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Discrete & Continuous Dynamical Systems - A

1999 , Volume 5 , Issue 2

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The optimal gap condition for invariant manifolds
Y. Latushkin and B. Layton
1999, 5(2): 233-268 doi: 10.3934/dcds.1999.5.233 +[Abstract](679) +[PDF](349.7KB)
Abstract:
We give optimal gap conditions using Lipschitz constants of the nonlinear terms and growth bounds of the linear terms that imply the existence of infinite dimensional Lipschitz invariant manifolds for systems of semilinear equations on Banach spaces. This result improves and generalizes recent theorems by C. Foias and by N. Castañeda and R. Rosa. The result is also shown to imply the existence of invariant manifolds for nonautonomous equations and semilinear skew-product flows. Also, generalizations for smoothness of invariant manifolds are given.
Evolution equations generated by subdifferentials in the dual space of $(H^1(\Omega))$
A. Damlamian and Nobuyuki Kenmochi
1999, 5(2): 269-278 doi: 10.3934/dcds.1999.5.269 +[Abstract](593) +[PDF](212.0KB)
Abstract:
This paper is concerned with the subdifferential operator approach to nonlinear (possibly degenerate and singular) parabolic PDE's of the form $u_t-\Delta \beta(u) \ni f$ formulated in the dual space of $H^1(\Omega)$, where $\beta$ is a maximal monotone graph in $\mathbf R\times \mathbf R$. In the set-up considered so far [8], some coerciveness condition has been required for $\beta$, corresponding at least to the fact that it is onto $\mathbf R$. In the present paper, we show that the subdifferential operator approach is possible for any maximal monotone graph $\beta$ without any growth condition.
Optimal control of systems governed by some elliptic equations
Urszula Ledzewicz and Stanislaw Walczak
1999, 5(2): 279-290 doi: 10.3934/dcds.1999.5.279 +[Abstract](530) +[PDF](205.7KB)
Abstract:
Based on variational methods sufficient conditions for the continuous dependence of the solution of a system governed by some elliptic equation on controls is discussed. Then these conditions are used to obtain an existence theorem for the optimal control problem of a system governed by nonlinear elliptic equations with controls.
Topologically transitive homeomorphisms of quotients of tori
Grant Cairns, Barry Jessup and Marcel Nicolau
1999, 5(2): 291-300 doi: 10.3934/dcds.1999.5.291 +[Abstract](461) +[PDF](194.6KB)
Abstract:
This paper considers the following question: for what finite subgroups $G\subset GL(n, \mathbb Z)$, does there exist an element $A\in GL(n, \mathbb Z)$ inducing a topologically transitive homeomorphism of $T^n$/$G$ We show that for $n = 2$ and 3, the only possibility is $G =\{\pm I\}$. Curiously, in higher dimension the structure is less restrictive. We give a variety of examples in dimension 4. Nevertheless, we show that in dimension $\geq 4$, there are relatively few irreducible examples.
Multiple solutions of Neumann elliptic problems with critical nonlinearity
Monica Musso and Donato Passaseo
1999, 5(2): 301-320 doi: 10.3934/dcds.1999.5.301 +[Abstract](582) +[PDF](267.3KB)
Abstract:
The paper is concerned with a class of Neumann elliptic problems, in bounded domains, involving the critical Sobolev exponent. Some conditions on the lower order term are given, sufficient to guarantee existence and multiplicity of positive solutions without any geometrical assumption on the boundary of the domain.
A neural network based anti-skid brake system
Sanjay K. Mazumdar and Cheng-Chew Lim
1999, 5(2): 321-338 doi: 10.3934/dcds.1999.5.321 +[Abstract](587) +[PDF](514.7KB)
Abstract:
The novel application of a neural network based adaptive control scheme to an anti-skid brake system (ABS) is presented in this paper. The anti-skid brake system represents a unique and challenging application for neural network based control schemes. The principal benefit of using neural networks in anti-skid brake systems is their ability to adapt to changes in the environmental conditions without a significant degradation in performance. In the proposed approach, the controller neural network is designed to produce a braking torque which regulates the wheel slip for the vehicle-brake system to a prespecified level. An enhanced reference model is proposed which generates the desired slip response and enables a sufficient condition for the convergence of the tracking error to be derived. Simulation studies are performed to demonstrate the effectiveness of the proposed neural network based anti-skid brake system (NN-ABS) for various road surface conditions, inclines in the road, and transition between road surfaces.
Hyperbolic sets for noninvertible maps and relations
Evelyn Sander
1999, 5(2): 339-357 doi: 10.3934/dcds.1999.5.339 +[Abstract](500) +[PDF](330.6KB)
Abstract:
This paper develops the theory of hyperbolic sets for relations, a generalization of both noninvertible and multivalued maps. We give proofs of shadowing and the stable manifold theorem in this context.
Bifurcation from degenerate homoclinics in periodically forced systems
Michal Fečkan
1999, 5(2): 359-374 doi: 10.3934/dcds.1999.5.359 +[Abstract](475) +[PDF](791.2KB)
Abstract:
Bifurcation of homoclinic solutions are investigated for ordinary differential equations with periodic perturbations possessing a degenerate homoclinic solution.
Uniqueness and long-time behavior for the conserved phase-field system with memory
Pierluigi Colli, Gianni Gilardi, Philippe Laurençot and Amy Novick-Cohen
1999, 5(2): 375-390 doi: 10.3934/dcds.1999.5.375 +[Abstract](501) +[PDF](209.9KB)
Abstract:
This paper is concerned with a conserved phase-field model with memory. We include memory by replacing the standard Fourier heat law with a constitutive assumption of Gurtin-Pipkin type, and the system is conservative in the sense that the initial mass of the order parameter as well as the energy are preserved during the evolution. A Cauchy-Neumann problem is investigated for this model which couples a Volterra integro-differential equation with fourth order dynamics for the phase field. A sharp uniqueness theorem is proven by demonstrating continuous dependence for a suitably weak formulation. With regard to the long-time behavior, the limit points of the trajectories are completely characterized.
Vortices for the nonlinear wave equation
Tai-Chia Lin
1999, 5(2): 391-398 doi: 10.3934/dcds.1999.5.391 +[Abstract](528) +[PDF](186.5KB)
Abstract:
We derive the governing equations of vortices for the nonlinear wave equation. The initial data is a small perturbation of the symmetric vortex solution in the steady state Ginzburg-Landau equation. Then by the well-posedness of the nonlinear wave equation (cf. [12]) and the spectrum of the linearized operator for the Ginzburg-Landau equation (cf. [9], [8]), we obtain the local dynamic laws of vortices in a short time.
Finite dimensional attractors for reaction-diffusion equations in $R^n$ with a strong nonlinearity
Messoud Efendiev and Alain Miranville
1999, 5(2): 399-424 doi: 10.3934/dcds.1999.5.399 +[Abstract](603) +[PDF](231.8KB)
Abstract:
Our aim in this article is to study the long time behavior of a class of reaction-diffusion equations in the whole space for which the nonlinearity depends explicitly on the gradient of the unknown function. We prove the existence of the global attractor and of exponential attractors for the semigroup associated with the equation. We also consider the nonautonomous case, and when the forcing term depends quasiperiodically on the time, we prove the existence of uniform and uniform exponential attractors.
Statistical properties of piecewise smooth hyperbolic systems in high dimensions
N. Chernov
1999, 5(2): 425-448 doi: 10.3934/dcds.1999.5.425 +[Abstract](543) +[PDF](307.5KB)
Abstract:
We study smooth hyperbolic systems with singularities and their SRB measures. Here we assume that the singularities are submanifolds, the hyperbolicity is uniform aside from the singularities, and one-sided derivatives exist on the singularities. We prove that the ergodic SRB measures exist, are finitely many, and mixing SRB measures enjoy exponential decay of correlations and a central limit theorem. These properties have been proved previously only for two-dimensional systems.
The relaxation to the drift-diffusion system for the 3-$D$ isentropic Euler-Poisson model for semiconductors
Corrado Lattanzio and Pierangelo Marcati
1999, 5(2): 449-455 doi: 10.3934/dcds.1999.5.449 +[Abstract](579) +[PDF](182.6KB)
Abstract:
In this paper we are concerned with the study of the relaxation limit of the 3-$D$ hydrodynamic model for semiconductors. We prove the convergence of the weak solutions to the Euler-Poisson system toward the solutions to the drift-diffusion system, as the relaxation time tends to zero.
Errata to "Stably ergodic skew products"
Roy Adler, Bruce Kitchens and Michael Shub
1999, 5(2): 456-456 doi: 10.3934/dcds.1999.5.456 +[Abstract](549) +[PDF](92.5KB)
Abstract:
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