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Discrete & Continuous Dynamical Systems - A

2015 , Volume 35 , Issue 5

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On the integrability of polynomial vector fields in the plane by means of Picard-Vessiot theory
Primitivo B. Acosta-Humánez, J. Tomás Lázaro, Juan J. Morales-Ruiz and Chara Pantazi
2015, 35(5): 1767-1800 doi: 10.3934/dcds.2015.35.1767 +[Abstract](373) +[PDF](628.0KB)
Abstract:
We study the integrability of polynomial vector fields using Galois theory of linear differential equations when the associated foliations is reduced to a Riccati type foliation. In particular we obtain integrability results for some families of quadratic vector fields, Liénard equations and equations related with special functions such as Hypergeometric and Heun ones. The Poincaré problem for some families is also approached.
The magnetic ray transform on Anosov surfaces
Gareth Ainsworth
2015, 35(5): 1801-1816 doi: 10.3934/dcds.2015.35.1801 +[Abstract](287) +[PDF](396.3KB)
Abstract:
Assume (M,g,$\Omega$) is a closed, oriented Riemannian surface equipped with an Anosov magnetic flow. We establish certain results on the surjectivity of the adjoint of the magnetic ray transform, and use these to prove the injectivity of the magnetic ray transform on sums of tensors of degree at most two. In the final section of the paper we give an application to the entropy production of magnetic flows perturbed by symmetric 2-tensors.
Actions of solvable Baumslag-Solitar groups on surfaces with (pseudo)-Anosov elements
Juan Alonso, Nancy Guelman and Juliana Xavier
2015, 35(5): 1817-1827 doi: 10.3934/dcds.2015.35.1817 +[Abstract](296) +[PDF](379.3KB)
Abstract:
Let $BS(1,n)= \langle a,b : a b a ^{-1} = b ^n\rangle$ be the solvable Baumslag-Solitar group, where $n \geq 2$. We study representations of $BS(1, n)$ by homeomorphisms of closed surfaces of genus $g\geq 1$ with (pseudo)-Anosov elements. That is, we consider a closed surface $S$ of genus $g\geq 1$, and homeomorphisms $f, h: S \to S$ such that $h f h^{-1} = f^n$, for some $ n\geq 2$. It is known that $f$ (or some power of $f$) must be homotopic to the identity. Suppose that $h$ is (pseudo)-Anosov with stretch factor $\lambda >1$. We show that $\langle f,h \rangle$ is not a faithful representation of $BS(1, n)$ if $\lambda > n$. We also show that there are no faithful representations of $BS(1, n)$ by torus homeomorphisms with $h$ an Anosov map and $f$ area preserving (regardless of the value of $\lambda$).
Lipschitz perturbations of expansive systems
Alfonso Artigue
2015, 35(5): 1829-1841 doi: 10.3934/dcds.2015.35.1829 +[Abstract](339) +[PDF](533.5KB)
Abstract:
We extend some known results from smooth dynamical systems to the category of Lipschitz homeomorphisms of compact metric spaces. We consider dynamical properties as robust expansiveness and structural stability allowing Lipschitz perturbations with respect to a hyperbolic metric. We also study the relationship between Lipschitz topologies and the $C^1$ topology on smooth manifolds.
A variational approach to reaction-diffusion equations with forced speed in dimension 1
Juliette Bouhours and Grégroie Nadin
2015, 35(5): 1843-1872 doi: 10.3934/dcds.2015.35.1843 +[Abstract](298) +[PDF](1067.6KB)
Abstract:
We investigate in this paper a scalar reaction diffusion equation with a nonlinear reaction term depending on $x-ct$. Here, $c$ is a prescribed parameter modelling the speed of climate change and we wonder whether a population will survive or not, that is, we want to determine the large-time behaviour of the associated solution.
    This problem has been solved recently when the nonlinearity is of KPP type. We consider in the present paper general reaction terms, that are only assumed to be negative at infinity. Using a variational approach, we construct two thresholds $0<\underline{c}\leq \overline{c} <\infty$ determining the existence and the non-existence of travelling waves. Numerics support the conjecture $\underline{c}=\overline{c}$. We then prove that any solution of the initial-value problem converges at large times, either to $0$ or to a travelling wave. In the case of bistable nonlinearities, where the steady state $0$ is assumed to be stable, our results lead to constrasting phenomena with respect to the KPP framework. Lastly, we illustrate our results and discuss several open questions through numerics.
Stability of singular limit cycles for Abel equations
José Luis Bravo, Manuel Fernández and Armengol Gasull
2015, 35(5): 1873-1890 doi: 10.3934/dcds.2015.35.1873 +[Abstract](346) +[PDF](464.8KB)
Abstract:
We obtain a criterion for determining the stability of singular limit cycles of Abel equations $x'=A(t)x^3+B(t)x^2$. This stability controls the possible saddle-node bifurcations of limit cycles. Therefore, studying the Hopf-like bifurcations at $x=0$, together with the bifurcations at infinity of a suitable compactification of the equations, we obtain upper bounds of their number of limit cycles. As an illustration of this approach, we prove that the family $x'=a t(t-t_A)x^3+b (t-t_B)x^2$, with $a ,b>0$, has at most two positive limit cycles for any $t_B,t_A$.
Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces
Xinru Cao
2015, 35(5): 1891-1904 doi: 10.3934/dcds.2015.35.1891 +[Abstract](322) +[PDF](412.9KB)
Abstract:
In this paper, the fully parabolic Keller-Segel system \begin{equation}\label{problemAbstract}\left\{\begin{array}{ll} u_t=\Delta u-\nabla\cdot(u\nabla v), &(x,t)\in \Omega\times (0,T),\\ v_t=\Delta v-v+u, &(x,t)\in\Omega\times (0,T),\\ \end{array}\right.\tag{$\star$} \end{equation} is considered under Neumann boundary conditions in a bounded domain $\Omega\subset\mathbb{R}^n$ with smooth boundary, where $n\ge 2$. We derive a smallness condition on the initial data in optimal Lebesgue spaces which ensure global boundedness and large time convergence. More precisely, we shall show that one can find $\varepsilon_0>0$ such that for all suitably regular initial data $(u_0,v_0)$ satisfying $\|u_0\|_{L^{\frac{n}{2}}(\Omega)}<\varepsilon_0$ and $\|\nabla v_0\|_{L^{n}(\Omega)}<\varepsilon_0$, the above problem possesses a global classical solution which is bounded and converges to the constant steady state $(m,m)$ with $m:=\frac{1}{|\Omega|}\int_\Omega u_0$.
    Our approach allows us to furthermore study a general chemotaxis system with rotational sensitivity in dimension 2, which is lacking the natural energy structure associated with ($\star$). For such systems, we prove a global existence and boundedness result under corresponding smallness conditions on the initially present total mass of cells and the chemical gradient.
$L^q$-Extensions of $L^p$-spaces by fractional diffusion equations
Der-Chen Chang and Jie Xiao
2015, 35(5): 1905-1920 doi: 10.3934/dcds.2015.35.1905 +[Abstract](423) +[PDF](428.7KB)
Abstract:
Based on the geometric-measure-theoretic analysis of a new $L^p$-type capacity defined in the upper-half Euclidean space, this note characterizes a nonnegative Randon measure $\mu$ on $\mathbb R^{1+n}_+$ such that the extension $R_\alpha L^p(\mathbb R^n)\subseteq L^q(\mathbb R^{1+n}_+,\mu)$ holds for $(\alpha,p,q)\in (0,1)\times(1,\infty)\times(1,\infty)$ where $u=R_\alpha f$ is the weak solution of the fractional diffusion equation $(\partial_t + (-\Delta_x)^\alpha)u(t, x) = 0$ in $\mathbb R^{1+n}_+$ subject to $u(0,x)=f(x)$ in $\mathbb R^n$.
Radial stability of periodic solutions of the Gylden-Meshcherskii-type problem
Jifeng Chu, Pedro J. Torres and Feng Wang
2015, 35(5): 1921-1932 doi: 10.3934/dcds.2015.35.1921 +[Abstract](298) +[PDF](411.4KB)
Abstract:
For the Gylden-Meshcherskii-type problem with a periodically cha-nging gravitational parameter, we prove the existence of radially periodic solutions with high angular momentum, which are Lyapunov stable in the radial direction.
Topological defects in the abelian Higgs model
Magdalena Czubak and Robert L. Jerrard
2015, 35(5): 1933-1968 doi: 10.3934/dcds.2015.35.1933 +[Abstract](281) +[PDF](670.2KB)
Abstract:
We give a rigorous description of the dynamics of the Nielsen-Olesen vortex line. In particular, given a worldsheet of a string, we construct initial data such that the corresponding solution of the abelian Higgs model will concentrate near the evolution of the string. Moreover, the constructed solution stays close to the Nielsen-Olesen vortex solution.
Integrability of potentials of degree $k \neq \pm 2$. Second order variational equations between Kolchin solvability and Abelianity
Guillaume Duval and Andrzej J. Maciejewski
2015, 35(5): 1969-2009 doi: 10.3934/dcds.2015.35.1969 +[Abstract](348) +[PDF](564.8KB)
Abstract:
In our previous paper [4], we tried to extract some particular structures of the higher variational equations (the $\mathrm{VE}_p$ for $p \geq 2$), along particular solutions of natural Hamiltonian systems with homogeneous potential of degree $k=\pm 2$. We investigate these variational equations in a framework of differential Galois theory. Our aim was to obtain new obstructions for complete integrability. In this paper we extend results of [4] to the complementary cases, when the homogeneous potential has integer degree of homogeneity $k\in\mathbb{Z}$, and $|k| \geq 3$. Since these cases are much more general and complicated, we restrict our study only to the second order variational equation $\mathrm{VE}_2$.
A note on the Cauchy problem of a modified Camassa-Holm equation with cubic nonlinearity
Ying Fu
2015, 35(5): 2011-2039 doi: 10.3934/dcds.2015.35.2011 +[Abstract](373) +[PDF](727.1KB)
Abstract:
Considered herein is the Cauchy problem for a modified Camassa-Holm equation with cubic nonlinearity. The local well-posedness in Besov space $B^s_{2,1}$ with the critical index $s=5/2$ is established. Then a lower bound for the maximal time of existence of its solutions is found. With analytic initial data, the solutions to this Cauchy problem are analytic in both variables, globally in space and locally in time, which extends the result of Himonas and Misiołek [A. Himonas, G. Misiołek, Analyticity of the Cauchy problem for an integrable evolution equation, Math. Ann. 327 (2003) 575---584] to more general $\mu$-version equations and systems.
Blow-up for the two-component Camassa--Holm system
Katrin Grunert
2015, 35(5): 2041-2051 doi: 10.3934/dcds.2015.35.2041 +[Abstract](360) +[PDF](386.7KB)
Abstract:
Following conservative solutions of the two-component Camassa--Holm system $u_t-u_{txx}+3uu_x-2u_xu_{xx}-uu_{xxx}+\rho\rho_x=0$, $\rho_t+(u\rho)_x=0$ along characteristics, we determine if wave breaking occurs in the nearby future or not, for initial data $u_0\in H^1(\mathbb{R})$ and $\rho_0\in L^2(\mathbb{R})$.
Short-wavelength instabilities of edge waves in stratified water
Delia Ionescu-Kruse
2015, 35(5): 2053-2066 doi: 10.3934/dcds.2015.35.2053 +[Abstract](310) +[PDF](463.0KB)
Abstract:
In this paper we make a detailed analysis of the short-wavelength instability method for barotropic incompressible fluids. We apply this method to edge waves in stratified water. These waves are unstable to short-wavelength perturbations if their steepness exceeds a specific threshold.
Wolff type potential estimates and application to nonlinear equations with negative exponents
Yutian Lei
2015, 35(5): 2067-2078 doi: 10.3934/dcds.2015.35.2067 +[Abstract](273) +[PDF](386.9KB)
Abstract:
In this paper, we are concerned with the positive continuous entire solutions of the Wolff type integral equation $$ u(x)=c(x)W_{\beta,\gamma}(u^{-p})(x), \quad u>0 ~in~ R^n, $$ where $n \geq 1$, $p>0$, $\gamma>1$, $\beta>0$ and $\beta\gamma \neq n$. In addition, $c(x)$ is a double bounded function. Such an integral equation is related to the study of the conformal geometry and nonlinear PDEs, such as $\gamma$-Laplace equations and $k$-Hessian equations with negative exponents. By some Wolff type potential integral estimates, we obtain the asymptotic rates and the integrability of positive solutions, and discuss the existence and nonexistence results of the radial solutions.
Projection methods and discrete gradient methods for preserving first integrals of ODEs
Richard A. Norton, David I. McLaren, G. R. W. Quispel, Ari Stern and Antonella Zanna
2015, 35(5): 2079-2098 doi: 10.3934/dcds.2015.35.2079 +[Abstract](343) +[PDF](627.3KB)
Abstract:
In this paper we study linear projection methods for approximating the solution and simultaneously preserving first integrals of autonomous ordinary differential equations. We show that each (linear) projection method is equivalent to a class of discrete gradient methods, in both single and multiple first integral cases, and known results for discrete gradient methods also apply to projection methods. Thus we prove that in the single first integral case, under certain mild conditions, the numerical solution for a projection method exists and is locally unique, and preserves the order of accuracy of the underlying method. Our results allow considerable freedom for the choice of projection direction and do not have a time step restriction close to critical points.
Unbounded regime for circle maps with a flat interval
Liviana Palmisano
2015, 35(5): 2099-2122 doi: 10.3934/dcds.2015.35.2099 +[Abstract](339) +[PDF](962.1KB)
Abstract:
We study $\mathcal{C}^2$ weakly order preserving circle maps with a flat interval. In particular we are interested in the geometry of the mapping near to the singularities at the boundary of the flat interval. Without any assumption on the rotation number we show that the geometry is degenerate when the degree of the singularities is less than or equal to two and becomes bounded when the degree goes to three. As an example of application, the result is applied to study Cherry flows.
One-parameter solutions of the Euler-Arnold equation on the contactomorphism group
Stephen C. Preston and Alejandro Sarria
2015, 35(5): 2123-2130 doi: 10.3934/dcds.2015.35.2123 +[Abstract](438) +[PDF](354.2KB)
Abstract:
We study solutions of the equation $$ g_t-g_{tyy} + 4g^2 - 4gg_{yy} = y gg_{yyy}-yg_yg_{yy}, \qquad y\in\mathbb{R},$$ which arises by considering solutions of the Euler-Arnold equation on a contactomorphism group when the stream function is of the form $f(t,x,y,z) = zg(t,y)$. The equation is analogous to both the Camassa-Holm equation and the Proudman-Johnson equation. We write the equation as an ODE in a Banach space to establish local existence, and we describe conditions leading to global existence and conditions leading to blowup in finite time.
Local integration by parts and Pohozaev identities for higher order fractional Laplacians
Xavier Ros-Oton and Joaquim Serra
2015, 35(5): 2131-2150 doi: 10.3934/dcds.2015.35.2131 +[Abstract](433) +[PDF](511.9KB)
Abstract:
We establish an integration by parts formula in bounded domains for the higher order fractional Laplacian $(-\Delta)^s$ with $s>1$. We also obtain the Pohozaev identity for this operator. Both identities involve local boundary terms, and they extend the identities obtained by the authors in the case $s\in(0,1)$.
    As an immediate consequence of these results, we obtain a unique continuation property for the eigenfunctions $(-\Delta)^s\phi=\lambda\phi$ in $\Omega$, $\phi\equiv0$ in $\mathbb{R}^n\setminus\Omega$.
On the least energy sign-changing solutions for a nonlinear elliptic system
Yohei Sato and Zhi-Qiang Wang
2015, 35(5): 2151-2164 doi: 10.3934/dcds.2015.35.2151 +[Abstract](477) +[PDF](439.3KB)
Abstract:
In this paper, as bound state solutions we consider least energy sign-changing solutions to a nonlinear elliptic system which consists of N-equations defined on a bounded domain $\Omega$. For any subset $K\subset \{1,\cdots, N\}$, we show the existence of sign-changing solution $\vec{u}=(u_1,\cdots,u_n)$ such that, for $i\in K$, $u_i$ are sign-changing functions that change sign exactly once in $\Omega$, and, for $i\notin K$, $u_i$ are one sign functions. We give a variational characterization of such solutions on modified Nehari type constrained sets.
Random backward iteration algorithm for Julia sets of rational semigroups
Rich Stankewitz and Hiroki Sumi
2015, 35(5): 2165-2175 doi: 10.3934/dcds.2015.35.2165 +[Abstract](328) +[PDF](370.3KB)
Abstract:
We provide proof that a random backward iteration algorithm for approximating Julia sets of rational semigroups, previously proven to work in the context of iteration of a rational function of degree two or more, extends to rational semigroups (of a certain type). We also provide some consequences of this result.
Center of planar quintic quasi--homogeneous polynomial differential systems
Yilei Tang, Long Wang and Xiang Zhang
2015, 35(5): 2177-2191 doi: 10.3934/dcds.2015.35.2177 +[Abstract](432) +[PDF](378.5KB)
Abstract:
In this paper we first characterize all quasi--homogeneous but non--homogeneous planar polynomial differential systems of degree five, and then among which we classify all the ones having a center at the origin. Finally we present the topological phase portrait of the systems having a center at the origin.
Global regularity of the two-dimensional magneto-micropolar fluid system with zero angular viscosity
Kazuo Yamazaki
2015, 35(5): 2193-2207 doi: 10.3934/dcds.2015.35.2193 +[Abstract](450) +[PDF](447.0KB)
Abstract:
We study the two-dimensional magneto-micropolar fluid system. Making use of the structure of the system, we show that with zero angular viscosity the solution triple remains smooth for all time.
Sharper estimates on the eigenvalues of Dirichlet fractional Laplacian
Selma Yildirim Yolcu and Türkay Yolcu
2015, 35(5): 2209-2225 doi: 10.3934/dcds.2015.35.2209 +[Abstract](428) +[PDF](2629.0KB)
Abstract:
This article is to analyze certain bounds for the sums of eigenvalues of the Dirichlet fractional Laplacian operator $(-\Delta)^{\alpha/2}|_{\Omega}$ restricted to a bounded domain $\Omega\subset{\mathbb R}^d$ with $d=2,$ $1\leq \alpha\leq 2$ and $d\geq 3,$ $0< \alpha\le 2$. A primary topic is the refinement of the Berezin-Li-Yau inequality for the fractional Laplacian eigenvalues. Our result advances the estimates recently established by Wei, Sun and Zheng in [34]. Another aspect of interest in this work is that we obtain some estimates for the sums of powers of the eigenvalues of the fractional Laplacian operator.
Minimal period problems for brake orbits of nonlinear autonomous reversible semipositive Hamiltonian systems
Duanzhi Zhang
2015, 35(5): 2227-2272 doi: 10.3934/dcds.2015.35.2227 +[Abstract](261) +[PDF](660.5KB)
Abstract:
In this paper, for any positive integer $n$, we study the Maslov-type index theory of $i_{L_0}$, $i_{L_1}$ and $i_{\sqrt{-1}}^{L_0}$ with $L_0 = \{0\}\times \mathbf{R}^n\subset \mathbf{R}^{2n}$ and $L_1=\mathbf{R}^n\times \{0\} \subset \mathbf{R}^{2n}$. As applications we study the minimal period problems for brake orbits of nonlinear autonomous reversible Hamiltonian systems. For first order nonlinear autonomous reversible Hamiltonian systems in $\mathbf{R}^{2n}$, which are semipositive, and superquadratic at zero and infinity, we prove that for any $T>0$, the considered Hamiltonian systems possesses a nonconstant $T$ periodic brake orbit $X_T$ with minimal period no less than $\frac{T}{2n+2}$. Furthermore if $\int_0^T H''_{22}(x_T(t))dt$ is positive definite, then the minimal period of $x_T$ belongs to $\{T,\;\frac{T}{2}\}$. Moreover, if the Hamiltonian system is even, we prove that for any $T>0$, the considered even semipositive Hamiltonian systems possesses a nonconstant symmetric brake orbit with minimal period belonging to $\{T,\;\frac{T}{3}\}$.
Dynamics of hyperbolic meromorphic functions
Jian-Hua Zheng
2015, 35(5): 2273-2298 doi: 10.3934/dcds.2015.35.2273 +[Abstract](313) +[PDF](474.6KB)
Abstract:
A definition of hyperbolic meromorphic functions is given and then we discuss the dynamical behavior and the thermodynamic formalism of hyperbolic functions on their Julia sets. We prove the important expanding properties for hyperbolic functions on the complex plane or with respect to the Euclidean metric. We establish the Bowen formula for hyperbolic functions on the complex plane, that is, the Poincare exponent equals to the Hausdorff dimension of the radial Julia set and furthermore, we prove that all the results in the Walters' theory hold for hyperbolic functions on the Riemann sphere.
Boundedness and blow-up for a chemotaxis system with generalized volume-filling effect and logistic source
Pan Zheng, Chunlai Mu and Xuegang Hu
2015, 35(5): 2299-2323 doi: 10.3934/dcds.2015.35.2299 +[Abstract](517) +[PDF](578.0KB)
Abstract:
This paper deals with a parabolic-elliptic chemotaxis system with generalized volume-filling effect and logistic source \begin{eqnarray*} \left\{ \begin{split}{} &u_t=\nabla\cdot(\varphi(u)\nabla u)-\chi\nabla\cdot(\psi(u)\nabla v)+f(u), &(x,t)\in \Omega\times (0,\infty), \\ &0=\Delta v-m(t)+u, &(x,t)\in \Omega\times (0,\infty), \end{split} \right. \end{eqnarray*} under homogeneous Neumann boundary conditions in a smooth bounded domain $\Omega\subset \mathbb{R}^{n}$ $(n\geq1)$, where $\chi>0$, $m(t)=\frac{1}{|\Omega|}\int_{\Omega}u(x,t)dx$, the nonlinear diffusivity $\varphi(u)$ and chemosensitivity $\psi(u)$ are supposed to extend the prototypes $$\varphi(u)=(u+1)^{-p},\quad \psi(u)=u(u+1)^{q-1}$$ with $p\geq0$, $q\in \mathbb{R}$, and $f(u)$ is assumed to generalize the standard logistic function $$f(u)=\lambda u-\mu u^{k},\;\text{with}\;\;\lambda\geq 0,\mu>0\;\text{and}\;k>1.$$ Under some different suitable assumptions on the nonlinearities $\varphi(u), \psi(u)$ and logistic source $f(u)$, we study the global boundedness and finite-time blow-up of solutions for the problem.

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