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Discrete & Continuous Dynamical Systems - A

December 2014 , Volume 34 , Issue 12

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Structural stability for the splash singularities of the water waves problem
Angel Castro, Diego Córdoba, Charles Fefferman, Francisco Gancedo and Javier Gómez-Serrano
2014, 34(12): 4997-5043 doi: 10.3934/dcds.2014.34.4997 +[Abstract](633) +[PDF](603.0KB)
Abstract:
In this paper we show a structural stability result for water waves. The main motivation for this result is that we aim to exhibit a water wave whose interface starts as a graph and ends in a splash. Numerical simulations lead to an approximate solution with the desired behaviour. The stability result will conclude that near the approximate solution to water waves there is an exact solution.
Remarks on geometric properties of SQG sharp fronts and $\alpha$-patches
Angel Castro, Diego Córdoba, Javier Gómez-Serrano and Alberto Martín Zamora
2014, 34(12): 5045-5059 doi: 10.3934/dcds.2014.34.5045 +[Abstract](540) +[PDF](639.9KB)
Abstract:
Guided by numerical simulations, we present the proof of two results concerning the behaviour of SQG sharp fronts and $\alpha$-patches. We establish that ellipses are not rotational solutions and we prove that initially convex interfaces may lose this property in finite time.
A transformation of Markov jump processes and applications in genetic study
Xian Chen and Zhi-Ming Ma
2014, 34(12): 5061-5084 doi: 10.3934/dcds.2014.34.5061 +[Abstract](806) +[PDF](449.3KB)
Abstract:
In this paper we provide a verifiable necessary and sufficient condition for a regular q-process to be again a q-process under a transformation of state space. The result as well as some other results on continuous states Markov jump processes is employed to investigate jump processes arising from the study in modeling genetic coalescent with recombination.
Mathematical theory of solids: From quantum mechanics to continuum models
Weinan E and Jianfeng Lu
2014, 34(12): 5085-5097 doi: 10.3934/dcds.2014.34.5085 +[Abstract](702) +[PDF](384.6KB)
Abstract:
N/A
Nonlinear Sturm global attractors: Unstable manifold decompositions as regular CW-complexes
Bernold Fiedler and Carlos Rocha
2014, 34(12): 5099-5122 doi: 10.3934/dcds.2014.34.5099 +[Abstract](593) +[PDF](572.3KB)
Abstract:
We study global attractors $\mathcal{A}_f$ of scalar partial differential equations $u_t=u_{xx}+f(x,u,u_x)$ on the unit interval with, say, Neumann boundary. Due to nodal properties of differences of solutions, which amount to a nonlinear Sturm property, we call $\mathcal{A}_f$ a Sturm global attractor. We assume all equilibria $v$ to be hyperbolic. Due to a gradient-like structure we can then write \begin{equation} \mathcal{A}_f = \bigcup\limits_{v}\, W^u(v)                                            (*) \end{equation} as a dynamic decomposition into finitely many disjoint invariant sets: the unstable manifolds $W^u(v)$ of the equilibria $v$. Based on our previous Schoenflies result [17], we prove that the dynamic decomposition $(*)$ is in fact a regular finite CW-complex with cells $W^u(v)$, in the Sturm case. We call this complex the regular dynamic complex or Sturm complex of the Sturm attractor $\mathcal{A}_f$.
    We characterize the planar Sturm complexes by bipolar orientations of their 1-skeletons. We also show that any regular finite CW-complex which is the closure of a single 3-cell arises as a Sturm complex. We include a preliminary discussion of the tetrahedron and the octahedron as Sturm complexes.
Modelling collective cell behaviour
Deborah C. Markham, Ruth E. Baker and Philip K. Maini
2014, 34(12): 5123-5133 doi: 10.3934/dcds.2014.34.5123 +[Abstract](823) +[PDF](688.6KB)
Abstract:
The classical mean-field approach to modelling biological systems makes a number of simplifying assumptions which typically lead to coupled systems of reaction-diffusion partial differential equations. While these models have been very useful in allowing us to gain important insights into the behaviour of many biological systems, recent experimental advances in our ability to track and quantify cell behaviour now allow us to build more realistic models which relax some of the assumptions previously made. This brief review aims to illustrate the type of models obtained using this approach.
Measuring the total amount of chaos in some Hamiltonian systems
Carles Simó
2014, 34(12): 5135-5164 doi: 10.3934/dcds.2014.34.5135 +[Abstract](694) +[PDF](10457.4KB)
Abstract:
We consider some simple Hamiltonian systems, variants or generalizations of the Hénon-Heiles system, in two and three degrees of freedom, around a positive definite elliptic point, in resonant and non-resonant cases. After reviewing some theoretical background, we determine a measure of the domain of chaoticity by looking at the frequency of positive Lyapunov exponents in a sample of initial conditions. The question we study is how this measure depends on the energy and parameters and which are the dynamical objects responsible for the observed behaviour.
On a Keller-Segel system with logarithmic sensitivity and non-diffusive chemical
Jaewook Ahn and Kyungkeun Kang
2014, 34(12): 5165-5179 doi: 10.3934/dcds.2014.34.5165 +[Abstract](860) +[PDF](441.4KB)
Abstract:
We consider a chemotactic system with a logarithmic sensitivity and a non-diffusing chemical. We establish local regular solutions in time and give some characterizations on parameters and initial data for global solutions and blow-up in a finite time. We also prove that there does not exist finite time self-similar solution of the backward type.
Multi-hump solutions of some singularly-perturbed equations of KdV type
J. W. Choi, D. S. Lee, S. H. Oh, S. M. Sun and S. I. Whang
2014, 34(12): 5181-5209 doi: 10.3934/dcds.2014.34.5181 +[Abstract](548) +[PDF](565.1KB)
Abstract:
This paper studies the existence of multi-hump solutions with oscillations at infinity for a class of singularly perturbed 4th-order nonlinear ordinary differential equations with $\epsilon > 0$ as a small parameter. When $\epsilon =0$, the equation becomes an equation of KdV type and has solitary-wave solutions. For $\epsilon > 0 $ small, it is proved that such equations have single-hump (also called solitary wave or homoclinic) solutions with small oscillations at infinity, which approach to the solitary-wave solutions for $\epsilon = 0$ as $\epsilon$ goes to zero. Furthermore, it is shown that for small $\epsilon > 0$ the equations have two-hump solutions with oscillations at infinity. These two-hump solutions can be obtained by patching two appropriate single-hump solutions together. The amplitude of the oscillations at infinity is algebraically small with respect to $\epsilon$ as $\epsilon \rightarrow 0$. The idea of the proof may be generalized to prove the existence of symmetric solutions of $2^n$-humps with $n=2,3,\dots,$ for the equations. However, this method cannot be applied to show the existence of general nonsymmetric multi-hump solutions.
Cocycle rigidity and splitting for some discrete parabolic actions
Danijela Damjanović and James Tanis
2014, 34(12): 5211-5227 doi: 10.3934/dcds.2014.34.5211 +[Abstract](508) +[PDF](475.7KB)
Abstract:
We prove trivialization of the first cohomology with coefficients in smooth vector fields, for a class of $\mathbb{Z}^2$ parabolic actions on $(SL(2, \mathbb R)\times SL(2, \mathbb R))/\Gamma$, where the lattice $\Gamma$ is irreducible and co-compact. We also obtain a splitting construction involving first and second coboundary operators in the cohomology with coefficients in smooth vector fields.
Some remarks on symmetric periodic orbits in the restricted three-body problem
Jungsoo Kang
2014, 34(12): 5229-5245 doi: 10.3934/dcds.2014.34.5229 +[Abstract](633) +[PDF](1264.3KB)
Abstract:
The planar circular restricted three-body problem (PCRTBP) is symmetric with respect to the line of masses and there is a corresponding anti-symplectic involution on the cotangent bundle of the 2-sphere in the regularized PCRTBP. Recently it turned out that each bounded component of an energy hypersurface with low energy for the regularized PCRTBP is fiberwise starshaped. This enables us to define a Lagrangian Rabinowitz Floer homology which is related to periodic orbits symmetric for the anti-symplectic involution in the regularized PCRTBP and hence to symmetric periodic orbits in the unregularized problem. We compute this homology and discuss the properties of the symmetric periodic orbits.
Stochastic adding machine and $2$-dimensional Julia sets
Ali Messaoudi and Rafael Asmat Uceda
2014, 34(12): 5247-5269 doi: 10.3934/dcds.2014.34.5247 +[Abstract](695) +[PDF](787.9KB)
Abstract:
In this work we define a stochastic adding machine associated to a quadratic base $(F_n)_{n \geq 0}$ formed by recurrent sequences of order 2. We obtain a Markov chain with states in $\mathbb{Z}^+$ and we prove that the spectrum of the transition operator associated to this Markov chain is connected to the filled Julia sets for a class of endomorphisms in $\mathbb{C}^2$ of which we study topological properties.
The existence and stability of nontrivial steady states for S-K-T competition model with cross diffusion
Wei-Ming Ni, Yaping Wu and Qian Xu
2014, 34(12): 5271-5298 doi: 10.3934/dcds.2014.34.5271 +[Abstract](727) +[PDF](534.8KB)
Abstract:
This paper concerns with the existence and stability properties of non-constant positive steady states in one dimensional space for the following competition system with cross diffusion $$\left\{ \begin{array}{ll} u_t=[(d_{1}+\rho_{12}v)u]_{xx}+u(a_{1}-b_{1}u-c_{1}v),&x\in(0,1), t>0, \\ v_t= d_{2}v_{xx}+v(a_{2}-b_{2}u-c_{2}v),& x\in(0,1),t>0,                    (1) \\ u_{x}=v_{x}=0, &x=0,1, t>0. \end{array}\right. $$ First, by Lyapunov-Schmidt method, we obtain the existence and the detailed structure of a type of small nontrivial positive steady states to the shadow system of (1) as $\rho_{12}\to \infty$ and when $d_2$ is near $a_2/\pi^2$, which also verifies some related existence results obtained earlier in [11] by a different method. Then, based on the detailed structure of the steady states, we further establish the stability of the small nontrivial positive steady states for the shadow system by spectral analysis. Finally, we prove the existence and stability of the corresponding nontrivial positive steady states for the original cross diffusion system (1) when $\rho_{12}$ is large enough and $d_2$ is near $a_2/\pi^2$.
Segregated peak solutions of coupled Schrödinger systems with Neumann boundary conditions
Zhongwei Tang
2014, 34(12): 5299-5323 doi: 10.3934/dcds.2014.34.5299 +[Abstract](557) +[PDF](493.1KB)
Abstract:
We consider the following two coupled Schrödinger systems in a bounded domain $\Omega\subset \mathbb{R}^N(N=2,3)$ with Neumann boundary conditions $$\left\{ \begin{array}{ll} -\epsilon^2 \triangle u + u = \mu_1 u^3+ \beta u v^2,\\ -\epsilon^2 \triangle v + v =\mu_2 v^3+ \beta u^2 v,\\ u>0,    v>0, \\ \partial u/\partial n = 0,\partial v/\partial n = 0, \mbox{on } \partial \Omega. \end{array}\right. $$ Suppose the mean curvature $H(P)$ of the boundary $\partial \Omega$ admits several local maximums( or local minimums), we obtain the existence of segregated solutions $(u_\epsilon,v_\epsilon)$ to the above system such that both of $u_\epsilon$ and $v_\epsilon$ admit more than one local maximums, furthermore as $\epsilon$ goes to zero, the maximum points of $u_\epsilon$ and $v_\epsilon$ concentrate at different local maximum points( or local minimum points) of the mean curvature $H(P)$ respectively.
Optimal parameter-dependent bounds for Kuramoto-Sivashinsky-type equations
Ralf W. Wittenberg
2014, 34(12): 5325-5357 doi: 10.3934/dcds.2014.34.5325 +[Abstract](636) +[PDF](1331.0KB)
Abstract:
We derive a priori estimates on the absorbing ball in $L^2$ for the stabilized and destabilized Kuramoto-Sivashinsky (KS) equations, and for a sixth-order analog, the Nikolaevskiy equation, and in each case obtain bounds whose parameter dependence is demonstrably optimal. This is done by extending a Lyapunov function construction developed by Bronski and Gambill (Nonlinearity 19 , 2023--2039 (2006)) to take into account the dependence on both large and small parameters in the system. In the case of the destabilized KS equation, the rigorous bound lim $\sup_{t \to \infty}|| u || \leq K \alpha L^{3/2}$ is sharp in both the large parameter $\alpha$ and the system size $L$. We also apply our methods to improve previous estimates on a nonlocal variant of the KS equation.
Realization of tangent perturbations in discrete and continuous time conservative systems
Hassan Najafi Alishah and João Lopes Dias
2014, 34(12): 5359-5374 doi: 10.3934/dcds.2014.34.5359 +[Abstract](547) +[PDF](595.4KB)
Abstract:
We prove that any perturbation of the symplectic part of the derivative of a Poisson diffeomorphism can be realized as the derivative of a $C^1$-close Poisson diffeomorphism. We also show that a similar property holds for the Poincaré map of a Hamiltonian on a Poisson manifold. These results are the conservative counterparts of the Franks lemma, a perturbation tool used in several contexts most notably in the theory of smooth dynamical systems.

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