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Discrete & Continuous Dynamical Systems - A

2014 , Volume 34 , Issue 9

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A Liouville-type theorem for higher order elliptic systems
Frank Arthur, Xiaodong Yan and Mingfeng Zhao
2014, 34(9): 3317-3339 doi: 10.3934/dcds.2014.34.3317 +[Abstract](592) +[PDF](458.6KB)
Abstract:
We prove there are no positive solutions to higher order elliptic system \begin{equation*} \left\{ \begin{array}{c} \left( -\Delta \right) ^{m}u=v^{p} \\ \left( -\Delta \right) ^{m}v=u^{q} \end{array} \text{ in }\mathbb{R}^{N}\right. \end{equation*} if $p\geq 1,$ $q\geq 1$, and $( p,q) \neq ( 1,1) $ satisfies $\frac{1}{p+1}+\frac{1}{q+1}>1-\frac{2m}{N}$ and $\max \left( \frac{2\left( p+1\right) }{pq-1},\frac{2\left( q+1\right) }{pq-1}\right) > \frac{N-2m-1}{m}.$ Moreover, if $N=2m+1$ or $N=2m+2,$ this system admits no positive solutions if $p\geq 1,$ $q\geq 1,$ $\left( p,q\right) \neq \left( 1,1\right) $ satisfies $\frac{1}{p+1}+\frac{1}{q+1}>1-\frac{2m}{N}.$
Density of fiberwise orbits in minimal iterated function systems on the circle
Pablo G. Barrientos, Abbas Fakhari and Aliasghar Sarizadeh
2014, 34(9): 3341-3352 doi: 10.3934/dcds.2014.34.3341 +[Abstract](604) +[PDF](374.8KB)
Abstract:
We study the minimality of almost every orbital branch of minimal iterated function systems (IFSs). We prove that this kind of minimality holds for forward and backward minimal IFSs generated by orientation-preserving homeomorphisms of the circle. We provide new examples of iterated functions systems where this behavior persists under perturbation of the generators.
Homoclinic orbits of first-order superquadratic Hamiltonian systems
Cyril Joel Batkam
2014, 34(9): 3353-3369 doi: 10.3934/dcds.2014.34.3353 +[Abstract](542) +[PDF](452.9KB)
Abstract:
In this article, we study the existence of homoclinic orbits for the first-order Hamiltonian system \begin{equation*} J\dot{u}(t)+\nabla H(t,u(t))=0,\quad t\in\mathbb{R}. \end{equation*} Under the Ambrosetti-Rabinowitz's superquadraticy condition, or no Ambrosetti-Rabinowitz's superquadracity condition, we present two results on the existence of infinitely many large energy homoclinic orbits when $H$ is even in $u$. We apply the generalized (variant) fountain theorems established recently by the author and Colin. Under no Ambrosetti-Rabinowitz's superquadracity condition, we also obtain the existence of a ground state homoclinic orbit by using the method of the generalized Nehari manifold for strongly indefinite functionals developed by Szulkin and Weth.
A sharper energy method for the localization of the support to some stationary Schrödinger equations with a singular nonlinearity
Pascal Bégout and Jesús Ildefonso Díaz
2014, 34(9): 3371-3382 doi: 10.3934/dcds.2014.34.3371 +[Abstract](399) +[PDF](417.4KB)
Abstract:
We prove the compactness of the support of the solution of some stationary Schrödinger equations with a singular nonlinear order term. We present here a sharper version of some energy methods previously used in the literature.
On the microscopic spacetime convexity principle of fully nonlinear parabolic equations I: Spacetime convex solutions
Chuanqiang Chen
2014, 34(9): 3383-3402 doi: 10.3934/dcds.2014.34.3383 +[Abstract](441) +[PDF](405.4KB)
Abstract:
Spacetime convexity is a basic geometric property of the solutions of parabolic equations. In this paper, we study microscopic convexity properties of spacetime convex solutions of fully nonlinear parabolic partial differential equations and give a new simple proof of the constant rank theorem in [11].
Justification of leading order quasicontinuum approximations of strongly nonlinear lattices
Christopher Chong, P.G. Kevrekidis and Guido Schneider
2014, 34(9): 3403-3418 doi: 10.3934/dcds.2014.34.3403 +[Abstract](384) +[PDF](392.7KB)
Abstract:
We consider the leading order quasicontinuum limits of a one-dimensional granular medium governed by the Hertz contact law under precompression. The approximate model which is derived in this limit is justified by establishing asymptotic bounds for the error with the help of energy estimates. The continuum model predicts the development of shock waves, which are also studied in the full system with the aid of numerical simulations. We also show that existing results concerning the Nonlinear Schrödinger (NLS) and Korteweg de-Vries (KdV) approximation of FPU models apply directly to a precompressed granular medium in the weakly nonlinear regime.
Generalized pitchfork bifurcation on a two-dimensional gaseous star with self-gravity and surface tension
Shengfu Deng
2014, 34(9): 3419-3435 doi: 10.3934/dcds.2014.34.3419 +[Abstract](587) +[PDF](414.8KB)
Abstract:
Travelling wave solutions of a two-dimensional gaseous star with self-gravity and surface tension are considered. The star rotates along a clockwise direction with a travelling speed. The governing equations on a whole unknown domain are changed to ones on the boundary using the Dirichlet-Neumann operator. The problem of the existence of its periodic solutions is equivalent to one of a functional equation. After applying the method of Lyapunov-Schmidt reduction, the reduced equation of this functional equation has a generalized Pitchfork bifurcation with the bifurcation parameter being the travelling speed. This shows that there exist two nontrivial periodic solutions.
On a generalized maximum principle for a transport-diffusion model with $\log$-modulated fractional dissipation
Hongjie Dong and Dong Li
2014, 34(9): 3437-3454 doi: 10.3934/dcds.2014.34.3437 +[Abstract](422) +[PDF](448.1KB)
Abstract:
We consider a transport-diffusion equation of the form $\partial_t \theta + v \cdot \nabla \theta + \nu \mathcal{A} \theta = 0$, where $v$ is a given time-dependent vector field on $\mathbb R^d$. The operator $\mathcal{A}$ represents log-modulated fractional dissipation: $\mathcal{A}=\frac {|\nabla|^{\gamma}}{\log^{\beta}(\lambda+|\nabla|)}$ and the parameters $\nu\ge 0$, $\beta\ge 0$, $0\le \gamma \le 2$, $\lambda>1$. We introduce a novel nonlocal decomposition of the operator $\mathcal{A}$ in terms of a weighted integral of the usual fractional operators $|\nabla|^{s}$, $0\le s \le \gamma$ plus a smooth remainder term which corresponds to an $L^1$ kernel. For a general vector field $v$ (possibly non-divergence-free) we prove a generalized $L^\infty$ maximum principle of the form $ \| \theta(t)\|_\infty \le e^{Ct} \| \theta_0 \|_{\infty}$ where the constant $C=C(\nu,\beta,\gamma)>0$. In the case $\text{div}(v)=0$ the same inequality holds for $\|\theta(t)\|_p$ with $1\le p \le \infty$. Under the additional assumption that $\theta_0\in L^2$, we show that $\|\theta(t)\|_p$ is uniformly bounded for $2\le p\le \infty$. At the cost of a possible exponential factor, this extends a recent result of Hmidi [7] to the full regime $d\ge 1$, $0\le \gamma \le 2$ and removes the incompressibility assumption in the $L^\infty$ case.
Periodic solutions of El Niño model through the Vallis differential system
Rodrigo Donizete Euzébio and Jaume Llibre
2014, 34(9): 3455-3469 doi: 10.3934/dcds.2014.34.3455 +[Abstract](547) +[PDF](357.8KB)
Abstract:
By rescaling the variables, the parameters and the periodic function of the Vallis differential system we provide sufficient conditions for the existence of periodic solutions and we also characterize their kind of stability. The results are obtained using averaging theory.
Quasiconformal Anosov flows and quasisymmetric rigidity of Hamenst$\ddot{a}$dt distances
Yong Fang
2014, 34(9): 3471-3483 doi: 10.3934/dcds.2014.34.3471 +[Abstract](369) +[PDF](394.6KB)
Abstract:
This paper deals with interactions between metric quasiconformal geometry and the rigidity of Anosov flows. In the first part of this article, we study a canonical time change of Anosov flows. Then we use it to obtain the thorough classification of volume-preserving quasiconformal Anosov flows.
    Given a $C^\infty$ Anosov flow $\varphi$, it is well-known that along the leaves of its strong unstable foliation there are two natural distances: the Hamenst$\ddot {\rm a}$dt distance and the induced Riemannian distance. In general these two distances are H$\ddot{\rm o}$lder equivalent. We prove the following rigidity result about quasisymmetric equivalence:
    Let $\varphi: M\to M$ be a $C^\infty$ transversely symplectic Anosov flow with $C^1$ weak distributions. We suppose that ${\rm dim}M\geq 5$.
    If over a leaf of the strong unstable foliation, the Hamenst$\ddot{\rm a}$dt distance is quasisymmetric equivalent to the Riemannian distance, then up to finite covers $\varphi$ is $C^\infty$ orbit equivalent either to the geodesic flow of a closed hyperbolic manifold, or to the suspension of an Anosov automorphism.
Dynamics of Klein-Gordon on a compact surface near a homoclinic orbit
Benoît Grébert, Tiphaine Jézéquel and Laurent Thomann
2014, 34(9): 3485-3510 doi: 10.3934/dcds.2014.34.3485 +[Abstract](509) +[PDF](545.0KB)
Abstract:
We consider the Klein-Gordon equation (KG) on a Riemannian surface $M$ $$ \partial^{2}_t u-\Delta u-m^{2}u+u^{2p+1} =0,\quad p\in \mathbb{N}^{*},\quad (t,x)\in \mathbb{R}\times M,$$ which is globally well-posed in the energy space. This equation has a homoclinic orbit to the origin, and in this paper we study the dynamics close to it. Using a strategy from Groves-Schneider, we get the existence of a large family of heteroclinic connections to the center manifold that are close to the homoclinic orbit during all times. We point out that the solutions we construct are not small.
Slowly oscillating wavefronts of the KPP-Fisher delayed equation
Karel Hasik and Sergei Trofimchuk
2014, 34(9): 3511-3533 doi: 10.3934/dcds.2014.34.3511 +[Abstract](425) +[PDF](625.5KB)
Abstract:
This paper concerns the semi-wavefronts (i.e. bounded solutions $u=\phi(x\cdot\nu +ct) >0,$ $ |\nu|=1, $ satisfying $\phi(-\infty)=0$) to the delayed KPP-Fisher equation $u_t(t,x) = \Delta u(t,x) + u(t,x)(1-u(t-\tau,x)), \ u \geq 0,\ x \in \mathbb{R}^m.$ First, we show that the profile $\phi$ of each semi-wavefront should be either monotone or eventually sine-like slowly oscillating around the positive equilibrium. Then a solution to the problem of existence of semi-wavefronts is provided. Next, we prove that the semi-wavefronts are in fact wavefronts (i.e. additionally $\phi(+\infty)=1$) if $c \geq 2$ and $\tau \leq 1$; our proof uses dynamical properties of an auxiliary one-dimensional map with the negative Schwarzian. However, we also show that, for $c \geq 2$ and $\tau \geq 1.87$, each semi-wavefront profile $\phi(t)$ should develop non-decaying oscillations around $1$ as $t \to +\infty$.
Some results for the Perelman LYH-type inequality
Shu-Yu Hsu
2014, 34(9): 3535-3554 doi: 10.3934/dcds.2014.34.3535 +[Abstract](421) +[PDF](170.1KB)
Abstract:
Let $(M,g(t))$, $0\le t\le T$, $\partial M\ne\phi$, be a compact $n$-dimensional manifold, $n\ge 2$, with metric $g(t)$ evolving by the Ricci flow such that the second fundamental form of $\partial M$ with respect to the unit outward normal of $\partial M$ is uniformly bounded below on $\partial M\times [0,T]$. We will prove a global Li-Yau gradient estimate for the solution of the generalized conjugate heat equation on $M\times [0,T]$. We will give another proof of Perelman's Li-Yau-Hamilton type inequality for the fundamental solution of the conjugate heat equation on closed manifolds without using the properties of the reduced distance. We will also prove various gradient estimates for the Dirichlet fundamental solution of the conjugate heat equation.
An averaging theorem for nonlinear Schrödinger equations with small nonlinearities
Guan Huang
2014, 34(9): 3555-3574 doi: 10.3934/dcds.2014.34.3555 +[Abstract](467) +[PDF](484.5KB)
Abstract:
Consider nonlinear Schrödinger equations with small nonlinearities \[\frac{d}{dt}u+i(-\triangle u+V(x)u)=\epsilon \mathcal{P}(\triangle u,\nabla u,u,x),           x\in \mathbb{T}^d.                     (*)\] Let $\{\zeta_1(x),\zeta_2(x),\dots\}$ be the $L_2$-basis formed by eigenfunctions of the operator $-\triangle +V(x)$. For any complex function $u(x)$, write it as $ u(x) = \sum_{k≥1} v_k \zeta_k (x)$ and set $I_k(u)=\frac{1}{2}|v_k|^2$. Then for any solution $u(t,x)$ of the linear equation $(*)_{\epsilon=0}$ we have $I(u(t,\cdot))=const$. In this work it is proved that if $(*)$ is well posed on time-intervals $t≲\epsilon^{-1}$ and satisfies there some mild a-priori assumptions, then for any its solution $u^{\epsilon}(t,x)$, the limiting behavior of the curve $I(u^{\epsilon}(t,\cdot))$ on time intervals of order $\epsilon^{-1}$, as $\epsilon\to0$, can be uniquely characterized by solutions of a certain well-posed effective equation.
Large time behaviour for a nonlocal diffusion - convection equation related with gas dynamics
Liviu I. Ignat and Ademir F. Pazoto
2014, 34(9): 3575-3589 doi: 10.3934/dcds.2014.34.3575 +[Abstract](506) +[PDF](396.4KB)
Abstract:
In this paper we consider a model that involves nonlocal diffusion and a classical convective term. Using a scaling argument and a new compactness argument we obtain the first term in the asymptotic behavior of the solutions. Such scaling argument is very common for the study of long time behavior of solutions to evolutionary problems where a scaling invariance of the main part of the operator is present.
Shadowing is generic---a continuous map case
Piotr Kościelniak, Marcin Mazur, Piotr Oprocha and Paweł Pilarczyk
2014, 34(9): 3591-3609 doi: 10.3934/dcds.2014.34.3591 +[Abstract](593) +[PDF](449.9KB)
Abstract:
We prove that shadowing (the pseudo-orbit tracing property), periodic shadowing (tracing periodic pseudo-orbits with periodic real trajectories), and inverse shadowing with respect to certain families of methods (tracing all real orbits of the system with pseudo-orbits generated by arbitrary methods from these families) are all generic in the class of continuous maps and in the class of continuous onto maps on compact topological manifolds (with or without boundary) that admit a decomposition (including triangulable manifolds and manifolds with handlebody).
The bang-bang property of time optimal controls for the Burgers equation
Karl Kunisch and Lijuan Wang
2014, 34(9): 3611-3637 doi: 10.3934/dcds.2014.34.3611 +[Abstract](541) +[PDF](514.7KB)
Abstract:
The bang-bang property of time optimal controls for the Burgers equations in dimension up to three, with homogeneous Dirichlet boundary conditions and distributed controls acting on an open subset of the domain is established. This relies on an observability estimate from a measurable set in time for linear parabolic equations, with potentials depending on both space and time variables. The proof of the bang-bang property relies on a Kakutani fixed point argument.
Invariant foliations for random dynamical systems
Ji Li, Kening Lu and Peter W. Bates
2014, 34(9): 3639-3666 doi: 10.3934/dcds.2014.34.3639 +[Abstract](610) +[PDF](434.8KB)
Abstract:
We prove the existence of invariant foliations of stable and unstable manifolds of a normally hyperbolic random invariant manifold. The normally hyperbolic random invariant manifold referred to here is that which was shown to exist in a previous paper when a deterministic dynamical system having a normally hyperbolic invariant manifold is subjected to a small random perturbation.
Weak-Painlevé property and integrability of general dynamical systems
Wenlei Li and Shaoyun Shi
2014, 34(9): 3667-3681 doi: 10.3934/dcds.2014.34.3667 +[Abstract](491) +[PDF](414.5KB)
Abstract:
The purpose of this paper is to investigate the connection between singular property and integrability for general dynamical systems. We will firstly present some methods to test the Painlevé property and weak-Painlevé property, then we will show the equivalence between the weak-Painlevé property and certain formal integrability for general dynamical systems.
Effectual leadership in flocks with hierarchy and individual preference
Zhuchun Li
2014, 34(9): 3683-3702 doi: 10.3934/dcds.2014.34.3683 +[Abstract](531) +[PDF](547.8KB)
Abstract:
We study the emergent flocking behavior in a group of interacting particles following a Cucker-Smale type model with hierarchical leadership and individual effects. In this model, each individual adjusts its velocity to match that of its leaders and in meantime has a preferred acceleration if there is no local velocity consensus. We give some sufficient conditions on the range of parameters and initial configurations to have an effectual leadership, i.e., to guarantee the asymptotic convergence to a velocity agreement with the leader. To do this, we explore a special matrix norm for the flocking under hierarchical leadership.
Spatially structured networks of pulse-coupled phase oscillators on metric spaces
Stilianos Louca and Fatihcan M. Atay
2014, 34(9): 3703-3745 doi: 10.3934/dcds.2014.34.3703 +[Abstract](536) +[PDF](1108.5KB)
Abstract:
The Winfree model describes finite networks of phase oscillators. Oscillators interact by broadcasting pulses that modulate the frequencies of connected oscillators. We study a generalization of the model and its fluid-dynamical limit for networks, where oscillators are distributed on some abstract $\sigma$-finite Borel measure space over a separable metric space. We give existence and uniqueness statements for solutions to the continuity equation for the oscillator phase densities. We further show that synchrony in networks of identical oscillators is locally asymptotically stable for finite, strictly positive measures and under suitable conditions on the oscillator response function and the coupling kernel of the network. The conditions on the latter are a generalization of the strong connectivity of finite graphs to abstract coupling kernels.
Memory loss for nonequilibrium open dynamical systems
Anushaya Mohapatra and William Ott
2014, 34(9): 3747-3759 doi: 10.3934/dcds.2014.34.3747 +[Abstract](472) +[PDF](407.3KB)
Abstract:
We introduce a notion of conditional memory loss for nonequilibrium open dynamical systems. We prove that this type of memory loss occurs at an exponential rate for nonequilibrium open systems generated by one-dimensional piecewise-differentiable expanding Lasota-Yorke maps. This result may be viewed as a prototype for time-dependent dynamical systems with holes.
Shadowing near nonhyperbolic fixed points
Alexey A. Petrov and Sergei Yu. Pilyugin
2014, 34(9): 3761-3772 doi: 10.3934/dcds.2014.34.3761 +[Abstract](441) +[PDF](309.8KB)
Abstract:
We use Lyapunov type functions to find conditions of finite shadowing in a neighborhood of a nonhyperbolic fixed point of a one-dimensional or two-dimensional homeomorphism or diffeomorphism. A new concept of shadowing in which we control the size of one-step errors is introduced in the case of a nonisolated fixed point.
Modulation of uniform motion in diatomic Frenkel-Kontorova model
Wen-Xin Qin
2014, 34(9): 3773-3788 doi: 10.3934/dcds.2014.34.3773 +[Abstract](487) +[PDF](475.1KB)
Abstract:
We study modulated structures of uniform motion in the diatomic Frenkel-Kontorova (FK) model with alternating light and heavy particles. By applying topological method for the damped and driven case and variational approach for the conservative case, we demonstrate for the diatomic FK model the existence of two different periodic modulation functions corresponding respectively to light and heavy particles.
Volume growth and entropy for $C^1$ partially hyperbolic diffeomorphisms
Radu Saghin
2014, 34(9): 3789-3801 doi: 10.3934/dcds.2014.34.3789 +[Abstract](478) +[PDF](372.8KB)
Abstract:
We show that the metric entropy of a $C^1$ diffeomorphism with a dominated splitting and with the dominating bundle uniformly expanding is bounded from above by the integrated volume growth of the dominating (expanding) bundle plus the maximal Lyapunov exponent from the dominated bundle multiplied by its dimension. We also discuss different types of volume growth that can be associated with an expanding foliation and relationships between them, specially when there exists a closed form non-degenerate on the foliation. Some consequences for partially hyperbolic diffeomorphisms are presented.
On resolving singularities of piecewise-smooth discontinuous vector fields via small perturbations
David J. W. Simpson
2014, 34(9): 3803-3830 doi: 10.3934/dcds.2014.34.3803 +[Abstract](412) +[PDF](714.6KB)
Abstract:
A two-fold singularity is a point on a discontinuity surface of a piecewise-smooth vector field at which the vector field is tangent to the surface on both sides. Due to the double tangency, forward evolution from a two-fold is typically ambiguous. This is an especially serious issue for two-folds that are reached by the forward orbits of a non-zero measure set of initial points. However, arbitrarily small perturbations of the vector field can make forward evolution well-defined, and from an applied perspective, such perturbations may represent additional model features that enhance the realism of a piecewise-smooth mathematical model. Three physically motivated forms of perturbation: hysteresis, time-delay, and noise, are analysed individually. The purpose of this paper is to characterise the perturbed dynamics in the limit that the size of the perturbation tends to zero. This concept is applied to a two-fold in two dimensions. In each case the limit leads to a novel probabilistic notion of forward evolution from the two-fold.
Blow-up of solutions to the one-dimensional semilinear wave equation with damping depending on time and space variables
Yuta Wakasugi
2014, 34(9): 3831-3846 doi: 10.3934/dcds.2014.34.3831 +[Abstract](556) +[PDF](415.8KB)
Abstract:
In this paper, we give a small data blow-up result for the one-dimensional semilinear wave equation with damping depending on time and space variables. We show that if the damping term can be regarded as perturbation, that is, non-effective damping in a certain sense, then the solution blows up in finite time for any power of nonlinearity. This gives an affirmative answer for the conjecture that the critical exponent agrees with that of the wave equation when the damping is non-effective in one space dimension.
Dimension estimates in non-conformal setting
Juan Wang, Yongluo Cao and Yun Zhao
2014, 34(9): 3847-3873 doi: 10.3934/dcds.2014.34.3847 +[Abstract](472) +[PDF](481.0KB)
Abstract:
Given an arbitrary subset of a non-conformal repeller of a $C^1$ map. Using directly the definitions of Hausdorff dimension and Box dimension and of pressure, this paper first proves that the zeros of the topological pressure on this set give its dimension estimates. And without using the estimate of pointwise dimension of an ergodic measure on a non-conformal repeller, it is showed that the zeros of non-additive measure-theoretic pressure give the lower and upper bound of dimension estimate for it. Some results from [22,41] are extended for $C^1$ maps or arbitrary subsets of a non-conformal repeller. And a remark on Rugh's result [34] is also given in this paper.
Positive steady state solutions of a diffusive Leslie-Gower predator-prey model with Holling type II functional response and cross-diffusion
Jun Zhou, Chan-Gyun Kim and Junping Shi
2014, 34(9): 3875-3899 doi: 10.3934/dcds.2014.34.3875 +[Abstract](610) +[PDF](1145.1KB)
Abstract:
In this paper we consider a diffusive Leslie-Gower predator-prey model with Holling type II functional response and cross-diffusion under zero Dirichlet boundary condition. By using topological degree theory, bifurcation theory, energy estimates and asymptotic behavior analysis, we prove the existence, uniqueness and multiplicity of positive steady states solutions under certain conditions on the parameters.

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