# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems - A

July 2011 , Volume 29 , Issue 3

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2011, 29(3): 693-736 doi: 10.3934/dcds.2011.29.693 +[Abstract](1658) +[PDF](960.3KB)
Abstract:
We prove that the $C^3$ diffeomorphisms on surfaces, exhibiting infinitely many sinks near the generic unfolding of a quadratic homoclinic tangency of a dissipative saddle, can be perturbed along an infinite dimensional manifold of $C^3$ diffeomorphisms such that infinitely many sinks persist simultaneously. On the other hand, if they are perturbed along one-parameter families that unfold generically the quadratic tangencies, then at most a finite number of those sinks have continuation.
2011, 29(3): 737-756 doi: 10.3934/dcds.2011.29.737 +[Abstract](1506) +[PDF](429.2KB)
Abstract:
In this paper we prove a global well-posedness result for tridimensional Navier-Stokes-Boussinesq system with axisymmetric initial data. This system couples Navier-Stokes equations with a transport equation governing the density.
2011, 29(3): 757-767 doi: 10.3934/dcds.2011.29.757 +[Abstract](1353) +[PDF](392.6KB)
Abstract:
A well-known result by Lazer and Leach establishes that if $g:\R\to \R$ is continuous and bounded with limits at infinity and $m\in \mathbb{N}$, then the resonant periodic problem

$u'' + m^2 u + g(u)=p(t),\qquad u(0)-u(2\pi)=u'(0)-u'(2\pi)=0$

admits at least one solution, provided that

$(\a_m(p)^2+$β$_m(p)^2$$)^\frac 1\2$< $\frac 2\pi |g(+\infty)-g(-\infty)|,$

where $\a_m(p)$ and β$_m(p)$ denote the $m$-th Fourier coefficients of the forcing term $p$.
In this article we prove that, as it occurs in the case $m=0$, the condition on $g$ may be relaxed. In particular, no specific behavior at infinity is assumed.

2011, 29(3): 769-801 doi: 10.3934/dcds.2011.29.769 +[Abstract](1876) +[PDF](505.7KB)
Abstract:
In this paper, we study the incompressible Navier-Stokes equations on a moving domain in $\mathbb{R}^{3}$ of finite depth, bounded above by the free surface and bounded below by a solid flat bottom. We prove that there exists a unique, global-in-time solution to the problem provided that the initial velocity field and the initial profile of the boundary are sufficiently small in Sobolev spaces.
2011, 29(3): 803-822 doi: 10.3934/dcds.2011.29.803 +[Abstract](1500) +[PDF](463.2KB)
Abstract:
In this paper, we consider codimension one Anosov actions of $\RR^k,\ k\geq 1,$ on closed connected orientable manifolds of dimension $n+k$ with $n\geq 3$. We show that the fundamental group of the ambient manifold is solvable if and only if the weak foliation of codimension one is transversely affine. We also study the situation where one $1$-parameter subgroup of $\RR^k$ admits a cross-section, and compare this to the case where the whole action is transverse to a fibration over a manifold of dimension $n$. As a byproduct, generalizing a Theorem by Ghys in the case $k=1$, we show that, under some assumptions about the smoothness of the sub-bundle $E^{ss}\oplus E$uu , and in the case where the action preserves the volume, it is topologically equivalent to a suspension of a linear Anosov action of $\mathbb{Z}^k$ on $\TT^{n}$.
2011, 29(3): 823-838 doi: 10.3934/dcds.2011.29.823 +[Abstract](1416) +[PDF](412.3KB)
Abstract:
We consider solutions of the Allen-Cahn equation in the whole Grushin plane and we show that if they are monotone in the vertical direction, then they are stable and they satisfy a good energy estimate.
However, they are not necessarily one-dimensional, as a counter-example shows.
2011, 29(3): 839-844 doi: 10.3934/dcds.2011.29.839 +[Abstract](1288) +[PDF](313.1KB)
Abstract:
In the study of the local dynamics of a germ of diffeomorphism fixing the origin in $\mathbb C$, an important problem is to determine the centralizer of the germ in the group Diff$(\mathbb C,0)$ of germs of diffeomorphisms fixing the origin. When the germ is not of finite order, then the centralizer is abelian, and hence a maximal abelian subgroup of Diff$(\mathbb C,0)$. Conversely any maximal abelian subgroup which contains an element of infinite order is equal to the centralizer of that element. A natural question is whether every maximal abelian subgroup contains an element of infinite order, or whether there exist maximal abelian torsion subgroups; we show that such subgroups do indeed exist, and moreover that any infinite subgroup of the rationals modulo the integers $\mathbb{Q/Z}$ can be embedded into Diff$(\mathbb C,0)$ as such a subgroup.
2011, 29(3): 845-871 doi: 10.3934/dcds.2011.29.845 +[Abstract](1324) +[PDF](481.0KB)
Abstract:
We consider two models for branched transport: the one introduced in Bernot et al. (Publ Mat 49:417-451, 2005), which makes use of a functional defined on measures over the space of Lipschitz paths, and the path functional model presented in Brancolini et al. (J Eur Math Soc 8:415-434, 2006), where one minimizes some suitable action functional defined over the space of measure-valued Lipschitz curves, getting sort of a Riemannian metric on the space of probabilities, favouring atomic measures, with a cost depending on the masses of each of their atoms. We prove that modifying the latter model according to Brasco (Ann Mat Pura Appl 189:95-125, 2010), then the two models turn out to be equivalent.
2011, 29(3): 873-891 doi: 10.3934/dcds.2011.29.873 +[Abstract](1956) +[PDF](457.8KB)
Abstract:
Consider a class of skew product transformations consisting of an ergodic or a periodic transformation on a probability space $(M, \B,\mu)$ in the base and a semigroup of transformations on another probability space (Ω,$\F,P)$ in the fibre. Under suitable mixing conditions for the fibre transformation, we show that the properties ergodicity, weakly mixing, and strongly mixing are passed on from the base transformation to the skew product (with respect to the product measure). We derive ergodic theorems with respect to the skew product on the product space.
The main aim of this paper is to establish uniform convergence with respect to the base variable for the series of ergodic averages of a function $F$ on $M\times$Ω along the orbits of such a skew product. Assuming a certain growth condition for the coupling function, a strong mixing condition on the fibre transformation, and continuity and integrability conditions for $F,$ we prove uniform convergence in the base and $\L^p(P)$-convergence in the fibre. Under an equicontinuity assumption on $F$ we further show $P$-almost sure convergence in the fibre. Our work has an application in information theory: It implies convergence of the averages of functions on random fields restricted to parts of stair climbing patterns defined by a direction.
2011, 29(3): 893-908 doi: 10.3934/dcds.2011.29.893 +[Abstract](1449) +[PDF](589.6KB)
Abstract:
We consider billiard trajectories in ideal hyperbolic polygons and present a conjecture about the minimality of the average length of cyclically related billiard trajectories in regular hyperbolic polygons. We prove this conjecture in particular cases, using geometric and algebraic methods from hyperbolic geometry.
2011, 29(3): 909-928 doi: 10.3934/dcds.2011.29.909 +[Abstract](1550) +[PDF](518.7KB)
Abstract:
We consider the mass concentration phenomenon for the $L^2$-critical nonlinear Schrödinger equations of higher orders. We show that any solution $u$ to $iu_{t} + (-\Delta)^{\frac\alpha 2} u =\pm |u|^\frac{2\alpha}{d}u$, $u(0,\cdot)\in L^2$ for $\alpha >2$, which blows up in a finite time, satisfies a mass concentration phenomenon near the blow-up time. We verify that as $\alpha$ increases, the size of region capturing a mass concentration gets wider due to the stronger dispersive effect.
2011, 29(3): 929-952 doi: 10.3934/dcds.2011.29.929 +[Abstract](1360) +[PDF](543.7KB)
Abstract:
It is known that for a rational map $f$ with a disconnected Julia set, the set of wandering Julia components is uncountable. We prove that all but countably many of them have a simple topology, namely having one or two complementary components. We show that the remaining countable subset $\Sigma$ is backward invariant. Conjecturally $\Sigma$ does not contain an infinite orbit. We give a very strong necessary condition for $\Sigma$ to contain an infinite orbit, thus proving the conjecture for many different cases. We provide also two sufficient conditions for a Julia component to be a point. Finally we construct several examples describing different topological structures of Julia components.
2011, 29(3): 953-978 doi: 10.3934/dcds.2011.29.953 +[Abstract](1241) +[PDF](556.2KB)
Abstract:
We prove a formula relating the fixed point index of rest points of a completely continuous semiflow defined on a (not necessarily locally compact) metric space in the interior of an isolating block $B$ to the Euler characteristic of the pair $(B,B^-)$, where $B^-$ is the exit set. The proof relies on a general concept of an approximate neighborhood extension space and a full fixed point index theory for self-maps of such spaces. As a consequence, a generalized Poincaré-Hopf type formula for the differential equation determined by a perturbation of the generator of a compact $C_0$ semigroup is obtained.
2011, 29(3): 979-999 doi: 10.3934/dcds.2011.29.979 +[Abstract](1420) +[PDF](434.2KB)
Abstract:
We show that there is a bijection between the renormalizations and proper completely invariant closed sets of expanding Lorenz map, which enables us to distinguish periodic and non-periodic renormalizations. Based on the properties of the periodic orbit with minimal period, the minimal completely invariant closed set is constructed. Topological characterizations of the renormalizations and $\alpha$-limit sets are obtained via consecutive renormalizations. Some properties of periodic renormalizations are collected in Appendix.
2011, 29(3): 1001-1030 doi: 10.3934/dcds.2011.29.1001 +[Abstract](1370) +[PDF](565.1KB)
Abstract:
In this paper we build models for short-term, mean-term and long-term dynamics of dune and megariple morphodynamics. They are models that are degenerated parabolic equations which are, moreover, singularly perturbed. We, then give an existence and uniqueness result for the short-term and mean-term models. This result is based on a time-space periodic solution existence result for degenerated parabolic equation that we set out. Finally the short-term model is homogenized.
2011, 29(3): 1031-1039 doi: 10.3934/dcds.2011.29.1031 +[Abstract](1317) +[PDF](348.2KB)
Abstract:
We study the cohomological equation for a smooth vector field on a compact manifold. We show that if the vector field is cohomology free, then it can be embedded continuously in a linear flow on an Abelian group.
2011, 29(3): 1041-1083 doi: 10.3934/dcds.2011.29.1041 +[Abstract](1215) +[PDF](746.2KB)
Abstract:
We consider a parabolic problem with discontinuous hysteresis on the boundary, arising in modelling various thermal control processes. By reducing the problem to an infinite dynamical system, sufficient conditions for the existence and uniqueness of a periodic solution are found. Global stability of the periodic solution is proved.
2011, 29(3): 1085-1096 doi: 10.3934/dcds.2011.29.1085 +[Abstract](1478) +[PDF](358.7KB)
Abstract:
This paper is mainly concerned with Euler-Poisson equations modeling Newtonian stars. We establish the existence of rotating star solutions for general compressible fluids with prescribed angular velocity law, which is the main point distinguished with the case with prescribed angular momentum per unit mass. The compactness of any minimizing sequence is established, which is important from the stability point of view.
2011, 29(3): 1097-1111 doi: 10.3934/dcds.2011.29.1097 +[Abstract](1589) +[PDF](403.5KB)
Abstract:
Let $\mathcal M$ be a smooth Riemannian manifold with the metric $(g_{ij})$ of dimension $n$, and let $H= 1/2 g^{ij}(q)p_ip_j+V(t,q)$ be a smooth Hamiltonian on $\mathcal M$, where $(g^{ij})$ is the inverse matrix of $(g_{ij})$. Under suitable assumptions we prove the existence of heteroclinic orbits of the induced Hamiltonian systems.
2011, 29(3): 1113-1139 doi: 10.3934/dcds.2011.29.1113 +[Abstract](1799) +[PDF](534.2KB)
Abstract:
In this paper we focus on the initial value problem for quasi-linear dissipative plate equation in multi-dimensional space $(n\geq2)$. This equation verifies the decay property of the regularity-loss type, which causes the difficulty in deriving the global a priori estimates of solutions. We overcome this difficulty by employing a time-weighted $L^2$ energy method which makes use of the integrability of $||$(∂$^2_xu_t,$∂$^3_xu)(t)||_{L^{\infty}}$. This $L^\infty$ norm can be controlled by showing the optimal $L^2$ decay estimates for lower-order derivatives of solutions. Thus we obtain the desired a priori estimate which enables us to prove the global existence and asymptotic decay of solutions under smallness and enough regularity assumptions on the initial data. Moreover, we show that the solution can be approximated by a simple-looking function, which is given explicitly in terms of the fundamental solution of a fourth-order linear parabolic equation.
2011, 29(3): 1141-1154 doi: 10.3934/dcds.2011.29.1141 +[Abstract](1436) +[PDF](396.1KB)
Abstract:
We consider the dependence on parameters of the solutions of cohomology equations over Anosov diffeomorphisms. We show that the solutions depend on parameters as smoothly as the data. As a consequence we prove optimal regularity results for the solutions of cohomology equations taking value in diffeomorphism groups. These results are motivated by applications to rigidity theory, dynamical systems, and geometry.
In particular, in the context of diffeomorphism groups we show: Let $f$ be a transitive Anosov diffeomorphism of a compact manifold $M$. Suppose that $\eta \in C$k+α$(M,$Diff$^r(N))$ for a compact manifold $N$, $k,r \in \N$, $r \geq 1$, and $0 < \alpha \leq \Lip$. We show that if there exists a $\varphi\in C$k+α$(M,$Diff$^1(N))$ solving

$\varphi_{f(x)} = \eta_x \circ \varphi_x$

then in fact $\varphi \in C$k+α$(M,$Diff$^r(N))$. The existence of this solutions for some range of regularities is studied in the literature.

2011, 29(3): 1155-1174 doi: 10.3934/dcds.2011.29.1155 +[Abstract](1345) +[PDF](476.7KB)
Abstract:
We obtain a large deviation function for the stationary measures of twisted Brownian motions associated to the Lagrangians $L_{\lambda}(p,v)=\frac{1}{2}g_{p}(v,v)- \lambda\omega_{p}(v)$, where $g$ is a $C^{\infty}$ Riemannian metric in a compact surface $(M,g)$ with nonpositive curvature, $\omega$ is a closed 1-form such that the Aubry-Mather measure of the Lagrangian $L(p,v)=\frac{1}{2}g_{p}(v,v)-\omega_{p}(v)$ has support in a unique closed geodesic $\gamma$; and the curvature is negative at every point of $M$ but at the points of $\gamma$ where it is zero. We also assume that the Aubry set is equal to the Mather set. The large deviation function is of polynomial type, the power of the polynomial function depends on the way the curvature goes to zero in a neighborhood of $\gamma$. This results has interesting counterparts in one-dimensional dynamics with indifferent fixed points and convex billiards with flat points in the boundary of the billiard. A previous estimate by N. Anantharaman of the large deviation function in terms of the Peierl's barrier of the Aubry-Mather measure is crucial for our result.
2011, 29(3): 1175-1189 doi: 10.3934/dcds.2011.29.1175 +[Abstract](1259) +[PDF](529.6KB)
Abstract:
Using semihyperbolicity as a basic tool, we provide a general computer assisted method for verifying hyperbolicity of a given set. As a consequence we obtain that the Hénon attractor is hyperbolic for some parameter values.
2011, 29(3): 1191-1195 doi: 10.3934/dcds.2011.29.1191 +[Abstract](1134) +[PDF](285.7KB)
Abstract:
We prove that a $C^1$ generic orientation-preserving diffeomorphism of a closed orientable surface either is Axiom A without cycles or the closures of the sets of saddles and of periodic points without real eigenvalues have nonempty intersection.
2011, 29(3): 1197-1204 doi: 10.3934/dcds.2011.29.1197 +[Abstract](1489) +[PDF](326.1KB)
Abstract:
Assume that $X$ is a hyperbolic basic set for $f:X\to X$. We show new examples of Lie group fibers $G$ for which, in the class of $C^r, r>0,$ $G$-extensions of $f$, those that are transitive are open and dense. The fibers are semidirect products of compact and nilpotent groups.
2011, 29(3): 1205-1244 doi: 10.3934/dcds.2011.29.1205 +[Abstract](1559) +[PDF](749.9KB)
Abstract:
We investigate the dynamics of semigroups generated by a family of polynomial maps on the Riemann sphere such that the postcritical set in the complex plane is bounded. The Julia set of such a semigroup may not be connected in general. We show that for such a polynomial semigroup, if $A$ and $B$ are two connected components of the Julia set, then one of $A$ and $B$ surrounds the other. From this, it is shown that each connected component of the Fatou set is either simply or doubly connected. Moreover, we show that the Julia set of such a semigroup is uniformly perfect. An upper estimate of the cardinality of the set of all connected components of the Julia set of such a semigroup is given. By using this, we give a criterion for the Julia set to be connected. Moreover, we show that for any $n\in N \cup \{ \aleph _{0}\} ,$ there exists a finitely generated polynomial semigroup with bounded planar postcritical set such that the cardinality of the set of all connected components of the Julia set is equal to $n.$ Many new phenomena of polynomial semigroups that do not occur in the usual dynamics of polynomials are found and systematically investigated.
2011, 29(3): 1245-1260 doi: 10.3934/dcds.2011.29.1245 +[Abstract](1523) +[PDF](392.5KB)
Abstract:
This paper is concerned with the analytic and numerical dissipativity of nonlinear neutral differential equations with proportional delay, the so-called generalized pantograph equations. A sufficient condition for the dissipativity of the systems is given. It is shown that the backward Euler method inherits the dissipativity of the underlying system. Numerical examples are given to confirm the theoretical results.
2011, 29(3): 1261-1275 doi: 10.3934/dcds.2011.29.1261 +[Abstract](1659) +[PDF](399.1KB)
Abstract:
We obtained the $C^{\alpha}$ continuity of weak solutions for a class of ultraparabolic equations with measurable coefficients of the form

$\partial_t \ u= \sum_{i,j=1}^{m_0}X_i(a_{ij}(x,t)X_j\ u )+X_0 u.$

By choosing a new cut-off function, we simplified and generalized the earlier arguments and proved the $C^{\alpha}$ regularity of weak solutions for more general ultraparabolic equations.

2011, 29(3): 1277-1290 doi: 10.3934/dcds.2011.29.1277 +[Abstract](1556) +[PDF](447.6KB)
Abstract:
In this paper, we first discuss the regular level set of a nonsingular Smale flow (NSF) on a 3-manifold. The main result about this topic is that a 3-manifold $M$ admits an NSF which has a regular level set homeomorphic to $(n+1)T^{2}$ $(n\in \mathbb{Z}, n\geq 0)$ if and only if $M=M'$#$n S^{1}\times S^{2}$. Then we discuss how to realize a template as a basic set of an NSF on a 3-manifold. We focus on the connection between the genus of the template $T$ and the topological structure of the realizing 3-manifold $M$.
2011, 29(3): 1291-1307 doi: 10.3934/dcds.2011.29.1291 +[Abstract](1358) +[PDF](2579.5KB)
Abstract:
This paper studies the $C^1$-perturbation problem of strictly $A$-coupled-expanding maps in finite-dimensional Euclidean spaces, where $A$ is an irreducible transition matrix with one row-sum no less than $2$. It is proved that under certain conditions strictly $A$-coupled-expanding maps are chaotic in the sense of Li-Yorke or Devaney under small $C^1$-perturbations. It is shown that strictly $A$-coupled-expanding maps are $C^1$ structurally stable in their chaotic invariant sets under certain stronger conditions. One illustrative example is provided with computer simulations.

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