By Courant, Hilbert
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Of course, not all dynamical systems will possess attractors, exponential attractors or inertial manifolds. In the sequel, we shall try to present a theory, by now quite well established, that provides a number of sufficient conditions on the system for at least some of these sets to exist. 17), this theory is really a natural extension of the classical theory of stability for ODEs. 4 Iterated Sequences Not surprisingly, many of the ideas (and difficulties) in the theory of continuous dynamical systems already surface in the context of discrete dynamical systems generated by ITERATED SEQUENCES.
27) with initial value u(0) = u0 , that is, u(t) = u0 . 1 + 2u20t 4. Inertial Manifolds. On the other hand, there are systems whose attractors do not present this type of difficulties, since they are imbedded into a finite dimensional Lipschitz manifold M of X , and the orbits converge to this manifold with a uniform exponential rate. Such a set M is called an INERTIAL MANIFOLD of the system (fig. 5). 5: Inertial Manifolds. manifold is governed by a finite system of ODEs, called the INERTIAL FORM of the semiflow.
7). 3 Attracting Sets 19 inertial manifold provides an indirect way to obtain the above mentioned desired system of ODEs. Moreover, the uniformity of the rate of convergence of the orbits to the manifold makes these systems extremely stable under perturbations and numerical approximations. Unfortunately, there are not many examples of systems which are known to admit an inertial manifold; among these, we mention the semiflows generated by a number of reaction-diffusion equations of “parabolic” type, and by the corresponding hyperbolic (small) perturbations of these equations.