# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems - A

June 2009 , Volume 24 , Issue 2

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2009, 24(2): 213-271 doi: 10.3934/dcds.2009.24.213 +[Abstract](2162) +[PDF](616.0KB)
Abstract:
This paper presents a geometric description on Lie algebroids of Lagrangian systems subject to nonholonomic constraints. The Lie algebroid framework provides a natural generalization of classical tangent bundle geometry. We define the notion of nonholonomically constrained system, and characterize regularity conditions that guarantee that the dynamics of the system can be obtained as a suitable projection of the unconstrained dynamics. The proposed novel formalism provides new insights into the geometry of nonholonomic systems, and allows us to treat in a unified way a variety of situations, including systems with symmetry, morphisms, reduction, and nonlinearly constrained systems. Various examples illustrate the results.
2009, 24(2): 273-313 doi: 10.3934/dcds.2009.24.273 +[Abstract](2051) +[PDF](444.0KB)
Abstract:
This paper studies the internal controllability and stabilizability of a family of Boussinesq systems recently proposed by J. L. Bona, M. Chen and J.-C. Saut to describe the two-way propagation of small amplitude gravity waves on the surface of water in a canal. The space of the controllable data for the associated linear system is determined for all values of the four parameters. As an application of this newly established exact controllability, some simple feedback controls are constructed such that the resulting closed-loop systems are exponentially stable. When the parameters are all different from zero, the local exact controllability and stabilizability of the nonlinear system are also established.
2009, 24(2): 315-329 doi: 10.3934/dcds.2009.24.315 +[Abstract](1409) +[PDF](210.1KB)
Abstract:
We prove that a $C^2$ diffeomorphism $f$ of a compact manifold $M$ satisfies Axiom A and the strong transversality condition if and only if it is Hölder stable, that is, any $C^1$ diffeomorphism $g$ of $M$ sufficiently $C^1$ close to $f$ is conjugate to $f$ by a homeomorphism which is Hölder on the whole manifold.
2009, 24(2): 331-348 doi: 10.3934/dcds.2009.24.331 +[Abstract](1740) +[PDF](274.3KB)
Abstract:
We establish a quenched Central Limit Theorem (CLT) for a smooth observable of random sequences of iterated linear hyperbolic maps on the torus. To this end we also obtain an annealed CLT for the same system. We show that, almost surely, the variance of the quenched system is the same as for the annealed system. Our technique is the study of the transfer operator on an anisotropic Banach space specifically tailored to use the cone condition satisfied by the maps.
2009, 24(2): 349-366 doi: 10.3934/dcds.2009.24.349 +[Abstract](1570) +[PDF](248.9KB)
Abstract:
We consider one parameter families of analytic vector fields and diffeomorphisms, including for a parameter value, say $\varepsilon = 0$, the product of rotations in $\R^{2m}\times \R^n$ such that for positive values of the parameter the origin is a hyperbolic point of saddle type. We address the question of determining the limit stable invariant manifold when $\varepsilon$ goes to zero as a subcenter invariant manifold when $\varepsilon = 0$.
2009, 24(2): 367-380 doi: 10.3934/dcds.2009.24.367 +[Abstract](1514) +[PDF](184.7KB)
Abstract:
We give an explicit formula for exponential decay properties of positive solutions for a class of semilinear elliptic equations with Hardy term in the whole space Rn.
2009, 24(2): 381-403 doi: 10.3934/dcds.2009.24.381 +[Abstract](1456) +[PDF](298.3KB)
Abstract:
Let $T_{f}$ : S1S1 be a circle homeomorphism with two break points ab, cb that means the derivative $Df$ of its lift $f\ :\ \mathbb{R}\rightarrow\mathbb{R}$ has discontinuities at the points ã b, ĉb, which are the representative points of ab, cb in the interval $[0,1)$, and irrational rotation number ρf. Suppose that $Df$ is absolutely continuous on every connected interval of the set [0,1]\{ãb, ĉb}, that DlogDf ∈ L1([0,1]) and the product of the jump ratios of $Df$ at the break points is nontrivial, i.e. $\frac{Df_{-}(\tilde{a}_{b})}{Df_{+}(\tilde{a}_{b})}\frac{Df_{-}(\tilde{c}_{b})}{Df_{+}(\tilde{c}_{b})} \ne1$. We prove, that the unique Tf - invariant probability measure $\mu_{f}$ is then singular with respect to Lebesgue measure on S1.
2009, 24(2): 405-440 doi: 10.3934/dcds.2009.24.405 +[Abstract](1682) +[PDF](440.8KB)
Abstract:
We consider nonlinear elliptic problems driven by the $p$-Laplacian with a nonsmooth potential depending on a parameter $\lambda\ >\ 0$. The main result guarantees the existence of two positive, two negative and a nodal (sign-changing) solution for the studied problem whenever $\lambda\ >\ 0$ belongs to a small interval (0, λ*) and $p$ ≥ 2. We do not impose any symmetry hypothesis on the nonlinear potential. The constant-sign solutions are obtained by using variational techniques based on nonsmooth critical point theory (minimization argument, Mountain Pass theorem, and a Brézis-Nirenberg type result for C1-minimizers), while the nodal solution is constructed by an upper-lower solutions argument combined with the Zorn lemma and a nonsmooth second deformation theorem.
2009, 24(2): 441-454 doi: 10.3934/dcds.2009.24.441 +[Abstract](1859) +[PDF](183.6KB)
Abstract:
In this paper, we consider the analytic reducibility problem of an analytic $d-$dimensional quasi-periodic cocycle $(\alpha,\ A)$ on $U(n)$ where $\alpha$ is a Diophantine vector. We prove that, if the cocycle is conjugated to a constant cocycle $(\alpha,\ C)$ by a measurable conjugacy $(0,\ B)$, then for almost all $C$ it is analytically conjugated to $(\alpha,\ C)$ provided that $A$ is sufficiently close to some constant. Moreover $B$ is actually analytic if it is continuous.
2009, 24(2): 455-470 doi: 10.3934/dcds.2009.24.455 +[Abstract](1633) +[PDF](210.0KB)
Abstract:
In this paper, a one-dimensional bipolar hydrodynamic model is considered. This system takes the form of Euler-Poisson with electric field and frictional damping added to the momentum equations. The large time behavior of L entropy solutions of the bipolar hydrodynamic model is firstly studied. Previous works on this topic are mainly concerned with the smooth solution in which no vacuum occurs and the initial data is small. It is proved in this paper that any bounded entropy solution strongly converges to the similarity solution of the porous media equation or the heat equation in L 2(R) with time decay rate. The initial data can contain vacuum and can be arbitrarily large. The method is also applied to improve the convergence rate of [F.Huang, R.Pan, Arch. Rational Mech. Anal.,166(2003),359-376] for compressible Euler equations with damping. As a by product, it is shown that the bounded L entropy solution of the bipolar hydrodynamic model converges to the entropy solution of Euler equations with damping as $t\rightarrow\infty$.
2009, 24(2): 471-487 doi: 10.3934/dcds.2009.24.471 +[Abstract](1591) +[PDF](213.9KB)
Abstract:
We consider Anosov thermostats on a closed surface and the X-ray transform on functions which are up to degree two in the velocities. We show that the subspace where the X-ray transform fails to be s-injective is finite dimensional. Furthermore, if the surface is negatively curved and the thermostat is pure Gaussian (i.e. no magnetic field is present), then the X-ray transform is s-injective.
2009, 24(2): 489-509 doi: 10.3934/dcds.2009.24.489 +[Abstract](1956) +[PDF](308.6KB)
Abstract:
This paper is concerned with the following Lotka-Volterra cross-diffusion system

ut = Δ[(1+kρ(x) v)u] +u(a-u-c(x)v) in Ω Χ (0, ∞),
τvt = Δv +v(b+d(x)u-v) in Ω Χ (0, ∞)

in a bounded domain Ω ⊂ RN with Neumann boundary conditions δvu = δvv = 0 on δΩ. In the previous paper [18], the author has proved that the set of positive stationary solutions forms a fishhook shaped branch Γ under a segregation of $\rho (x)$ and $d(x)$. In the present paper, we give some criteria on the stability of solutions on Γ. We prove that the stability of solutions changes only at every turning point of Γ if τ is large enough. In a different case that $c(x)\ >\ 0$ is large enough, we find a parameter range such that multiple Hopf bifurcation points appear on Γ.

2009, 24(2): 511-521 doi: 10.3934/dcds.2009.24.511 +[Abstract](1589) +[PDF](185.9KB)
Abstract:
We study critical threshold phenomena in a dynamic continuum traffic flow model known as the Payne and Whitham (PW) model. This model is a quasi-linear hyperbolic relaxation system, and when equilibrium velocity is specifically associated with pressure, the equilibrium characteristic speed resonates with one characteristic speed of the full relaxation system. For a scenario of physical interest we identify a lower threshold for finite time singularity in solutions and an upper threshold for the global existence of the smooth solution. The set of initial data leading to global smooth solutions is large, in particular allowing initial velocity of negative slope.
2009, 24(2): 523-545 doi: 10.3934/dcds.2009.24.523 +[Abstract](4807) +[PDF](262.8KB)
Abstract:
In this paper, we consider a non-autonomous stochastic Lotka-Volterra competitive system $dx_i (t) = x_i(t)$[($b_i(t)$-$\sum_{j=1}^{n} a_{ij}(t)x_j(t))$$dt$$+ \sigma_i(t) d B_i(t)]$, where $B_i(t)$($i=1 ,\ 2,\cdots,\ n$) are independent standard Brownian motions. Some dynamical properties are discussed and the sufficient conditions for the existence of global positive solutions, stochastic permanence, extinction as well as global attractivity are obtained. In addition, the limit of the average in time of the sample paths of solutions is estimated.
2009, 24(2): 547-565 doi: 10.3934/dcds.2009.24.547 +[Abstract](1675) +[PDF](248.2KB)
Abstract:
We prove that the Cauchy problem for the three-dimensional Zakharov-Kuznetsov equation is locally well-posed for data in $H^s(\R^3)$, s > $\frac{9}{8}$.
2009, 24(2): 567-587 doi: 10.3934/dcds.2009.24.567 +[Abstract](1393) +[PDF](254.0KB)
Abstract:
We are interested in the planar Lorentz process with a periodic configuration of strictly convex obstacles and with finite horizon. Its recurrence comes from a criteria of Conze in [8] or of Schmidt in [15] and from the central limit theorem for the billiard in the torus ([2,4,19]) Another way to prove recurrence is given by Szász and Varjú in [18]. Total ergodicity follows from these results (see [16] and [12]). In this paper we answer a question of Szász about the asymptotic behaviour of the number of visited cells when the time goes to infinity. It is not more difficult to study the asymptotic of the number of obstacles hit by the particle when the time goes to infinity. We give an estimate for the expectation and a result of almost sure convergence. For the simple random walk in Z2, this question has been studied by Dvoretzky and Erdös in [10]. We adapt the proof of Dvoretzky and Erdös. The lack of independence is compensated by a strong decorrelation result due to Chernov ([6])and by some refinement (got in [14])of the local limit theorem proved by Szász and Varjú in [18].
2009, 24(2): 589-611 doi: 10.3934/dcds.2009.24.589 +[Abstract](1780) +[PDF](301.1KB)
Abstract:
Skew-product semiflows induced by semi-convex and type-K competitive almost periodic delay differential equations are studied. If $M$ is a compact positively invariant subset of the skew-product semiflow, then continuous separation of the skew-product semiflow on $M$ holds. Furthermore, if two minimal subsets $M_{1}$ and $M_{2}$ of the skew-product semiflow satisfying completely strongly type-K ordering $M_{1}$«$^C_K M_{2}$, then $M_{1}$ is an attractor. Finally, these results are applied to a nonautonomous delayed Hopfield-type neural networks with the diagonal-nonnegative type-K monotone interconnection matrix and sufficient conditions are obtained for the existence of global or partial attractors.
2009, 24(2): 613-624 doi: 10.3934/dcds.2009.24.613 +[Abstract](1506) +[PDF](187.3KB)
Abstract:
We give an explicit geometric description of the $\times2$, $\times3$ system, and use this to study a uniform family of Markov partitions related to those of Wilson and Abramov. The behaviour of these partitions is stable across expansive cones and transitions in this behaviour detect the non-expansive lines.
2009, 24(2): 625-651 doi: 10.3934/dcds.2009.24.625 +[Abstract](1489) +[PDF](362.9KB)
Abstract:
We study a free boundary problem modelling the growth of non-necrotic tumors with fluid-like tissues. The fluid velocity satisfies Stokes equations with a source determined by the proliferation rate of tumor cells which depends on the concentration of nutrients, subject to a boundary condition with stress tensor effected by surface tension. It is easy to prove that this problem has a unique radially symmetric stationary solution. By using a functional approach, we prove that there exists a threshold value γ* > 0 for the surface tension coefficient $\gamma$, such that in the case γ > γ* this radially symmetric stationary solution is asymptotically stable under small non-radial perturbations, whereas in the opposite case it is unstable.

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