# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems - A

October 2001 , Volume 7 , Issue 4

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2001, 7(4): 663-674 doi: 10.3934/dcds.2001.7.663 +[Abstract](1165) +[PDF](1349.2KB)
Abstract:
The elliptical stadium is a plane region bounded by a curve constructed by joining two half-ellipses with axes $a > 1$ and $b = 1$, by two parallel segments of equal length $2h$. The corresponding billiard problem defines a two-parameter family of discrete dynamical systems through the maps $T_{a,h}$.
We investigate the existence of elliptic islands for a special family of periodic orbits of $T_{a,h}$. The hyperbolic character of those orbits were studied in [2] for $1 < a < \sqrt 2$ and here we look for the elliptical character for every $a > 1$.
We prove that, for $a < \sqrt 2$, the lower bound for chaos $h = H(a)$ found in [2] is the upper bound of ellipticity for this special family. For $a > \sqrt 2$ we prove that there is no upper bound on h for the existence of elliptic islands.
2001, 7(4): 675-702 doi: 10.3934/dcds.2001.7.675 +[Abstract](1364) +[PDF](279.7KB)
Abstract:
A method of constructing solutions of semilinear dissipative equations in bounded domains is proposed. It allows to calculate the higher-order long-time asymptotics. The application of this approach is given for solving the first initialboundary value problem for the damped Boussinesq equation

$u_{t t} - 2b\Delta u_t = -\alpha \Delta^2 u+ \Delta u + \beta\Delta(u^2)$

in a unit ball $B$. Homogeneous boundary conditions and small initial data are examined. The existence of mild global-in-time solutions is established in the space $C^0([0,\infty), H^s_0(B)), s < 3/2$, and the solutions are constructed in the form of the expansion in the eigenfunctions of the Laplace operator in $B$. For $-3/2 +\varepsilon \leq s <3/2$, where $\varepsilon > 0$ is small, the uniqueness is proved. The second-order long-time asymptotics is calculated which is essentially nonlinear and shows the nonlinear mode multiplication.

2001, 7(4): 703-718 doi: 10.3934/dcds.2001.7.703 +[Abstract](1234) +[PDF](208.5KB)
Abstract:
In this paper we state some existence results for the semilinear elliptic equation $-\Delta u(x)-\lambda u(x) = W(x)f(u)$ where $W(x)$ is a function possibly changing sign, $f$ has a superlinear growth and $\lambda$ is a positive real parameter. We discuss both the cases of subcritical and critical growth for $f$, and prove the existence of Linking type solutions.
2001, 7(4): 719-735 doi: 10.3934/dcds.2001.7.719 +[Abstract](1334) +[PDF](200.1KB)
Abstract:
A strongly damped semilinear wave equation on the whole space is considered. Existence and uniqueness results are provided, together with the existence of an absorbing set, which is uniform as the external force is allowed to run in a certain functional set. In the autonomous case, the equation is shown to possess a universal attractor.
2001, 7(4): 737-745 doi: 10.3934/dcds.2001.7.737 +[Abstract](1101) +[PDF](148.1KB)
Abstract:
This paper addresses the oscillation death in systems of coupled neural oscillators. The coupling is assumed to be transferable and such transferable structure includes the nearest-neighbor coupling and the multiple-neighbor coupling. The death solution is obtained as a limit of upper solutions and lower solutions. We investigate a coupled cyclic chain of oscillators, in which the coupling is transferred in one direction and with a time lag. To obtain the asymptotic stability of the death solution, we establish the necessary and sufficient conditions to ensure the zeros for a class of exponential polynomials to lie to the left of the imaginary axis.
2001, 7(4): 747-762 doi: 10.3934/dcds.2001.7.747 +[Abstract](1165) +[PDF](218.2KB)
Abstract:
The bifurcation of subharmonics for resonant nonautonomous equations of the second order is studied. The set of subharmonics is defined by principal homogeneous parts of the nonlinearities provided these parts are not polynomials. Analogous statements are proved for bifurcations of $p$-periodic orbits of a planar dynamical system. The analysis is based on topological methods and harmonic linearization.
2001, 7(4): 763-780 doi: 10.3934/dcds.2001.7.763 +[Abstract](1146) +[PDF](187.6KB)
Abstract:
In this paper, we consider a $2\times 2$ hyperbolic system originates from the theory of phase dynamics. This one-phase problem can be obtained by using the Catteneo-Fourier law which is a variant of the standard Fourier law in one dimensional space. A new classical existence and uniqueness result is established by some a priori estimates using the characteristic method. The convergence of the solutions to the one of classical Stefan problems is also obtained.
2001, 7(4): 781-786 doi: 10.3934/dcds.2001.7.781 +[Abstract](1153) +[PDF](140.8KB)
Abstract:
Let $S$ be an increasing sequence of positive integers and let $\omega$ be an $\omega$–limit set of a continuous interval map $f$. We prove that $h_S(f|\omega) = 0$ if $h(f) = 0$, where $h_S(f)$ denotes the topological sequence entropy of $f$.
2001, 7(4): 787-800 doi: 10.3934/dcds.2001.7.787 +[Abstract](1310) +[PDF](189.0KB)
Abstract:
Let $H_\mu, 0 < \mu < < 1$ be a small perturbation of size $\mu$ of an initially hyperbolic Hamiltonian system. We prove that Graff tori satisfy the transitivity property : if $T_1, T_2$ and $T_3$ are three Graff tori such that $W^+(T_1)$ (resp. $W^+(T_2))$ and $W^-$ $(T_2)$ (resp. $W^-$ $(T_3))$ intersect transversally in a given energy level $H$ with an angle of order $\mu$, then $W^+(T_1)$ and $W^-$ $(T_3)$ intersect transversally in $H$ with an angle of order $\mu- c(\mu)$, with $c(\mu)$ exponentially small in $\mu$. The proof is based on a quantitative version of the $\lambda$-lemma for Graff tori called the transfer lemma. This result allows us to compute the Arnold diffusion time along transition chains for initially hyperbolic Hamiltonian systems. We prove that this time is polynomial in the inverse of the perturbing parameter.
2001, 7(4): 801-808 doi: 10.3934/dcds.2001.7.801 +[Abstract](1025) +[PDF](140.6KB)
Abstract:
The method of generalized quasilinearization is extended to semilinear degenerate elliptic boundary value problems.
2001, 7(4): 809-820 doi: 10.3934/dcds.2001.7.809 +[Abstract](1020) +[PDF](151.5KB)
Abstract:
Forced periodic oscillations of coupled systems of semilinear neutral type functional differential equations are investigated. Explicit conditions for existence, uniqueness and stability of periodic solutions are derived. These conditions are formulated in terms of the roots of characteristic polynomials.
In addition, estimates for periodic solutions and their derivatives are established.
2001, 7(4): 821-836 doi: 10.3934/dcds.2001.7.821 +[Abstract](1145) +[PDF](235.5KB)
Abstract:
In this paper, we review several notions from thermodynamic formalism, like topological pressure and entropy and show how they can be employed, in order to obtain information about stable and unstable sets of holomorphic endomorphisms of $\mathbb P^2$ with Axiom A.
In particular, we will consider the non-wandering set of such a mapping and its "saddle" part $S_1$, i.e the subset of points with both stable and unstable directions. Under a derivative condition, the stable manifolds of points in S1 will have a very "thin" intersection with $S_1$, from the point of view of Hausdorff dimension. While for diffeomorphisms there is in fact an equality between $HD(W^s_\varepsilon(x)\cap S_1)$ and the unique zero of $P(t \cdot \phi^s)$ (Verjovsky-Wu [VW]) in the case of endomorphisms this will not be true anymore; counterexamples in this direction will be provided. We also prove that the unstable manifolds of an endomorphism depend Hölder continuously on the corresponding prehistory of their base point and employ this in the end to give an estimate of the Hausdorff dimension of the global unstable set of $S_1$. This set could be à priori very large, since, unlike in the case of H´enon maps, there is an uncountable collection of local unstable manifolds passing through each point of $S_1$.
2001, 7(4): 837-853 doi: 10.3934/dcds.2001.7.837 +[Abstract](1150) +[PDF](185.6KB)
Abstract:
We construct a class of $3\times 3$ systems of conservation laws with all characteristic fields genuinely nonlinear, and we show the existence of entropy solutions for these that blow up in sup-norm in finite time. The solutions are constructed by considering wave patterns where infinitely many shock waves are produced in finite time. We also consider the role of entropies as a mechanism for preventing this type of singular behavior.
2001, 7(4): 855-871 doi: 10.3934/dcds.2001.7.855 +[Abstract](1386) +[PDF](224.9KB)
Abstract:
It is shown that a Hamiltonian system in the neighbourhood of an equilibrium may be given a special normal form in case four of the eigenvalues of the linearized system are of the form $\lambda_1, -\lambda_1, \lambda_2, -\lambda_2,$ with $\lambda_1$ and $\lambda_2$ independent over the reals, i.e., $\lambda_1/\lambda_2 \notin \mathbf R$. That is, for a real Hamiltonian system and concerning the variables $x_1, y_1, x_2, y_2$ the equilibrium is of either type center–saddle or complex–saddle. The normal form exhibits the existence of a four–parameter family of solutions which has been previously investigated by Moser. This paper completes Moser's result in that the convergence of the transformation of the Hamiltonian to a normal form is proven.
2001, 7(4): 873-882 doi: 10.3934/dcds.2001.7.873 +[Abstract](905) +[PDF](151.4KB)
Abstract:
We consider $L^2$ bounds on the gradient of solutions of the Navier-Stokes equations on a general bounded 2D domain with Dirichlet boundary conditions. We obtain an upper bound for this norm on any compact subset of a given domain. We show that the bound is uniform on the global attractor and depends polynomially on the Grashof number.
2001, 7(4): 883-893 doi: 10.3934/dcds.2001.7.883 +[Abstract](1527) +[PDF](187.1KB)
Abstract:
The influence of the driving system on a skew-product flow generated by a triangular system of differential equations can be perturbed in two ways, directly by perturbing the vector field of the driving system component itself or indirectly by perturbing its input variable in the vector field of the coupled component. The effect of such perturbations on a nonautonomous attractor of the driven component is investigated here. In particular, it is shown that a perturbed nonautonomous attractor with nearby components exists in the indirect case if the driven system has an inflated nonautonomous attractor and that the direct case can be reduced to this case if the driving system is shadowing.

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