# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems - A

2005 , Volume 12 , Issue 3

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2005, 12(3): 377-385 doi: 10.3934/dcds.2005.12.377 +[Abstract](169) +[PDF](197.5KB)
Abstract:
We consider conformal structures invariant under a volume-preserving Anosov system. We show that if such a structure is in $L^p$ for sufficiently large $p$, then it is continuous.
2005, 12(3): 387-402 doi: 10.3934/dcds.2005.12.387 +[Abstract](175) +[PDF](249.8KB)
Abstract:
The well-posedness of the Cauchy problem for a generalized nonlinear dispersive equation is studied. Local well-posedness for data in $H^s(\mathbb R)(s>-\frac{1}{8})$ and the global result for data in $L^{2}(\mathbb{R})$ are obtained if $l=2$. Moreover, for $l=3$, the problem is locally well-posed for data in $H^s(s>\frac{1}{4}).$ The main idea is to use the Fourier restriction norm method.
2005, 12(3): 403-412 doi: 10.3934/dcds.2005.12.403 +[Abstract](145) +[PDF](212.8KB)
Abstract:
In this note we derive an upper bound for the Hausdorff and box dimension of the stable and local stable set of a hyperbolic set $\Lambda$ of a $C^2$ diffeomorphisms on a $n$-dimensional manifold. As a consequence we obtain that dim$_H W^s(\Lambda)=n$ is equivalent to the existence of a SRB-measure. We also discuss related results for expanding maps.
Dong Ye and Feng Zhou
2005, 12(3): 413-424 doi: 10.3934/dcds.2005.12.413 +[Abstract](150) +[PDF](234.2KB)
Abstract:
We consider semilinear elliptic equations $\Delta u \pm \rho(x)f(u) = 0$, or more generally $\Delta u + \varphi(x, u) = 0$, posed in $\R^N$ ($N\geq 3$). We prove that the existence of entire bounded positive solutions is closely related to the existence of bounded solution for $\Delta u + \rho(x) = 0$ in $\mathbb R^N$. Many sufficient conditions which are invariant under the isometry group of $\mathbb R^N$ are established. Our proofs use the standard barrier method, but our results extend many earlier works in this direction. Our ideas can also be applied for the existence of large solutions, for the exterior domain problem and for the system situations.
2005, 12(3): 425-436 doi: 10.3934/dcds.2005.12.425 +[Abstract](172) +[PDF](212.5KB)
Abstract:
We consider the coupled Euler-Bernoulli viscoelastic system with boundary damping. The existence of global solutions is proved by means of the Faedo-Galerkin method and the uniform decay rate of the energy is obtained by following the perturbed energy method.
2005, 12(3): 437-463 doi: 10.3934/dcds.2005.12.437 +[Abstract](155) +[PDF](265.3KB)
Abstract:
In this article we study a class of robust control problems in fluid mechanics recently proposed in [5]. Using a method of [23], we provide another proof of the existence and the uniqueness of solutions to the robust control problems under weaker assumptions as compared to [5]. We also study the Newton method for the numerical solution of these control problems. We prove the convergence of the method and we obtain an estimate of the convergence rate.
2005, 12(3): 465-480 doi: 10.3934/dcds.2005.12.465 +[Abstract](123) +[PDF](244.8KB)
Abstract:
Let $f: M \rightarrow M$ denote a diffeomorphism of a smooth manifold $M$. Let $p \in M$ be its hyperbolic fixed point with stable and unstable manifolds $W_S$ and $W_U$ respectively. Assume that $W_S$ is a curve. Suppose that $W_U$ and $W_S$ have a degenerate homoclinic crossing at a point $B\ne p$, i.e., they cross at $B$ tangentially with a finite order of contact.
It is shown that, subject to $C^1$-linearizability and certain conditions on the invariant manifolds, a transverse homoclinic crossing will arise arbitrarily close to $B$. This proves the existence of a horseshoe structure arbitrarily close to $B$, and extends a similar planar result of Homburg and Weiss [10].
2005, 12(3): 481-500 doi: 10.3934/dcds.2005.12.481 +[Abstract](149) +[PDF](258.3KB)
Abstract:
For a scalar delayed differential equation $\dot x(t)=f(t,x_t)$, we give sufficient conditions for the global attractivity of its zero solution. Some technical assumptions are imposed to insure boundedness of solutions and attractivity of non-oscillatory solutions. For controlling the behaviour of oscillatory solutions, we require a very general condition of Yorke type, together with a 3/2-condition. The results are particularly interesting when applied to scalar differential equations with delays which have served as models in populations dynamics, and can be written in the general form $\dot x(t)=(1+x(t))F(t,x_t)$. Applications to several models are presented, improving known results in the literature.
2005, 12(3): 501-522 doi: 10.3934/dcds.2005.12.501 +[Abstract](199) +[PDF](264.5KB)
Abstract:
We analyze the asymptotic behavior of a nonlinear mathematical model of cellular proliferation which describes the production of blood cells in the bone marrow. This model takes the form of a system of two maturity structured partial differential equations, with a retardation of the maturation variable and a time delay depending on this maturity. We show that the stability of this system depends strongly on the behavior of the immature cell population. We obtain conditions for the global stability and the instability of the trivial solution.
2005, 12(3): 523-530 doi: 10.3934/dcds.2005.12.523 +[Abstract](146) +[PDF](207.4KB)
Abstract:
We prove that the following properties are $C^0$ generic in the space of discrete dynamical systems on a compact smooth manifold $M$: periodic shadowing (Theorem 1.1) and, assuming dim$M\leq 3$, $\mathcal T_C$-inverse shadowing (Theorem 1.2).
2005, 12(3): 531-554 doi: 10.3934/dcds.2005.12.531 +[Abstract](128) +[PDF](510.8KB)
Abstract:
We study a singularly perturbed scalar reaction-diffusion equation on a bounded interval with a spatially inhomogeneous bistable nonlinearity. For certain nonlinearities, which are piecewise constant in space on $k$ subintervals, it is possible to characterize all stationary solutions for small $\varepsilon$ by means of sequences of $k$ symbols, indicating the behavior of the solution in each subinterval. Determining also Morse indices and zero numbers of the equilibria in terms of the symbol sequences, we are able to give a criterion for heteroclinic connections and a description of the associated global attractor for all $k$.
2005, 12(3): 555-565 doi: 10.3934/dcds.2005.12.555 +[Abstract](296) +[PDF](232.6KB)
Abstract:
We establish lower bounds for the topological entropy expressed in terms of the exponential growth rate of $k$-volumes. This approach provides the sharpest possible bounds when no further geometric information is available. In particular, our methods apply to (partially) volume-expanding dynamics with not necessarily compact phase space, including a large class of geodesic flows. As an application, we conclude that the topological entropy of these systems is positive.

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