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1078-0947

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## Discrete & Continuous Dynamical Systems - A

2012 , Volume 32 , Issue 5

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2012, 32(5): 1449-1463
doi: 10.3934/dcds.2012.32.1449

*+*[Abstract](576)*+*[PDF](475.3KB)**Abstract:**

By performing estimates on the integral of the absolute value of vorticity along a local vortex line segment, we establish a relatively sharp dynamic growth estimate of maximum vorticity under some assumptions on the local geometric regularity of the vorticity vector. Our analysis applies to both the 3D incompressible Euler equations and the surface quasi-geostrophic model (SQG). As an application of our vorticity growth estimate, we apply our result to the 3D Euler equation with the two anti-parallel vortex tubes initial data considered by Hou-Li [12]. Under some additional assumption on the vorticity field, which seems to be consistent with the computational results of [12], we show that the maximum vorticity can not grow faster than double exponential in time. Our analysis extends the earlier results by Cordoba-Fefferman [6, 7] and Deng-Hou-Yu [8, 9].

2012, 32(5): 1465-1479
doi: 10.3934/dcds.2012.32.1465

*+*[Abstract](448)*+*[PDF](388.0KB)**Abstract:**

In [1] and [2] upperbounds have been given for the number of large amplitude limit cycles in polynomial Liénard systems of type $(m,n)$ with $m<2n+1$, $m$ and $n$ odd. In the current paper we improve the upperbounds from [1] and [2] by one unity, obtaining sharp results. We therefore introduce the "method of cloning variables" that might be useful in other cyclicity problems.

2012, 32(5): 1481-1502
doi: 10.3934/dcds.2012.32.1481

*+*[Abstract](546)*+*[PDF](437.4KB)**Abstract:**

In this paper we study an integro-differential equation describing slow erosion, in a model of granular flow. In this equation the flux is non local and depends on $x$, $t$. We define approximate solutions by using a front tracking technique, adapted to this special equation. Convergence of the approximate solutions is established by means of suitable a priori estimates. In turn, these yield the global existence of entropy solutions in BV. Such entropy solutions are shown to be unique.

We also prove the continuous dependence on initial data and on the erosion function, for the approximate as well as for the exact solutions. This establishes the well-posedness of the Cauchy problem.

2012, 32(5): 1503-1535
doi: 10.3934/dcds.2012.32.1503

*+*[Abstract](481)*+*[PDF](548.4KB)**Abstract:**

A 3D-2D dimensional reduction analysis for supremal functionals is performed in the realm of $\Gamma^*$-convergence. We show that the limit functional still admits a supremal representation, and we provide a precise identification of its density in some particular cases. Our results rely on an abstract representation theorem for the $\Gamma^*$-limit of a family of supremal functionals.

2012, 32(5): 1537-1555
doi: 10.3934/dcds.2012.32.1537

*+*[Abstract](504)*+*[PDF](392.1KB)**Abstract:**

We obtain stable invariant manifolds with optimal $C^k$ regularity for a nonautonomous dynamics with discrete time. The dynamics is obtained from a sufficiently small perturbation of a nonuniform exponential dichotomy, which includes the notion of (uniform) exponential dichotomy as a very special case. We emphasize that we do not require the dynamics to be of class $C^{k+\epsilon}$, in strong contrast to former results in the context of nonuniform hyperbolicity. We use the fiber contraction principle to establish the smoothness of the invariant manifolds. In addition, our method also allows linear perturbations, and thus the results readily apply to the robustness problem of nonuniform exponential dichotomies.

2012, 32(5): 1557-1574
doi: 10.3934/dcds.2012.32.1557

*+*[Abstract](506)*+*[PDF](412.3KB)**Abstract:**

This paper is devoted to the existence and multiplicity of periodic and subharmonic solutions for a superlinear Duffing equation with a singularity. In this manner, various preceding theorems are improved and sharpened. Our proof is based on a generalized version of the Poincaré-Birkhoff twist theorem.

2012, 32(5): 1575-1595
doi: 10.3934/dcds.2012.32.1575

*+*[Abstract](481)*+*[PDF](472.4KB)**Abstract:**

We deal with $m$-periodic, $n$-th order difference equations and study whether they can be globally linearized. We give an affirmative answer when $m=n+1$ and for most of the known examples appearing in the literature. Our main tool is a refinement of the Montgomery-Bochner Theorem.

2012, 32(5): 1597-1626
doi: 10.3934/dcds.2012.32.1597

*+*[Abstract](506)*+*[PDF](556.6KB)**Abstract:**

Let $(A_n(\omega))$ be a stationary process in ${\mathcal M}_d^*(\mathbb{Z})$. For a Hölder function $f$ on $\mathbb{T}^d$ we consider the sums $\sum_{k=1}^n f(^t\hskip -3pt A_k(\omega) \, ^t\hskip -3pt A_{k-1}(\omega) \cdots ^t\hskip -3pt A_1(\omega) \, x {\rm \ mod \ } 1)$ and prove a Central Limit Theorem for a.e. $\omega$ in different situations in particular for "kicked" stationary processes. We use the method of multiplicative systems of Komlòs and the Multiplicative Ergodic Theorem.

2012, 32(5): 1627-1637
doi: 10.3934/dcds.2012.32.1627

*+*[Abstract](565)*+*[PDF](334.0KB)**Abstract:**

In this paper we study the center--focus problem in families of rigid systems. We give explicit necessary and sufficient conditions to the unique equilibrium to be a center. We also study small amplitude limit cycles in these families of systems.

2012, 32(5): 1639-1655
doi: 10.3934/dcds.2012.32.1639

*+*[Abstract](511)*+*[PDF](288.4KB)**Abstract:**

We show that for every expanding self-map of a connected, closed $p$-dimensional manifold $M$, and for every codimension $q\geq p+1$, there exists a corresponding $(p,q)$-type DE attractor realized by a compactly-supported self-diffeomorphsm of $\mathbb{R}^{p+q}$. Moreover, when $M$ is the standard smooth $p$-dimensional torus $T^p$, the codimension $q$ can be taken as two. As a key ingredient of the construction, for the standard unknotted embedding $\imath_p:T^p\hookrightarrow\mathbb{R}^{p+2}$, we show the automorphisms that diffeomorphically extend over $\mathbb{R}^{p+2}$ form a subgroup of $Aut(T^p)$ of index at most $2^p-1$.

2012, 32(5): 1657-1674
doi: 10.3934/dcds.2012.32.1657

*+*[Abstract](531)*+*[PDF](423.5KB)**Abstract:**

Let $X$ be an infinite compact metric space and let $Z$ be a compact metric space admitting an arc-wise connected group $\mathcal H_0(Z)$ of homeomorphisms whose natural action on $Z$ is topologically transitive. We show that every map $f$ on $X$ with a hypertransitive property $\Lambda$ admits a skew product extension $F=(f,g_x)$ on $X\times Z$ which also has the property $\Lambda$ and whose all fibre maps $g_x$ lie in the closure $\overline{\mathcal H_0(Z)}$ of $\mathcal H_0(Z)$ in the space $\mathcal H(Z)$ of all homeomorphisms on $Z$.

If we additionally assume that both the map $f$ and the action of $\mathcal H_0(Z)$ on $Z$ are minimal then we can guarantee the existence of such an extension $F$ in the class of minimal maps. In particular case when $\Lambda$= topological transitivity, such a theorem was known before (for invertible $f$ it was proved by Glasner and Weiss already in 1979).

Finally, we show that if one imposes further restrictions on the group $\mathcal H_0(Z)$ then the analogues of the mentioned results for hypertransitive properties $\Lambda$ hold also for $\Lambda$= strong mixing.

2012, 32(5): 1675-1707
doi: 10.3934/dcds.2012.32.1675

*+*[Abstract](578)*+*[PDF](442.3KB)**Abstract:**

In this paper we study nonnegative, measure-valued solutions of the initial value problem for one-dimensional drift-diffusion equations when the nonlinear diffusion is governed by a strictly increasing $C^1$ function $\beta$ with $\lim_{r\to +\infty} \beta(r)<+\infty$. By using tools of optimal transport, we will show that this kind of problems is well posed in the class of nonnegative Borel measures with finite mass $m$ and finite quadratic momentum and it is the gradient flow of a suitable entropy functional with respect to the so called $L^2$-Wasserstein distance.

Due to the degeneracy of diffusion for large densities, concentration of masses can occur, whose support is transported by the drift. We shall show that the large-time behavior of solutions depends on a critical mass $m_{\rm c}$, which can be explicitly characterized in terms of $\beta$ and of the drift term. If the initial mass is less then $m_{\rm c}$, the entropy has a unique minimizer which is absolutely continuous with respect to the Lebesgue measure. Conversely, when the total mass $m$ of the solutions is greater than the critical one, the stationary solution has a singular part in which the exceeding mass $m- m_{\rm c}$ is accumulated.

2012, 32(5): 1709-1721
doi: 10.3934/dcds.2012.32.1709

*+*[Abstract](534)*+*[PDF](172.8KB)**Abstract:**

The aim of this work is to describe the set of fixed point free homeomorphisms of the plane (preserving orientation or not) under certain expansive conditions. We find necessary and sufficient conditions for a fixed point free homeomorphism of the plane to be topologically conjugate to a translation.

2012, 32(5): 1723-1746
doi: 10.3934/dcds.2012.32.1723

*+*[Abstract](564)*+*[PDF](525.0KB)**Abstract:**

We consider a nonlocal parabolic equation associated with Dirichlet boundary and initial conditions arising in MEMS control. First, we investigate the structure of the associated steady-state problem for a general star-shaped domain. Then we classify radially symmetric stationary solutions and their radial Morse indices. Finally, we study under which circumstances the solution of the time-dependent problem is global-in-time or

*quenches*in finite time.

2012, 32(5): 1747-1761
doi: 10.3934/dcds.2012.32.1747

*+*[Abstract](566)*+*[PDF](428.9KB)**Abstract:**

We prove existence of standing waves solutions for electrostatic Klein-Gordon-Maxwell systems in arbitrary dimensional compact Riemannian manifolds with boundary for zero Dirichlet boundary conditions. We prove that phase compensation holds true when the dimension $n = 3$ or $4$. In these dimensions, existence of a solution is obtained when the mass of the particle field, balanced by the phase, is small in a geometrically quantified sense. In particular, existence holds true for sufficiently large phases. When $n \ge 5$, existence of a solution is obtained when the mass of the particle field is sufficiently small.

2012, 32(5): 1763-1774
doi: 10.3934/dcds.2012.32.1763

*+*[Abstract](508)*+*[PDF](383.1KB)**Abstract:**

The Schubart-like orbits in the collinear four-body problem are similar to those discovered numerically by Schubart[12] in the collinear three-body problem. Schubart-like orbits are periodic solutions with exactly two binary collisions and one simultaneous binary collision per period. The proof of the existence of these orbits given in this paper is based on the direct method of Calculus of Variations. We exploit the variational structure of the problem and show that the minimizers of the Lagrangian action functional in a suitably chosen space have the desired properties.

2012, 32(5): 1775-1799
doi: 10.3934/dcds.2012.32.1775

*+*[Abstract](536)*+*[PDF](547.4KB)**Abstract:**

Existence of large-amplitude time-periodic breathers localized near a single site is proved for the discrete Klein--Gordon equation, in the case when the derivative of the on-site potential has a compact support. Breathers are obtained at small coupling between oscillators and under nonresonance conditions. Our method is different from the classical anti-continuum limit developed by MacKay and Aubry, and yields in general branches of breather solutions that cannot be captured with this approach. When the coupling constant goes to zero, the amplitude and period of oscillations at the excited site go to infinity. Our method is based on near-identity transformations, analysis of singular limits in nonlinear oscillator equations, and fixed-point arguments.

2012, 32(5): 1801-1833
doi: 10.3934/dcds.2012.32.1801

*+*[Abstract](540)*+*[PDF](633.9KB)**Abstract:**

We prove the existence of multiple periodic solutions for scalar-valued state-dependent delay equations of the form $x'(t) = f(x(t - d(x_t)))$, where $d(x_t)$ is given by a threshold condition and $f$ is close, in a suitable sense, to the step function $h(x) = -\mbox{sign}(x)$. We construct maps whose fixed points correspond to periodic solutions and show that these maps have nontrivial fixed points via homotopy to constant maps.

We also describe part of the global dynamics of the model equation $x'(t) = h(x(t - d(x_t)))$.

2012, 32(5): 1835-1855
doi: 10.3934/dcds.2012.32.1835

*+*[Abstract](506)*+*[PDF](398.0KB)**Abstract:**

In this paper, we study the three-dimensional(3D) compressible magnetohydrodynamic equations. Firstly, we obtain a blow-up criterion for the local strong solutions in terms of the gradient of the velocity, which is similar to the Beal-Kato-Majda criterion(see [1]) for the ideal incompressible flow. Secondly, we extend the well-known Serrin's blow-up criterion for the 3D incompressible Navier-Stokes equations to the 3D compressible magnetohydrodynamic equations. In addition, initial vacuum is allowed in our cases.

2012, 32(5): 1857-1879
doi: 10.3934/dcds.2012.32.1857

*+*[Abstract](701)*+*[PDF](466.1KB)**Abstract:**

In this paper we consider the problem of existence of conformal metrics with prescribed scalar curvature on n-dimensional Riemannian manifolds, $n \geq 5 $. Using precise estimates on the losses of compactness, we characterize the critical points at infinity of the associated variational problem and we prove existence results for curvatures satisfying an assumption of Bahri-Coron type.

2012, 32(5): 1881-1899
doi: 10.3934/dcds.2012.32.1881

*+*[Abstract](539)*+*[PDF](8724.1KB)**Abstract:**

We study periodic perturbations of planar quadratic vector fields having infinite heteroclinic cycles, consisting of an invariant straight line joining two saddle points at infinity and an arc of orbit also at infinity. The global study concerning the infinity of the perturbed system is performed by means of the Poincaré compactification in polar coordinates, from which we obtain a system defined on a set equivalent to a solid torus in $\mathbb{R}^3$, whose boundary plays the role of the infinity. It is shown that for certain type of periodic perturbation, there exist two differentiable curves in the parameter space for which the perturbed system presents heteroclinic tangencies and transversal intersections between the stable and unstable manifolds of two normally hyperbolic lines of singularities at infinity. The transversality of the manifolds is proved using the Melnikov method and implies, via the Birkhoff-Smale Theorem, in a complex dynamical behavior of the perturbed system solutions in a finite part of the phase space. Numerical simulations are performed for a particular example in order to illustrate this behavior, which could be called "the chaos arising from infinity", because it depends on the global structure of the quadratic system, including the points at infinity.

2012, 32(5): 1901-1914
doi: 10.3934/dcds.2012.32.1901

*+*[Abstract](740)*+*[PDF](419.0KB)**Abstract:**

This paper deals with a boundary-value problem in two-dimensional smoothly bounded domains for the coupled chemotaxis-fluid model $$ \left\{ \begin{array}{l} n_t+ u\cdot \nabla n=\Delta n^m - \nabla \cdot (n\chi(c)\nabla c)\\ c_t+ u\cdot \nabla c=\Delta c-nf(c)\\ u_t +\nabla P-\eta \Delta u+n \nabla \phi=0 \\ \nabla \cdot u=0, \end{array} \right. $$ which describes the motion of oxygen-driven swimming bacteria in an incompressible fluid. The given functions $\chi$ and $f$ are supposed to be sufficiently smooth and such that $f(0)=0$.

It is proved that global bounded weak solutions exist whenever $m>1$ and the initial data $(n_0,c_0,u_0)$ are sufficiently regular satisfying $n_0 \ge 0$ and $c_0\ge 0$. This extends a recent result by Di Francesco, Lorz and Markowich (

*Discrete Cont. Dyn. Syst. A*

**28**(2010)) which asserts global existence of weak solutions under the constraint $m \in (\frac{3}{2},2]$.

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