conformal maps around parabolic fixed points in any dimension. We also establish continuity of numerical values of Hausdorff measures for some families of infinite $1$-dimensional iterated function systems.
Global asymptotic dynamics of a class of nonlinearly coupled neural networks with delays
Jui-Pin Tseng
2013, 33(10): 4693-4729 doi: 10.3934/dcds.2013.33.4693 +[Abstract](72) +[PDF](1008.8KB)
This work presents an effective approach to the study of the global asymptotic dynamics of general coupled systems. Under the developed framework, the problem of establishing global synchronization or global convergence reduces to solving a corresponding system of linear equations. We illustrate this approach with a class of neural networks that consist of a pair of sub-networks under various types of nonlinear and delayed couplings. We study both the synchronization and the asymptotic synchronous phases of the dynamics, including global convergence to zero, global convergence to multiple synchronous equilibria, and global synchronization with nontrivial synchronous periodic solutions. Our investigation also provides theoretical support to some numerical findings, and improves or extend some results in the literature.
Expansion growth, entropy and invariant measures of distal groups and pseudogroups of homeo- and diffeomorphisms
Paweł G. Walczak
2013, 33(10): 4731-4742 doi: 10.3934/dcds.2013.33.4731 +[Abstract](90) +[PDF](352.6KB)
We estimate expansion growth types (in the sense of Egashira) of certain distal groups of homeomorphisms and manifold diffeomorphisms.The estimate implies zero entropy (in the sense of Ghys, Langevin and the author) and existence of invariant measures for such groups. We prove also existence of invariant measures for pseudogroups satisfying some conditions of distality type.
Zero-electron-mass limit of Euler-Poisson equations
Jiang Xu and Ting Zhang
2013, 33(10): 4743-4768 doi: 10.3934/dcds.2013.33.4743 +[Abstract](80) +[PDF](512.5KB)
We study the limit of vanishing ratio of the electron mass to the ion mass (zero-electron-mass limit) in the scaled Euler-Poisson equations. As the first step of this justification, we construct the uniform global classical solutions in critical Besov spaces with the aid of ``Shizuta-Kawashima" skew-symmetry. Then we establish frequency-localization estimates of Strichartz-type for the equation of acoustics according to the semigroup formulation. Finally, it is shown that the uniform classical solutions converge towards that of the incompressible Euler equations (for ill-preparedinitial data) in a refined way as the scaled electron-mass tends to zero. In comparison with the classical zero-mach-number limit in [7,23], we obtain different dispersive estimates due to the coupled electric field.
Ray and heteroclinic solutions of Hamiltonian systems with 2 degrees of freedom
Guowei Yu
2013, 33(10): 4769-4793 doi: 10.3934/dcds.2013.33.4769 +[Abstract](82) +[PDF](548.5KB)
In this paper, we study a class of Hamiltonian system with 2 degrees of freedom. We show that at any energy level above a certain critical value of each system, there are ray and heteroclinic solutions between any two periodic neighboring minimal solutions with any prescribed non-trivial homotopy class. Our proof is based on an elementary variational method.

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