By Jack K. Hale, Luis T. Magalhães, Waldyr M. Oliva (auth.)
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Additional resources for An Introduction to Infinite Dimensional Dynamical Systems — Geometric Theory
7. If FEEt closed and, moreover, if ~r is an RFDE on a compact manifold M,~ fI(F)!!. is one-to-one on A(F), then fI(F) is invariant. Most of the results in this section are valid in a more abstract setting. We state the results without proof, for maps, and the extension to flows is easy to accomplish. Throughout the discussion is continuous. 8. is the £-neighborhood of J. If T: X + X is continuous and there is a compact set which attracts compact sets of X and J = nn K TnK, then (i) J is independent of K; (ii) J is maximal, cOmpact, invariant; (iii) J is stable and attracts compact sets of X.
1), the periodic solution and its period depend continuously on the perturbation. t* > 1, j J there are r. t* and changing continuously with the J = 1,2,3. perturbation, for j have periods near NI N2t* J The orbits of period near Njt*, j also = N3t*, and, therefore, r l = r 2 = r3 d~f r. Since Nl ,N 2 are relatively prime we have that the period of r depends continuously on the perturbation. 1) of periods To prove that j12(K,A) = 1,2 < A and lying in 913/ 2 (K,A), it is sufficient to make K. -34- a small perturbation in a neighborhood of each periodic solution.
3. If F ESrI is a point dissipative RFDE on a connected mani- fold M and the corresponding solution map, compact subsets of [O,m), then A(F) ~t' is uniformly bounded on is the maximal compact invariant set of F, it is connected, uniformly asymptotically stable, attracts all bounded sets of cO and = n~o ~nl A(F) where Proof: K is any compact set which attracts all compact sets of CO. 2, there is a compact set K which attracts all compact ° sets of C , and the set , / = nn>0 ~nr K is the maximal compact invariant set of F.