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1078-0947

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1553-5231

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## Discrete & Continuous Dynamical Systems

February 2013 , Volume 33 , Issue 2

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2013, 33(2): 391-412
doi: 10.3934/dcds.2013.33.391

*+*[Abstract](2078)*+*[PDF](487.5KB)**Abstract:**

In this paper we study a certain regularity property of $C^2$ Axiom A flows $\phi_t$ over basic sets $Λ$ related to diameters of balls in Bowen's metric, which we call regular distortion along unstable manifolds. The motivation to investigate the latter comes from the study of spectral properties of Ruelle transfer operators in [21]. We prove that if the bottom of the spectrum of $d\phi_t$ over $E^u_{|Λ}$ is point-wisely pinched and integrable, then the flow has regular distortion along unstable manifolds over $Λ$. In the process, under the same conditions, we show that locally the flow is Lipschitz conjugate to its linearization over the `pinched part' of the unstable tangent bundle.

2013, 33(2): 413-463
doi: 10.3934/dcds.2013.33.413

*+*[Abstract](2271)*+*[PDF](793.2KB)**Abstract:**

We delve deeper into the compelling regularizing effect of the Brownian-time Brownian motion density,$ \mathbb{K}^{{BTBM}^d}_{t;x,y}$ , on the space-time-white-noise-driven stochastic integral equation we call BTBM SIE: \begin{equation} U(t,x)=\int_{{\mathbb R}^{d}}{{\mathbb K}}^{\text{BTBM}^d}_{t;x,y} u_0(y) dy+ \int_{{\mathbb R}^{d}}\int_0^t{{\mathbb K}}^{\text{BTBM}^d}_{t-s;x,y} a(U(s,y))\mathscr W(ds\times dy), (0.1) \end{equation} which we recently introduced in [3].In sharp contrast to traditional second order heat-operator-based SPDEs---whose real-valued mild solutions are confined to $d=1$---we prove the existence of solutions to (0.1) in $d=1,2,3$ with dimension-dependent and striking Hölder regularity, under both less than Lipschitz and Lipschitz conditions on $a$. In space, we show an unprecedented nearly local Lipschitz regularity for $d=1,2$---roughly, $U$ is spatially twice as regular as the Brownian sheet in these dimensions---and we prove nearly local Hölder $1/2$ regularity in $d=3$. In time, our solutions are locally $\gamma$-Hölder continuous with exponent $γ∈(0, \frac{4-d}{8})$,$1≤d≤3$. To investigate (0.1) under less than Lipschitz conditions on $a$, we (a) introduce the Brownian-time random walk---a special case of lattice processes we call Brownian-time chains---and we use it to formulate the spatial lattice version of (0.1); and (b) develop a delicate variant of Stroock-Varadhan martingale approach, the K-martingale approach, tailor-made for a wide variety of kernel SIEs including (0.1) and the mild forms of many SPDEs of different orders on the lattice. Solutions to (0.1) are defined as limits of their lattice version. Along the way, we prove interesting aspects of Brownian-time random walk, including a fourth order differential-difference equation connection. We also prove existence, pathwise uniqueness, and the same Hölder regularity for (0.1), without discretization, in the Lipschitz case. The SIE (0.1) is intimately connected to intriguing fourth order SPDEs in two ways. First, we show that (0.1) is connected to the diagonals of a new unconventional fourth order SPDE we call parametrized BTBM SPDE. Second, replacing $ \mathbb {K}^{{BTBM}^d}_{t;x,y}$ , by the intimately connected kernel of our recently-introduced imaginary-Brownian-time-Brownian-angle process (IBTBAP), (0.1) becomes the mild form of a Kuramoto-Sivashinsky (KS) SPDE with linearized PDE part. Ideas and tools developed here are adapted in separate papers to give an entirely new approach, via our explicit IBTBAP representation, to many linear and nonlinear KS-type SPDEs in multi-spatial dimensions.

2013, 33(2): 465-482
doi: 10.3934/dcds.2013.33.465

*+*[Abstract](2559)*+*[PDF](433.2KB)**Abstract:**

In the setting of continuous piecewise monotone interval maps, we study the analytic properties of the zeta function of a periodic nonautonomous dynamical system. Based upon these properties we discuss the relationship between topological entropy and growth number of periodic points of a periodic nonautonomous dynamical system.

2013, 33(2): 483-503
doi: 10.3934/dcds.2013.33.483

*+*[Abstract](2078)*+*[PDF](421.3KB)**Abstract:**

Using the Lyapunov-Perron method, we prove the existence of an inertial manifold for the process associated to a class of non-autonomous semilinear parabolic equations with finite delay, where the linear principal part is positive definite with a discrete spectrum having a sufficiently large distance between some two successive point spectra, and the Lipschitz coefficient of the nonlinear term may depend on time and belongs to some admissible function spaces.

2013, 33(2): 505-525
doi: 10.3934/dcds.2013.33.505

*+*[Abstract](2260)*+*[PDF](651.2KB)**Abstract:**

We prove that a flow without singular points of index zero on a compact surface is expansive if and only if the singularities are of saddle type and the union of their separatrices is dense. Moreover we show that such flows are obtained by surgery on the suspension of minimal interval exchange maps.

2013, 33(2): 527-553
doi: 10.3934/dcds.2013.33.527

*+*[Abstract](3408)*+*[PDF](542.5KB)**Abstract:**

We study various aspects of the dynamics induced by integer matrices on the invariant rational lattices of the torus in dimension $2$ and greater. Firstly, we investigate the orbit structure when the toral endomorphism is not invertible on the lattice, characterising the

*pretails*of eventually periodic orbits. Next we study the nature of the symmetries and reversing symmetries of toral automorphisms on a given lattice, which has particular relevance to (quantum) cat maps.

2013, 33(2): 555-577
doi: 10.3934/dcds.2013.33.555

*+*[Abstract](1984)*+*[PDF](476.9KB)**Abstract:**

This paper deals with the elliptic equation Δu+K(|x|)

*u*

^{p}= 0 in

*$\mathbb{R}$*

^{n}\{0} when $r^{-l}K(r)$ for $l>-2$ behaves monotonically near $r=0$ or $\infty$ with $r=|x|$. By the method of phase plane, we present a new proof for the structure of positive radial solutions, and analyze the asymptotic behavior at $\infty$. We also employ the approach to classify singular solutions in terms of the asymptotic behavior at $0$. In particular, when $p=\frac{n+2+2l}{n-2}$, we establish the uniqueness of solutions with asymptotic self-similarity at $0$ and at $\infty$, and the existence of multiple solutions of Delaunay-Fowler type at $0$ and $\infty$.

2013, 33(2): 579-597
doi: 10.3934/dcds.2013.33.579

*+*[Abstract](2624)*+*[PDF](489.4KB)**Abstract:**

We introduce a technique for establishing pure discrete spectrum for substitution tiling systems of Pisot family type and illustrate with several examples.

2013, 33(2): 599-628
doi: 10.3934/dcds.2013.33.599

*+*[Abstract](2332)*+*[PDF](521.4KB)**Abstract:**

The present essay is concerned with a model for the propagation of three-dimensional, surface water waves. Of especial interest will be long-crested waves such as those sometimes observed in canals and in near-shore zones of large bodies of water. Such waves propagate primarily in one direction, taken to be the $x-$direction in a Cartesian framework, and variations in the horizontal direction orthogonal to the primary direction, the $y-$direction, say, are often ignored. However, there are situations where weak variations in the secondary horizontal direction need to be taken into account.

Our results are developed in the context of Boussinesq models, so they are applicable to waves that have small amplitude and long wavelength when compared with the undisturbed depth. Included in the theory are well-posedness results on the long, Boussinesq time scale. As mentioned, particular interest is paid to 1000 the lateral dynamics, which turn out to satisfy a reduced Boussinesq system. Waves corresponding to disturbances which are localized in the $x-$direction as well as bore-like disturbances that have infinite energy are taken up in the discussion.

2013, 33(2): 629-642
doi: 10.3934/dcds.2013.33.629

*+*[Abstract](2146)*+*[PDF](345.3KB)**Abstract:**

The no invariant line fields conjecture is one of the main outstanding problems in traditional complex dynamics. In this paper we consider non-autonomous iteration where one works with compositions of sequences of polynomials with suitable bounds on the degrees and coefficients. We show that the natural generalization of the no invariant line fields conjecture to this setting is not true. In particular, we construct a sequence of quadratic polynomials whose iterated Julia sets all have positive area and which has an invariant sequence of measurable line fields whose supports are these iterated Julia sets with at most countably many points removed.

2013, 33(2): 643-662
doi: 10.3934/dcds.2013.33.643

*+*[Abstract](3595)*+*[PDF](505.6KB)**Abstract:**

We study the behaviour of the solutions to the Cauchy problem $$ \left\{\begin{array}{ll} \rho(x)u_t=\Delta u+u^p,&\quad x\in\mathbb{R}^N ,\;t\in(0,T),\\ u(x,\, 0)=u_0(x),&\quad x\in\mathbb{R}^N , \end{array}\right. $$ with $p>0$ and a positive density $\rho$ satisfying $\rho(x)\sim|x|^{-\sigma}$ for large $|x|$, $0<\sigma<2< N$. We consider both the cases of a bounded density and the singular density $\rho(x)=|x|^{-\sigma}$. We are interested in describing sharp decay conditions on the data at infinity that guarantee local/global existence of solutions, which are unique in classes of functions with the same decay. We prove that larger data give rise to instantaneous complete blow-up. We also deal with the occurrence of finite-time blow-up. We prove that the global existence exponent is $p_0=1$, while the Fujita exponent depends on $\sigma$, namely $p_c=1+\frac2{N-\sigma}$.

We show that instantaneous blow-up at space infinity takes place when $p\le1$.

We also briefly discuss the case $2<\sigma< N$: we prove that the Fujita exponent in this case does not depend on $\sigma$, $\tilde{p}_c=1+\frac2{N-2}$, and for initial values not too small at infinity a phenomenon of instantaneous complete blow-up occurs in the range $1< p < \tilde{p}_c$

2013, 33(2): 663-700
doi: 10.3934/dcds.2013.33.663

*+*[Abstract](2862)*+*[PDF](876.8KB)**Abstract:**

This work studies the rotation-generalized Benjamin-Ono equation which is derived from the theory of weakly nonlinear long surface and internal waves in deep water under the presence of rotation. It is shown that the solitary-wave solutions are orbitally stable for certain wave speeds.

2013, 33(2): 701-721
doi: 10.3934/dcds.2013.33.701

*+*[Abstract](3192)*+*[PDF](472.1KB)**Abstract:**

In this paper, we study the transverse stability of random dynamical systems (RDS). Suppose a RDS on a Riemann manifold possesses a non-random invariant submanifold, what conditions can guarantee that a random attractor of the RDS restrained on the invariant submanifold is a random attractor with respect to the whole manifold? By the linearization technique, we prove that if all the normal Lyapunov exponents with respect to the tangent space of the submanifold are negative, then the attractor on the submanifold is also a random attractor of the whole manifold. This result extends the idea of the transverse stability analysis of deterministic dynamical systems in [1,3]. As an explicit example, we discuss the complete synchronization in network of coupled maps with both stochastic topologies and maps, which extends the well-known master stability function (MSF) approach for deterministic cases to stochastic cases.

2013, 33(2): 723-737
doi: 10.3934/dcds.2013.33.723

*+*[Abstract](2433)*+*[PDF](401.1KB)**Abstract:**

We prove global existence of strong solutions for the Vlasov-Poisson system in a convex bounded domain in the plasma physics case assuming homogeneous Dirichlet boundary conditions for the electric potential and the specular reflection boundary conditions for the distribution density.

2013, 33(2): 739-755
doi: 10.3934/dcds.2013.33.739

*+*[Abstract](2399)*+*[PDF](421.0KB)**Abstract:**

In this paper, we prove a limiting uniqueness criterion to harmonic map heat flows and liquid crystal flows. We firstly establish the uniqueness of harmonic map heat flows from $R^n$ to a smooth, compact Riemannian manifold $N$ in the class $C([0,T),BMO_T(R^n,N))\cap L^\infty_{loc}((0,T);\dot{W}^{1,\infty}(R^n))$ for $0< T ≤ +\infty.$ For the nematic liquid crystal flows $(v,d)$, we show that the mild solution is unique under the class $C([0,T),BMO_T^{-1}(R^n))\cap L^\infty_{loc}((0,T);L^\infty(R^n))\times C([0,T),BMO_T(R^n,S^2))\cap L^\infty_{loc}((0,T);\dot{W}^{1,\infty}(R^n))$ for $0< T ≤ +\infty.$

2013, 33(2): 757-788
doi: 10.3934/dcds.2013.33.757

*+*[Abstract](2516)*+*[PDF](478.9KB)**Abstract:**

We study a simplified system for the compressible fluid of Nematic Liquid Crystals in a bounded domain in three Euclidean space and prove the global existence of the finite energy weak solutions.

2013, 33(2): 789-801
doi: 10.3934/dcds.2013.33.789

*+*[Abstract](2115)*+*[PDF](347.2KB)**Abstract:**

We study the Cauchy problem for cubic Schrödinger equations modelling ultra-short laser pulses propagating along the line. The global existence, blow-up, and scattering of solutions is described exclusively in the charge space $L^2({\bf R})$ without any approximating arguments.

2013, 33(2): 803-817
doi: 10.3934/dcds.2013.33.803

*+*[Abstract](2154)*+*[PDF](466.1KB)**Abstract:**

In this work we analyze the convergence of solutions of the Poisson equation with Neumann boundary conditions in a two-dimensional thin domain with highly oscillatory behavior. We consider the case where the height of the domain, amplitude and period of the oscillations are all of the same order, and given by a small parameter $\epsilon>0$. Using an appropriate corrector approach, we show strong convergence and give error estimates when we replace the original solutions by the first-order expansion through the Multiple-Scale Method.

2013, 33(2): 819-835
doi: 10.3934/dcds.2013.33.819

*+*[Abstract](2266)*+*[PDF](453.7KB)**Abstract:**

Partial differential equations with discrete (concentrated) state-dependent delays are studied. The existence and uniqueness of solutions with initial data from a wider linear space is proven first and then a subset of the space of continuously differentiable (with respect to an appropriate norm) functions is used to construct a dynamical system. This subset is an analogue of

*the solution manifold*proposed for ordinary equations in [H.-O. Walther, The solution manifold and $C_{1}$-smoothness for differential equations with state-dependent delay, J. Differential Equations, 195(1), (2003) 46--65]. The existence of a compact global attractor is proven. As far as applications are concerned, we consider the well known Mackey-Glass-type equations with diffusion, the Lasota-Wazewska-Czyzewska model, and the delayed diffusive Nicholson's blowflies equation, all with

*state-dependent*delays.

2013, 33(2): 837-859
doi: 10.3934/dcds.2013.33.837

*+*[Abstract](2967)*+*[PDF](442.0KB)**Abstract:**

We establish existence and regularity results, as well as a priori estimates of Gidas-Spruck type for nonlinear problems involving the fractional power of the Dirichlet Laplacian.

2013, 33(2): 861-878
doi: 10.3934/dcds.2013.33.861

*+*[Abstract](2134)*+*[PDF](436.7KB)**Abstract:**

We consider in dimension four weakly convergent sequences of approximate biharmonic maps into sphere with bi-tension fields bounded in $L^p$ for $p>1$. We prove an energy identity that accounts for the loss of hessian energies by the sum of hessian energies over finitely many nontrivial biharmonic maps on $\mathbb R^4$. As a corollary, we obtain an energy identity for the heat flow of biharmonic maps into sphere at time infinity.

2013, 33(2): 879-884
doi: 10.3934/dcds.2013.33.879

*+*[Abstract](1969)*+*[PDF](326.4KB)**Abstract:**

In this note, we improve a combinatorial sifting-type lemma obtained in [11].More precisely, we sift out a continuous infinite "$(\xi_1,\xi_2)$-Liao string" sequence for any real sequence $\{a_i\}_1^\infty$ with $\limsup_{n\to\infty}{n}^{-1}\sum_{i=1}^na_i=\xi\in(\xi_1,\xi_2)$.

2013, 33(2): 885-903
doi: 10.3934/dcds.2013.33.885

*+*[Abstract](2307)*+*[PDF](438.9KB)**Abstract:**

A discrete stochastic Razumikhin-type theorem is established to investigate whether the Euler--Maruyama (EM) scheme can reproduce the moment exponential stability of exact solutions of stochastic functional differential equations (SFDEs). In addition, the Chebyshev inequality and the Borel-Cantelli lemma are applied to show the almost sure stability of the EM approximate solutions of SFDEs. To show our idea clearly, these results are used to discuss stability of numerical solutions of two classes of special SFDEs, including stochastic delay differential equations (SDDEs) with variable delay and stochastically perturbed equations.

2013, 33(2): 905-920
doi: 10.3934/dcds.2013.33.905

*+*[Abstract](2239)*+*[PDF](423.0KB)**Abstract:**

Let $f$ be a continuous transformation of a compact metric space $(X,d)$ and $\varphi$ any continuous function on $X$. In this paper, under the hypothesis that $f$ satisfies the specification property, we determine the topological entropy of the following sets: $$K_{I}=\Big\{x\in X: A\big(\frac{1}{n}\sum_{i=0}^{n-1}\varphi(f^{i}(x))\big)=I\Big\},$$ where $I$ is a closed subinterval of $\mathbb{R}$ and $A(a_{n})$ denotes the set of accumulation points of the sequence $\{a_{n}\}_{n}$. Our result generalizes the classical result of Takens and Verbitskiy ( Ergod. Th. Dynam. Sys.,

*23*(2003), 317-348 ). As an application, we present another concise proof of the fact that the irregular set has full topological entropy if $f$ satisfies the specification property.

2013, 33(2): 921-946
doi: 10.3934/dcds.2013.33.921

*+*[Abstract](2553)*+*[PDF](507.7KB)**Abstract:**

This paper is concerned with traveling fronts and entire solutions for a class of monostable partially degenerate reaction-diffusion systems. It is known that the system admits traveling wave solutions. In this paper, we first prove the monotonicity and uniqueness of the traveling wave solutions, and the existence of spatially independent solutions. Combining traveling fronts with different speeds and a spatially independent solution, the existence and various qualitative features of entire solutions are then established by using comparison principle. As applications, we consider a reaction-diffusion model with a quiescent stage in population dynamics and a man-environment-man epidemic model in physiology.

2013, 33(2): 947-964
doi: 10.3934/dcds.2013.33.947

*+*[Abstract](1968)*+*[PDF](416.3KB)**Abstract:**

Let Σ be a $C^2$ compact strictly convex hypersurface in

*R*

^{2n}with $n\ge 2$. Suppose $PΣ=Σ$ with $P$ being a $2n\times 2n$ symplectic and orthogonal matrix and $P^r=I_{2n}$. We prove that there are at least two geometrically distinct $P$-cyclic symmetric closed characteristics $(\tau_j,x_j)$ on Σ in the sense that $x_j(t+\frac{\tau_j}{r})=Px_j(t)$ for all $t∈R$ with $j=1,2$. As a corollary we obtain the existence of two geometrically distinct central symmetric closed characteristics on any $C^2$ central symmetric compact convex hypersurface in

*R*

^{2n}with $n\ge 2$.

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