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Discrete & Continuous Dynamical Systems - A

1996 , Volume 2 , Issue 3

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Continuity of admissible trajectories for state constraints control problems
M. Arisawa and  P.-L. Lions
1996, 2(3): 297-305 doi: 10.3934/dcds.1996.2.297 +[Abstract](40) +[PDF](177.9KB)
We consider here deterministic control problems with state constraints that is problems where admissible controls are those which keep the state of the system in a given region for all times. We prove that, given an initial state and an admissible control for that state, it is possible to construct admissible controls for all initial states such that the control and the corresponding trajectory are Lipschitz (in convenient norms) with respect to the initial condition.
Some remarks on a variational approach to Arnold's diffusion
Massimiliano Berti
1996, 2(3): 307-314 doi: 10.3934/dcds.1996.2.307 +[Abstract](36) +[PDF](1263.6KB)
In this paper we consider the following class of Lagrangian systems:

$\qquad\qquad \qquad\qquad L_{\epsilon,\mu}(q,\dot{q},Q,\dot{Q},t) = \frac{\dot{Q}^2}2+\frac{\dot{q}^2}2 + \epsilon(1-\cos q)+ \mu f(q,\dot{q},Q,\dot{Q},t,\mu) $

which has been studied by many authors in connection with Arnold's diffusion. Extending [2] prove, by variational means, that, for suitable perturbations including for example:

$\qquad\qquad \qquad\qquad f(q,\dot{q},Q,\dot{Q}, t,\mu)=(1-\cos q)(\cos Q+\cos t) + \mu^{p-1} \sin(q+Q) \quad (p>2) $

if $\mu$ is small enough, exists a diffusion orbit of $L_{\epsilon, \mu}$ such that $\dot{Q}(t)$ undergoes a variation of order $1$ in a time $t_d$ polinomial in $\mu$, $t_d\approx \frac 1{\mu^2}$.

Exponential attractors for the slightly compressible 2D-Navier-Stokes
Pierre Fabrie and  C. Galusinski
1996, 2(3): 315-348 doi: 10.3934/dcds.1996.2.315 +[Abstract](24) +[PDF](4267.4KB)
The equations of a slightly compressible fluid have been introduced to approach, when the parameter of compressibility $\epsilon$ is small, the incompressible Navier-Stokes equations. The object of this article is to prove the existence of exponential attractors in the 2D case for this partially dissipative system: The equations of a slightly compressible fluid. Furthermore, we establish an upper-bound of the fractal dimension of the exponential attractors described by the variable $(u^\epsilon, \sqrt{\epsilon}p^\epsilon)$; $u^\epsilon$ being the velocity and $p^\epsilon$ the pressure. Furthermore, a lower-semicontinuity result of these exponential attractors to the one of incompressible Navier-Stokes equations is obtained. These properties are linked to the existence of uniform absorbing sets with respect to $\epsilon$ for $\epsilon \leq \epsilon_0$ in the variable $(u^\epsilon, \sqrt{\epsilon}p^\epsilon)$ ($\epsilon_0$ fixed).
Stably ergodic skew products
Roy Adler , Bruce Kitchens and  Michael Shub
1996, 2(3): 349-350 doi: 10.3934/dcds.1996.2.349 +[Abstract](32) +[PDF](302.8KB)
In [PS] it is conjectured that among the volume preserving $C^2$ diffeomorphisms of a closed manifold which have some hyperbolicity, the ergodic ones contain an open and dense set. In this paper we prove an analogous statement for skew products of Anosov diffeomorphisms of tori and circle rotations. Thus this paper may be seen as an example of the phenomenon conjectured in [PS]. The corresponding theorem for skew products of Anosov diffeomorphisms and translations of arbitrary compact groups is an interesting open problem.
Discretizations of dynamical systems with a saddle-node homoclinic orbit
W.-J. Beyn and  Y.-K Zou
1996, 2(3): 351-365 doi: 10.3934/dcds.1996.2.351 +[Abstract](30) +[PDF](2056.5KB)
We consider a parametrized dynamical system with a homoclinic orbit that connects the center manifold of a saddle node to its strongly stable manifold. This is a codimension 2 homoclinic bifurcation with a well known unfolding. We show that the map obtained by discretizing such a system with a one-step method (the centered Euler scheme), inherits a discrete saddle-node homoclinic orbit. This orbit occurs on the line of saddle nodes and, as the numerical results suggest, there is actually a closed curve of such orbits and almost all of them consist of transversal homoclinic points. Our results complement those of [1], [5] on homoclinic discretization effects in the hyperbolic case.
Exact controllability of the wave equation for domains with slits and for mixed boundary conditions
Arnaud Heibig and  Mohand Moussaoui
1996, 2(3): 367-386 doi: 10.3934/dcds.1996.2.367 +[Abstract](37) +[PDF](254.4KB)
We discuss the problem of exact controllability for the wave equation in a plane domain with cracks and for mixed bonudary conditions. We use the Hilbert Uniqueness Method of J.-L. Lions. We recover some results due to P. Grisvard under some less restrictive geometrical hypotheses on the domain.
A result in global bifurcation theory using the Conley index
Todd Young
1996, 2(3): 387-396 doi: 10.3934/dcds.1996.2.387 +[Abstract](34) +[PDF](1397.0KB)
We study a saddle-node bifurcation in a Lipschitz family of diffeomorphisms on a manifold, in the case that the stable set and unstable set of the fixed point intersect transversally in a countable collection of one-dimensional manifolds diffeomorphic to circles. We formulate generic conditions on the circles stated in terms of standard coordinates, a recently defined tool for the study of saddle-node bifurcations. Under the conditions, it is shown that there is a decreasing sequence of intervals $[\underline{\mu_j},\overline{\mu_j}]$ of parameter values for which the diffeomorphism is semi-conjugated to shift dynamics on the space of binary sequences. The semi-conjugacy is implied by a recent result in the Conley index theory.
Hyperbolic measures and commuting maps in low dimension
Anatole Katok
1996, 2(3): 397-411 doi: 10.3934/dcds.1996.2.397 +[Abstract](26) +[PDF](3014.3KB)
We study invariant measures with non-vanishing Lyapunov characteristic exponents for commuting diffeomorphisms of compact manifolds. In particular we show that for $k=2,3$ no faithful $\mathbb{Z}^k$ real-analytic action on a $k$-dimensional manifold preserves a hyperbolic measure. In the smooth case similar statements hold for actions faithful on the support of the measure. Generalizations to higher dimension are proved under certain non-degeneracy conditions for the Lyapunov exponents.

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