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Discrete & Continuous Dynamical Systems - A

1996 , Volume 2 , Issue 2

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Closed geodesic distribution for manifolds of non-positive curvature
Mark Pollicott
1996, 2(2): 153-161 doi: 10.3934/dcds.1996.2.153 +[Abstract](143) +[PDF](168.8KB)
In this paper we give a generalization of Bowen's equidistribution result for closed geodesics on negatively curved manifolds to rank one manifolds.
Lyapunov functions and attractors under variable time-step discretization
Peter E. Kloeden and Björn Schmalfuss
1996, 2(2): 163-172 doi: 10.3934/dcds.1996.2.163 +[Abstract](171) +[PDF](185.8KB)
A one-step numerical scheme with variable time--steps is applied to an autonomous differential equation with a uniformly asymptotically stable set, which is compact but otherwise of arbitrary geometric shape. A Lyapunov function characterizing this set is used to show that the resulting nonautonomous difference equation generated by the numerical scheme has an absorbing set. The existence of a cocycle attractor consisting of a family of equivariant sets for the associated discrete time cocycle is then established and shown to be close in the Hausdorff separation to the original stable set for sufficiently small maximal time-steps.
On the critical decay and power for semilinear wave equtions in odd space dimensions
Hideo Kubo
1996, 2(2): 173-190 doi: 10.3934/dcds.1996.2.173 +[Abstract](105) +[PDF](2186.0KB)
In this paper we study global behaviors of solutions of initial value problem to wave equations with power nonlinearity. We shall derive space-time decay estimates according to decay rates of the initial data with low regularity (in classical sense). Indeed we can control $L^\infty$-norm of a solution in high dimension, provided the initial data are radially symmetric. This enables us to construct a global solution under suitable assumptions and to obtain an optimal estimate for a lifespan of a local solution.
A free boundary problem arising from a stress-driven diffusion in polymers
Hong-Ming Yin
1996, 2(2): 191-202 doi: 10.3934/dcds.1996.2.191 +[Abstract](100) +[PDF](1630.3KB)
In this paper we study a free boundary problem arising from a stress-driven diffusion in polymers. The main feature of the problem is that the mass flux of the penetrant is proportional to the gradient of the concentration and the gradient of the stress. A Maxwell-like viscoelastic relationship is assumed between the stress and the concentration. The phase change takes place on the interface between the glassy and rubbery states of the polymer and a Stefan-type of free boundary condition is imposed on the free boundary. It is shown that under certain conditions the problem has a unique weak solution.
Indestructibility of invariant locally non-unique manifolds
George Osipenko
1996, 2(2): 203-219 doi: 10.3934/dcds.1996.2.203 +[Abstract](76) +[PDF](2975.5KB)
The Sacker-Neimark-Mane result on persistent manifolds of autonomous systems is well-known: an invariant manifold is persistent iff it is normally hyperbolic. The persistent manifolds have the property of the local uniqueness. The paper gives conditions for the indestructibility of an invariant manifold without the supposition of its local uniqueness. These conditions are wider than the normally hyperbolicity conditions. Some examples are considered.
On the existence and profile of multi-peaked solutions to singularly perturbed semilinear Dirichlet problems
Daomin Cao, Norman E. Dancer, Ezzat S. Noussair and Shunsen Yan
1996, 2(2): 221-236 doi: 10.3934/dcds.1996.2.221 +[Abstract](165) +[PDF](1900.6KB)
Multi-peaked solutions to a singularly perturbed elliptic equation on a bounded domain $\Omega$ are constructed, provided the distance function $d(x, \delta\Omega)$ has more than one strict local maximum.
Existence and non-existence of homoclinic trajectories of the Liénard system
Jitsuro Sugie and Tadayuki Hara
1996, 2(2): 237-254 doi: 10.3934/dcds.1996.2.237 +[Abstract](103) +[PDF](2487.8KB)
In this paper we discuss the problem when the Liénard system $\dot{x}=y-F(x)$ and $\dot{y}=-g(x)$ has homoclinic trajectories or not. Some new criteria for the existence of periodic solutions of this system are also presented.
Pattern formation in reaction-diffusion systems with $D_2$-symmetric kinetics
Joseph G. Yan and Dong-Ming Hwang
1996, 2(2): 255-270 doi: 10.3934/dcds.1996.2.255 +[Abstract](111) +[PDF](2490.0KB)
We study the bifurcations of stationary solutions in a class of coupled reaction-diffusion systems on 1-dimensional space where the steady-state system is $D_2$-symmetric and reversible with respect to two involutions.
Because the stationary patterns of such reaction-diffusion systems are the symmetric cycles of its steady-state system, we investigate the bifurcations of manifolds of symmetric cycles near equilibria in general $D_2$-symmetric reversible systems. This is done through an analysis of the bifurcation regimes at strong resonances using 1-dimensional universal unfoldings of $D_2$-symmetric reversible normal forms. We prove there are two disjoint manifolds at "odd" resonance and four disjoint manifolds at "even" resonance. The number of these disjoint manifolds, in turn, determines the number of different types of stationary patterns.
Applications of our analysis to the study of pattern formation in reaction-diffusion systems are illustrated with a predator-prey model arising from mathematical ecology. Numerical results are obtained as a verification of our analysis.
Multiplicity and stability result for semilinear parabolic equations
Norimichi Hirano and Wen Se Kim
1996, 2(2): 271-280 doi: 10.3934/dcds.1996.2.271 +[Abstract](105) +[PDF](1420.7KB)
In this paper, we show the existence of stable and unstable periodic solutions for a semilinear parabolic equation

$\qquad\qquad \frac{\partial u}{\partial t}-\Delta_x u -\lambda_1 u +g(u) =s \phi + h$ in $ R\times \Omega$

$\qquad\qquad u(t,x) =0 $ on $R\times \partial \Omega$

$\qquad\qquad u(0,x)=u(2\pi, x)$ on $\Omega$

where $g$ is a continuous function on $R$, $\phi$ denotes the positve normalized eigenfunction corresponding to the first eigenvalue $\lambda_1$ of problem (L), $s \in R$, and $h \in C([0,2\pi],C^1_0(\overline{\Omega})).$

Control of plate vibrations by means of piezoelectric actuators
Marius Tucsnak
1996, 2(2): 281-293 doi: 10.3934/dcds.1996.2.281 +[Abstract](89) +[PDF](204.5KB)
We consider initial and boundary value problems modelling the vibrations of a plate with piezoelectric actuator. The simplest model leads to the Bernoulli-Euler plate equation with right hand side given by a distribution concentrated in an interior curve multiplied by a real valued time function representing the voltage applied to the actuator. We prove that, generically with respect to the curve, the plate vibrations can be strongly stabilized and approximatively controlled by means of the voltage applied to the actuator.

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