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Discrete & Continuous Dynamical Systems - A

2012 , Volume 32 , Issue 4

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A convex-concave elliptic problem with a parameter on the boundary condition
J. García-Melián , Julio D. Rossi and  José Sabina de Lis
2012, 32(4): 1095-1124 doi: 10.3934/dcds.2012.32.1095 +[Abstract](35) +[PDF](545.2KB)
In this paper we study existence and multiplicity of nonnegative solutions to $$ \begin{equation} \left\{\begin{array}{ll} \Delta u = u^p + u^q \qquad & \mbox{in }\Omega, \\ \frac{\partial u }{\partial \nu} =\lambda u \qquad & \mbox{on }\partial \Omega. \end{array}\right. \end{equation} $$ Here $\Omega$ is a smooth bounded domain of $\mathbb{R}^N$, $\nu$ stands for the outward unit normal and $p$, $q$ are in the convex-concave case, that is $0 < q < 1 < p$. We prove that there exists $\Lambda^* >0$ such that there are no nonnegative solutions for $\lambda < \Lambda^*$, and there is a maximal nonnegative solution for $\lambda \ge \Lambda^{*}$. If $\lambda$ is large enough, then there exist at least two nonnegative solutions. We also study the asymptotic behavior of solutions when $\lambda\to \infty$ and the occurrence of dead cores. In the particular case where $\Omega$ is the unit ball of $\mathbb{R}^N$ we show exact multiplicity of radial nonnegative solutions when $\lambda$ is large enough, and also the existence of nonradial nonnegative solutions.
Second order approximations of quasistatic evolution problems in finite dimension
Virginia Agostiniani
2012, 32(4): 1125-1167 doi: 10.3934/dcds.2012.32.1125 +[Abstract](40) +[PDF](653.6KB)
In this paper, we study the limit, as $\epsilon$ goes to zero, of a particular solution of the equation $\epsilon^2A\ddot u^{\epsilon}(t)+\epsilon B\dot u^{\epsilon}(t)+\nabla_xf(t,u^{\epsilon}(t))=0$, where $f(t,x)$ is a potential satisfying suitable coerciveness conditions. The limit $u(t)$ of $u^{\epsilon}(t)$ is piece-wise continuous and verifies $\nabla_xf(t,u(t))=0$. Moreover, certain jump conditions characterize the behaviour of $u(t)$ at the discontinuity times. The same limit behaviour is obtained by considering a different approximation scheme based on time discretization and on the solutions of suitable autonomous systems.
Abstract settings for stabilization of nonlinear parabolic system with a Riccati-based strategy. Application to Navier-Stokes and Boussinesq equations with Neumann or Dirichlet control
Mehdi Badra
2012, 32(4): 1169-1208 doi: 10.3934/dcds.2012.32.1169 +[Abstract](54) +[PDF](717.0KB)
Let $-A: \mathcal{D}(A)\to H$ be the generator of an analytic semigroup and $B : U \to [\mathcal{D}(A^*)]'$ a relatively bounded control operator such that $(A-\sigma,B)$ is stabilizable for some $\sigma>0$. In this paper, we consider the stabilization of the nonlinear system $y'+Ay+G(y,u)=Bu$ by means of a feedback or a dynamical control $u$. The control is obtained from the solution to a Riccati equation which is related to a low-gain optimal quadratic minimization problem. We provide a general abstract framework to define exponentially stable solutions which is based on the contruction of Lyapunov functions. We apply such a theory to stabilize, around an unstable stationary solution, the 2D or 3D Navier-Stokes equations with a Neumann control and the 2D or 3D Boussinesq equations with a Dirichlet control.
Symbolic approach and induction in the Heisenberg group
Jean-Francois Bertazzon
2012, 32(4): 1209-1229 doi: 10.3934/dcds.2012.32.1209 +[Abstract](27) +[PDF](795.7KB)
We associate a homomorphism in the Heisenberg group to each hyperbolic unimodular automorphism of the free group on two generators. We show that the first return-time of some flows in "good" sections, are conjugate to niltranslations, which have the property of being self-induced.
Dense area-preserving homeomorphisms have zero Lyapunov exponents
Mário Bessa and  César M. Silva
2012, 32(4): 1231-1244 doi: 10.3934/dcds.2012.32.1231 +[Abstract](37) +[PDF](428.2KB)
We give a new definition (different from the one in [14]) for a Lyapunov exponent (called new Lyapunov exponent) associated to a continuous map. Our first result states that these new exponents coincide with the usual Lyapunov exponents if the map is differentiable. Then, we apply this concept to prove that there exists a $C^0$-dense subset of the set of the area-preserving homeomorphisms defined in a compact, connected and boundaryless surface such that any element inside this residual subset has zero new Lyapunov exponents for Lebesgue almost every point. Finally, we prove that the function that associates an area-preserving homeomorphism, equipped with the $C^0$-topology, to the integral (with respect to area) of its top new Lyapunov exponent over the whole surface cannot be upper-semicontinuous.
On isotopy and unimodal inverse limit spaces
Henk Bruin and  Sonja Štimac
2012, 32(4): 1245-1253 doi: 10.3934/dcds.2012.32.1245 +[Abstract](45) +[PDF](379.0KB)
We prove that every self-homeomorphism $h : K_s \to K_s$ on the inverse limit space $K_s$ of tent map $T_s$ with slope $s \in (\sqrt 2, 2]$ is isotopic to a power of the shift-homeomorphism $\sigma^R : K_s \to K_s$.
Slow motion of particle systems as a limit of a reaction-diffusion equation with half-Laplacian in dimension one
María del Mar González and  Regis Monneau
2012, 32(4): 1255-1286 doi: 10.3934/dcds.2012.32.1255 +[Abstract](31) +[PDF](541.4KB)
We consider a reaction-diffusion equation with a half-Laplacian. In the case where the solution is independent on time, the model reduces to the Peierls-Nabarro model describing dislocations as transition layers in a phase field setting. We introduce a suitable rescaling of the evolution equation, using a small parameter $\varepsilon$. As $\varepsilon$ goes to zero, we show that the limit dynamics is characterized by a system of ODEs describing the motion of particles with two-body interactions. The interaction forces are in $1/x$ and correspond to the well-known interaction between dislocations.
Box dimension and bifurcations of one-dimensional discrete dynamical systems
Lana Horvat Dmitrović
2012, 32(4): 1287-1307 doi: 10.3934/dcds.2012.32.1287 +[Abstract](57) +[PDF](590.1KB)
This paper is devoted to study the box dimension of the orbits of one-dimensional discrete dynamical systems and their bifurcations in nonhyperbolic fixed points. It is already known that there is a connection between some bifurcations in a nonhyperbolic fixed point of one-dimensional maps, and the box dimension of the orbits near that point. The main purpose of this paper is to generalize that result to the one-dimensional maps of class $C^{k}$ and apply it to one and two-parameter bifurcations of maps with the generalized nondegeneracy conditions. These results show that the value of the box dimension changes at the bifurcation point, and depends only on the order of the nondegeneracy condition. Furthermore, we obtain the reverse result, that is, the order of the nondegeneracy of a map in a nonhyperbolic fixed point can be obtained from the box dimension of a orbit near that point. This reverse result can be applied to the continuous planar dynamical systems by using the Poincaré map, in order to get the multiplicity of a weak focus or nonhyperbolic limit cycle. We also apply the main result to the bifurcations of nonhyperbolic periodic orbits in the plane.
Computation of whiskered invariant tori and their associated manifolds: New fast algorithms
Gemma Huguet , Rafael de la Llave and  Yannick Sire
2012, 32(4): 1309-1353 doi: 10.3934/dcds.2012.32.1309 +[Abstract](52) +[PDF](506.7KB)
We present efficient (low storage requirement and low operation count) algorithms for the computation of several invariant objects for Hamiltonian dynamics, namely KAM tori (i.e diffeomorphic copies of tori such that the motion on them is conjugated to a rigid rotation) both Lagrangian tori(of maximal dimension) and whiskered tori (i.e. tori with hyperbolic directions which, together with the tangents to the torus and the symplectic conjugates span the whole tangent space). We also present algorithms to compute the invariant splitting and the invariant manifolds of whiskered tori. We present the algorithms for both discrete-time dynamical systems and differential equations.
    The algorithms do not require that the system is presented in action-angle variables nor that it is close to integrable and are backed up by rigorous a-posteriori bounds. We will report on the implementation results elsewhere.
Solutions to a fluid-structure interaction free boundary problem
Igor Kukavica and  Amjad Tuffaha
2012, 32(4): 1355-1389 doi: 10.3934/dcds.2012.32.1355 +[Abstract](156) +[PDF](487.4KB)
Our main result is the existence of solutions to the free boundary fluid-structure interaction system. The system consists of a Navier-Stokes equation and a wave equation defined in two different but adjacent domains. The interaction is captured by stress and velocity matching conditions on the free moving boundary lying in between the two domains. We prove the local existence of a solution when the initial velocity of the fluid belongs to $H^{3}$ while the velocity of the elastic body is in $H^{2}$.
Transition layers for a spatially inhomogeneous Allen-Cahn equation in multi-dimensional domains
Fang Li and  Kimie Nakashima
2012, 32(4): 1391-1420 doi: 10.3934/dcds.2012.32.1391 +[Abstract](60) +[PDF](733.4KB)
In this paper, we study a spatially inhomogeneous Allen-Cahn equation in multi-dimensional domains. By upper and lower solution method, we obtain a sufficient condition for a hypersurface $S$ in the domain $\Omega$ to support stable transition layers, and a necessary condition for $S$ in $\Omega$ to support transition layers, not necessarily stable. In addition, sharp estimates on depths of transition layers have also been derived.
Dominated splitting and Pesin's entropy formula
Wenxiang Sun and  Xueting Tian
2012, 32(4): 1421-1434 doi: 10.3934/dcds.2012.32.1421 +[Abstract](57) +[PDF](400.2KB)
Let $M$ be a compact manifold and $f:\,M\rightarrow M$ be a $C^1$ diffeomorphism on $M$. If $\mu$ is an $f$-invariant probability measure which is absolutely continuous relative to Lebesgue measure and for $\mu$ $a.\,\,e.\,\,x\in M,$ there is a dominated splitting $T_{orb(x)}M=E\oplus F$ on its orbit $orb(x)$, then we give an estimation through Lyapunov characteristic exponents from below in Pesin's entropy formula, i.e., the metric entropy $h_\mu(f)$ satisfies $$h_{\mu}(f)\geq\int \chi(x)d\mu,$$ where $\chi(x)=\sum_{i=1}^{dim\,F(x)}\lambda_i(x)$ and $\lambda_1(x)\geq\lambda_2(x)\geq\cdots\geq\lambda_{dim\,M}(x)$ are the Lyapunov exponents at $x$ with respect to $\mu.$
    Consequently, we obtain that Pesin's entropy formula always holds for (1) volume-preserving Anosov diffeomorphisms, (2) volume-preserving partially hyperbolic diffeomorphisms with one-dimensional center bundle, (3) volume-preserving diffeomorphisms far away from homoclinic tangency, and (4) generic volume-preserving diffeomorphisms.
Partially hyperbolic sets with positive measure and $ACIP$ for partially hyperbolic systems
Pengfei Zhang
2012, 32(4): 1435-1447 doi: 10.3934/dcds.2012.32.1435 +[Abstract](38) +[PDF](406.8KB)
In [23] Xia introduced a simple dynamical density basis for partially hyperbolic sets of volume preserving diffeomorphisms. We apply the density basis to the study of the topological structure of partially hyperbolic sets. We show that if $\Lambda$ is a strongly partially hyperbolic set with positive volume, then $\Lambda$ contains the global stable manifolds over ${\alpha}(\Lambda^d)$ and the global unstable manifolds over ${\omega}(\Lambda^d)$.
    We give several applications of the dynamical density to partially hyperbolic maps that preserve some $acip$. We show that if $f$ is essentially accessible and $\mu$ is an $acip$ of $f$, then $\text{supp}(\mu)=M$, the map $f$ is transitive, and $\mu$-a.e. $x\in M$ has a dense orbit in $M$. Moreover if $f$ is accessible and center bunched, then either $f$ preserves a smooth measure or there is no $acip$ at all.

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