Normal Hyperbolicity and Continuity of Global Attractors for a Nonlocal Evolution Equations

We show the normal hyperbolicity property for the equilibria of the evolution equation∂m(r,t)/∂t=-m(r,t)+g(βJ*m(r,t)+βh), h,β≥0,and using the normal hyperbolicity property we prove the continuity (upper semicontinuity and lower semicontinuity) of the global attractors of the flow generated by this equation, with respect to functional parameterJ.

BIFURCATIONS OF NORMALLY HYPERBOLIC INVARIANT MANIFOLDS IN ANALYTICALLY TRACTABLE MODELS AND CONSEQUENCES FOR REACTION DYNAMICS

In this paper, we study the breakdown of normal hyperbolicity and its consequences for reaction dynamics; in particular, the dividing surface, the flux through the dividing surface (DS), and the gap time distribution. Our approach is to study these questions using simple, two degree-of-freedom Hamiltonian models where calculations for the different geometrical and dynamical quantities can be carried out exactly. For our examples, we show that resonances within the normally hyperbolic invariant manifold may, or may not, lead to a "loss of normal hyperbolicity". Moreover, we show that the onset of such resonances results in a change in topology of the dividing surface, but does not affect our ability to define a DS. The flux through the DS varies continuously with energy, even as the energy is varied in such a way that normal hyperbolicity is lost. For our examples, the gap time distributions exhibit singularities at energies corresponding to the existence of homoclinic orbits in the DS, but these singularities are not associated with loss of normal hyperbolicity.

Stability and Persistence of Synchronization in a System with a Diffusive-Time-Lag Coupling

We study synchronization in the framework of invariant manifold theory for systems with a time lag. Normal hyperbolicity and its persistence in infinite dimensional dynamical systems in Banach spaces is applied to give general results on synchronization and its stability.

Synchronization and persistence in Diffusively Coupled Lattice Oscillators

We consider the synchronization and persistence of a system of identical lattice oscillators that are diffusively coupled to their nearest neighbours. Each subsystem has a compact global attractor. This is done in the framework of invariant manifold theory. Normal hyperbolicity and its persistence are applied to obtain general conditions for the stability and robustness of the synchronization manifold. AMS(MOS) Subject classifications: 37C80, 37D10.