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## Discrete & Continuous Dynamical Systems - A

2013 , Volume 33 , Issue 7

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2013, 33(7): 2621-2629
doi: 10.3934/dcds.2013.33.2621

*+*[Abstract](304)*+*[PDF](333.7KB)**Abstract:**

In this paper, we prove that stable ergodicity is $C^1$-dense among conservative partially hyperbolic systems which, in a stable way, have two ergodic measures such that one has all center Lyapunov exponents non-negative and the other one has all center Lyapunov exponents non-positive.

2013, 33(7): 2631-2650
doi: 10.3934/dcds.2013.33.2631

*+*[Abstract](452)*+*[PDF](399.5KB)**Abstract:**

We consider nonautonomous linear equations $x'=A(t)x$ in a Banach space, and we give a complete characterization of those admitting

*nonuniform exponential contractions*in terms of

*strict Lyapunov functions*. The uniform contractions are a very particular case of nonuniform exponential contractions. In addition, we establish ``inverse theorems'' that give explicitly a strict Lyapunov function for each nonuniform contraction. These functions are constructed in terms of Lyapunov norms, which transform the nonuniform behavior of the contraction into a uniform exponential behavior. Moreover, we use the characterization of nonuniform exponential contractions in terms of strict Lyapunov functions to establish in a very simple manner, in comparison with former works, the persistence of the asymptotic stability under sufficiently small linear and nonlinear perturbations.

2013, 33(7): 2651-2665
doi: 10.3934/dcds.2013.33.2651

*+*[Abstract](343)*+*[PDF](473.6KB)**Abstract:**

We investigate the possibility of modeling the

*delayed afterdepolarization*(DAD) occurrence in the framework of the classical FitzHugh-Nagumo (FN) dynamical system, as well as in more recent electromechanically-coupled cardiac models. Within the FN model, we identify the domain in the constitutive parameters' space for which orbits exist which exhibit a sufficiently strong secondary impulse. We then address the question whether a locally-induced secondary pulse succeeds or not in originating a self-propagating traveling impulse. Our results evidence that, in the range where secondary impulses exceed the physiological threshold for DAD onset, a local impulse almost certainly causes a traveling impulse (mechanism known as

*all-or-none*). We then consider a recently proposed electromechanically-coupled generalization of the FN model, and show that the mechanical coupling stabilizes the system, in the sense that the more strong the coupling, the less likely is DAD to occur.

2013, 33(7): 2667-2679
doi: 10.3934/dcds.2013.33.2667

*+*[Abstract](384)*+*[PDF](371.5KB)**Abstract:**

In this paper we consider all possible period sets $P(f)$ for self-maps of the Cantor set, $f:C\to C$. We prove that the possible period sets are completely unrestricted

*provided that, in addition, one allows points that are not periodic*. However, if every point is periodic, we show that a surprising finiteness condition is imposed on $P(f)$: namely, there is a finite subset $B$ of $P(f)$ such that every element of $P(f)$ is divisible by at least one element of $B$.

2013, 33(7): 2681-2710
doi: 10.3934/dcds.2013.33.2681

*+*[Abstract](339)*+*[PDF](504.0KB)**Abstract:**

We prove the global existence of finite energy weak solutions to the general liquid crystals system. The problem is studied in bounded domain of $\mathbb{R}^3$ with Dirichlet boundary conditions and the whole space $\mathbb{R}^3$.

2013, 33(7): 2711-2755
doi: 10.3934/dcds.2013.33.2711

*+*[Abstract](434)*+*[PDF](738.5KB)**Abstract:**

We consider the $\mathbf{U}(1)$-invariant nonlinear Klein-Gordon equation in discrete space and discrete time, which is the discretization of the nonlinear continuous Klein-Gordon equation. To obtain this equation, we use the energy-conserving finite-difference scheme of Strauss-Vazquez. We prove that each finite energy solution converges as $T → ± ∞$ to the finite-dimensional set of all multifrequency solitary wave solutions with one, two, and four frequencies. The components of the solitary manifold corresponding to the solitary waves of the first two types are generically two-dimensional, while the component corresponding to the last type is generically four-dimensional. The attraction to the set of solitary waves is caused by the nonlinear energy transfer from lower harmonics to the continuous spectrum and subsequent radiation. For the proof, we develop the well-posedness for the nonlinear wave equation in discrete space-time, apply the technique of quasimeasures, and also obtain the version of the Titchmarsh convolution theorem for distributions on the circle.

2013, 33(7): 2757-2776
doi: 10.3934/dcds.2013.33.2757

*+*[Abstract](404)*+*[PDF](530.9KB)**Abstract:**

The article deals with the time-dependent Oseen system in a 3D exterior domain. It is shown that the velocity part of a weak solution to that system decays as $\bigl(\, |x| \cdot (1+|x|-x_1) \,\bigr) ^{-1}$, and its spatial gradient as $\bigl(\, |x| \cdot (1+|x|-x_1) \,\bigr) ^{-3/2}$, for $|x|\to \infty $. This result is obtained for data that need not have compact support.

2013, 33(7): 2777-2790
doi: 10.3934/dcds.2013.33.2777

*+*[Abstract](360)*+*[PDF](409.9KB)**Abstract:**

We deal with the asymptotic behavior of the $s$-perimeter of a set $E$ inside a domain $\Omega$ as $s\searrow0$. We prove necessary and sufficient conditions for the existence of such limit, by also providing an explicit formulation in terms of the Lebesgue measure of $E$ and $\Omega$. Moreover, we construct examples of sets for which the limit does not exist.

2013, 33(7): 2791-2808
doi: 10.3934/dcds.2013.33.2791

*+*[Abstract](500)*+*[PDF](472.5KB)**Abstract:**

It was recently proven by De Lellis, Kappeler, and Topalov in [19] that the periodic Cauchy problem for the Camassa-Holm equations is locally well-posed in the space $Lip (\mathbb{T})$ endowed with the topology of $H^1 (\mathbb{T})$. We prove here that the Lagrangian flow of these solutions are analytic with respect to time and smooth with respect to the initial data.

These results can be adapted to the higher-order Camassa-Holm equations describing the exponential curves of the manifold of orientation preserving diffeomorphisms of $\mathbb{T}$ using the Riemannian structure induced by the Sobolev inner product $H^l (\mathbb{T})$, for $l ∈ \mathbb{N}$, $l\geq 2$ (the classical Camassa-Holm equation corresponds to the case $l=1$): the periodic Cauchy problem is locally well-posed in the space $ W^{2l-1,\infty} (\mathbb{T})$ endowed with the topology of $H^{2l-1} (\mathbb{T})$ and the Lagrangian flows of these solutions are analytic with respect to time with values in $ W^{2l-1,\infty} (\mathbb{T})$ and smooth with respect to the initial data.

These results extend some earlier results which dealt with more regular solutions. In particular our results cover the case of peakons, up to the first collision.

2013, 33(7): 2809-2827
doi: 10.3934/dcds.2013.33.2809

*+*[Abstract](494)*+*[PDF](529.9KB)**Abstract:**

We study stability of solutions of the Cauchy problem on the line for the Camassa--Holm equation $u_t-u_{xxt}+3uu_x-2u_xu_{xx}-uu_{xxx}=0$ with initial data $u_0$. In particular, we derive a new Lipschitz metric $d_D$ with the property that for two solutions $u$ and $v$ of the equation we have $d_D(u(t),v(t))\le e^{Ct} d_D(u_0,v_0)$. The relationship between this metric and the usual norms in $H^1$ and $L^\infty$ is clarified. The method extends to the generalized hyperelastic-rod equation $u_t-u_{xxt}+f(u)_x-f(u)_{xxx}+(g(u)+\frac12 f''(u)(u_x)^2)_x=0$ (for $f$ without inflection points).

2013, 33(7): 2829-2859
doi: 10.3934/dcds.2013.33.2829

*+*[Abstract](473)*+*[PDF](722.3KB)**Abstract:**

In this paper we consider general nearly integrable analytic Hamiltonian systems of one and a half degrees of freedom which are a trigonometric polynomial in the angular state variable. In the resonances of these systems generically appear hyperbolic periodic orbits. We study the possible transversal intersections of their invariant manifolds, which is exponentially small, and we give an asymptotic formula for the measure of the splitting. We see that its asymptotic first order is of the form $K \varepsilon^{\beta} \text{e}^{-a/\varepsilon}$ and we identify the constants $K,\beta,a$ in terms of the system features. We compare our results with the classical Melnikov Theory and we show that, typically, in the resonances of nearly integrable systems Melnikov Theory fails to predict correctly the constants $K$ and $\beta$ involved in the formula.

2013, 33(7): 2861-2883
doi: 10.3934/dcds.2013.33.2861

*+*[Abstract](369)*+*[PDF](507.2KB)**Abstract:**

In this paper, we study the bifurcation of isolated closed orbits from degenerated singularity of $3$-dimensional polynomial system $dx/dt = Q(x)$. For some types of $Q(x)$, we get the lower bound for the number of these isolated closed orbits. In particular cases, an explicit (sometimes sharp) upper bound is obtained. Using these results, we investigate degenerated Hopf bifurcation and give a sufficient condition for the existence of isolated closed orbits. Also we show that the $3$ species model of degree $3$ admits $2$ isolated closed orbits bifurcating from origin.

2013, 33(7): 2885-2900
doi: 10.3934/dcds.2013.33.2885

*+*[Abstract](439)*+*[PDF](407.0KB)**Abstract:**

Gierer-Meinhardt system is a mathematical model to describe biological pattern formation due to activator and inhibitor. Turing pattern is expected in the presence of local self-enhancement and long-range inhibition. The long-time behavior of the solution, however, has not yet been clarified mathematically. In this paper, we study the case when its ODE part takes periodic-in-time solutions, that is, $\tau=\frac{s+1}{p-1}$. Under some additional assumptions on parameters, we show that the solution exists global-in-time and absorbed into one of these ODE orbits. Thus spatial patterns eventually disappear if those parameters are in a region without local self-enhancement or long-range inhibition.

2013, 33(7): 2901-2909
doi: 10.3934/dcds.2013.33.2901

*+*[Abstract](522)*+*[PDF](390.6KB)**Abstract:**

We study shadowing property for a partially hyperbolic diffeomorphism $f$. It is proved that if $f$ is dynamically coherent then any pseudotrajectory can be shadowed by a pseudotrajectory with ``jumps'' along the central foliation. The proof is based on the Tikhonov-Shauder fixed point theorem.

2013, 33(7): 2911-2938
doi: 10.3934/dcds.2013.33.2911

*+*[Abstract](462)*+*[PDF](534.4KB)**Abstract:**

It is well known that a single nonlinear Schrödinger (NLS) equation with a potential $V$ and a small parameter $\varepsilon $ may have a unique positive solution that is concentrated at the nondegenerate minimum point of $V$ . However, the uniqueness may fail for two-component systems of NLS equations with a small parameter $\varepsilon $ and potentials $V_{1}$ and $V_{2}$ having the same nondegenerate minimum point. In this paper, we will use energy estimates and category theory to prove the nonuniqueness theorem.

2013, 33(7): 2939-2990
doi: 10.3934/dcds.2013.33.2939

*+*[Abstract](366)*+*[PDF](768.2KB)**Abstract:**

The Gromoll-Meyer's generalized Morse lemma (so called splitting lemma) near degenerate critical points on Hilbert spaces, which is one of key results in infinite dimensional Morse theory, is usually stated for at least $C^2$-smooth functionals. It obstructs one using Morse theory to study most of variational problems of form $F(u)=\int_\Omega f(x, u,\cdots, D^mu)dx$ as in (1.1). In this paper we establish a splitting theorem and a shifting theorem for a class of continuously directional differentiable functionals (lower than $C^1$) on a Hilbert space $H$ which have higher smoothness (but lower than $C^2$) on a densely and continuously imbedded Banach space $X\subset H$ near a critical point lying in $X$. (This splitting theorem generalize almost all previous ones to my knowledge). Moreover, a new theorem of Poincaré-Hopf type and a relation between critical groups of the functional on $H$ and $X$ are given. Different from the usual implicit function theorem method and dynamical system one our proof is to combine the ideas of the Morse-Palais lemma due to Duc-Hung-Khai [19] with some techniques from [27,43,46]. Our theory is applicable to the Lagrangian systems on compact manifolds and boundary value problems for a large class of nonlinear higher order elliptic equations.

2013, 33(7): 2991-3009
doi: 10.3934/dcds.2013.33.2991

*+*[Abstract](517)*+*[PDF](441.6KB)**Abstract:**

Let $f$ be a $C^1$-regular map of a closed $C^{\infty}$ manifold $M$ and $\Lambda$ be a locally maximal closed invariant set of $f$. We show that $f|_{\Lambda}$ satisfies the $C^1$-stable specification property if and only if $\Lambda$ is a hyperbolic elementary set. We also prove that there exists a residual subset $\mathcal{R}$ in the space of $C^1$-regular maps endowed with the $C^1$-topology such that for $f \in \mathcal{R}$, $f|_{\Lambda}$ satisfies the specification property if and only if $\Lambda$ is a hyperbolic elementary set.

2013, 33(7): 3011-3042
doi: 10.3934/dcds.2013.33.3011

*+*[Abstract](731)*+*[PDF](1882.8KB)**Abstract:**

We give a detailed analytical description of the global dynamics of $N$ points interacting through the singular logarithmic potential and subject to the following symmetry constraint: at each instant they form an orbit of the dihedral group $D_l$ of order $2l$. The main device in order to achieve our results is a technique very popular in Celestial Mechanics, usually referred to as

*McGehee transformation.*After performing this change of coordinates that regularizes the total collision, we study the rest-points of the flow, the invariant manifolds and, with the help of a computer algebra system, we derive interesting information about the global dynamics for $l=2$. We observe that our problem is equivalent to studying the geometry of stationary configurations of nearly-parallel vortex filaments in three dimensions in the LIA approximation.

2013, 33(7): 3043-3056
doi: 10.3934/dcds.2013.33.3043

*+*[Abstract](416)*+*[PDF](379.0KB)**Abstract:**

In this note we show that in metric measure spaces satisfying the reduced curvature-dimension condition $CD^*(K,N)$ we always have geodesics in the Wasserstein space of probability measures that satisfy the critical convexity inequality of $CD^*(K,N)$ also for intermediate times and in addition the measures along these geodesics have an upper-bound on their densities. This upper-bound depends on the bounds for the densities of the end-point measures, the lower-bound $K$ for the Ricci-curvature, the upper-bound $N$ for the dimension, and on the diameter of the union of the supports of the end-point measures.

2013, 33(7): 3057-3084
doi: 10.3934/dcds.2013.33.3057

*+*[Abstract](450)*+*[PDF](464.6KB)**Abstract:**

The aim of this paper is to obtain new criteria for the existence of the dichotomies of dynamical systems on the half-line. We associate to a discrete dynamical system an input-output system between two abstract sequence spaces. We deduce conditions for the existence of ordinary dichotomy and exponential dichotomy of the initial discrete system, by using certain admissibility properties of the associated input-output system. We establish the axiomatic structures of the input and output spaces, in each case, clarifying the underlying hypotheses as well as the generality of the proposed method. Next, we present a new and direct proof for the equivalence between the exponential dichotomy of an evolution family on the half-line and the exponential dichotomy of the associated discrete dynamical system. Finally, we apply our main results to the study of the exponential dichotomy of evolution families on the half-line.

2013, 33(7): 3085-3108
doi: 10.3934/dcds.2013.33.3085

*+*[Abstract](413)*+*[PDF](469.7KB)**Abstract:**

This paper is concerned with bifurcations of canard limit cycles in several singularly perturbed generalized polynomial Liénard systems. By analyzing the multiplicities of the zeroes of the slow divergence integrals and their complete unfolding, the upper bounds of canard limit cycles bifurcating from the suitable limit periodic sets through respectively the generic Hopf breaking mechanism, the generic jump breaking mechanism and a succession of the Hopf and jump mechanisms in these polynomial Liénard systems are obtained.

2013, 33(7): 3109-3134
doi: 10.3934/dcds.2013.33.3109

*+*[Abstract](412)*+*[PDF](584.8KB)**Abstract:**

We prove a necessary and sufficient criterion for the exponential stability of periodic solutions of delay differential equations with large delay. We show that for sufficiently large delay the Floquet spectrum near criticality is characterized by a set of curves, which we call asymptotic continuous spectrum, that is independent on the delay.

2013, 33(7): 3135-3152
doi: 10.3934/dcds.2013.33.3135

*+*[Abstract](414)*+*[PDF](451.3KB)**Abstract:**

We study topological entropy of exactly Devaney chaotic maps on totally regular continua, i.e. on (topologically) rectifiable curves. After introducing the so-called $P$-Lipschitz maps (where $P$ is a finite invariant set) we give an upper bound for their topological entropy. We prove that if a non-degenerate totally regular continuum $X$ contains a free arc which does not disconnect $X$ or if $X$ contains arbitrarily large generalized stars then $X$ admits an exactly Devaney chaotic map with arbitrarily small entropy. A possible application for further study of the best lower bounds of topological entropies of transitive/Devaney chaotic maps is indicated.

2013, 33(7): 3153-3170
doi: 10.3934/dcds.2013.33.3153

*+*[Abstract](594)*+*[PDF](455.3KB)**Abstract:**

We prove interior Harnack's inequalities for solutions of fractional nonlocal equations. Our examples include fractional powers of divergence form elliptic operators with potentials, operators arising in classical orthogonal expansions and the radial Laplacian. To get the results we use an analytic method based on a generalization of the Caffarelli--Silvestre extension problem, the Harnack's inequality for degenerate Schrödinger operators proved by C. E. Gutiérrez, and a transference method. In this manner we apply local PDE techniques to nonlocal operators. On the way a maximum principle and a Liouville theorem for some fractional nonlocal equations are obtained.

2013, 33(7): 3171-3188
doi: 10.3934/dcds.2013.33.3171

*+*[Abstract](394)*+*[PDF](385.0KB)**Abstract:**

A new method due to Fokas for explicitly solving boundary-value problems for linear partial differential equations is extended to equations with mixed partial derivatives. The Benjamin-Bona-Mahony equation is used as an example: we consider the Robin problem for this equation posed both on the half line and on the finite interval. For specific cases of the Robin boundary conditions the boundary-value problem is found to be ill posed.

2013, 33(7): 3189-3209
doi: 10.3934/dcds.2013.33.3189

*+*[Abstract](409)*+*[PDF](490.4KB)**Abstract:**

In this paper, we consider the upper semicontinuity of pullback attractors for a nonautonomous Kirchhoff wave model with strong damping. For this purpose, some necessary abstract results are established.

2013, 33(7): 3211-3223
doi: 10.3934/dcds.2013.33.3211

*+*[Abstract](451)*+*[PDF](362.2KB)**Abstract:**

In this paper, we mainly study persistence properties and infinite propagation for the modified 2-component Camassa--Holm equation. We first prove that persistence properties of the solution to the equation provided the initial potential satisfies a certain sign condition. Finally, we get the infinite propagation if the initial datas satisfy certain compact conditions, while the solution to system (1.1) instantly loses compactly supported, the solution has exponential decay as $|x|$ goes to infinity.

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