By Jack K. Hale, Luis T. Magalhães, Waldyr M. Oliva (auth.)

ISBN-10: 0387909311

ISBN-13: 9780387909318

ISBN-10: 1475744935

ISBN-13: 9781475744934

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**Additional info for An Introduction to Infinite Dimensional Dynamical Systems — Geometric Theory**

**Example text**

1) lying in [O,A], we get small perturbations the proof that ~(K,A) (f+g) € K and having periods in ~(K,A) is dense in ~/2(K,A). of f. This finishes -37- We are now in the situation of being able to use the induction procedure introduced by Peixoto for ordinary differential equations. ~(K. A) (by 5)). A) B < A. ~ of f constant periodic solutions of x(t) bounded below by some £ > O. in in {f ~2 is dense in f € ~(K). £). implying that is dense in ~ for all is dense in ~ (by 2)). A) for all ~ ~ = g(x t ).

4. t* > O. 1) of least period Then. for sufficiently large t ~ N. • x*(t-r)) is a one-to-one regular (that is the derivative of the reals mod t* into mn(N+l). # 0 everywhere) mapping -38- Proof: If the statement is not true, there would exist arbitrarily large N such that either: x*(t 2-kr/N) ° for all 1) there are ° for all < k < N. ~ k ~ that m -+ and t. (N ) -+ T. J m J as If 1) holds and co. for all least period of x*. co, x*(tl-kr/N) x*(t 3-kr/N) ° Consequently, one could find a sequence of integers -rkm/Nm -+ 9 as m -+ t3 with m -+ ° 9 E [-r,O], there exists a sequence X*(T l +9) = X*(T 2+9) as t2 (mod t*) with N, or 2) there is and take convergent subsequences hand, for any t tl < - t On the other km -< Nm such T2 (mod t*), then 9 E [-r,O], contradicting that If 1) holds and x(T 1+9) Tl co.

Tj' IPj Therefore, the set and OJ' A(F) =,/ is stable. 4. then A(F) If F E ~l is an RFDE on a connected compact manifold M, is the maximal compact invariant set of F, it is connected, uniformly asymptotically stable, attracts all bounded sets of CO and A(F) Proof: Noting that K = CR. 3. Due to the above properties of the set A(F), it is natural to call -51- it the attractor set of F. Most of the following sections are dedicated to studying properties of this set. The set A(F) dependence on F. has certain continuity properties in relation to the If M is compact.

### An Introduction to Infinite Dimensional Dynamical Systems — Geometric Theory by Jack K. Hale, Luis T. Magalhães, Waldyr M. Oliva (auth.)

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