Dynamics in Infinite Dimensions

by Hale, Jack K.; Magalhaes, Luis T.; Oliva, Waldyr M.

• ISBN13: 9780387954639
• ISBN10: 0387954635
• Edition: 2nd
• Format: Hardcover
• Copyright: 2002-07-01
• Publisher: Copernicus Books

Summary

This book presents an introduction to the geometric theory of infinite dimensional dynamical systems. Many of the fundamental results are presented for asymptotically smooth dynamical systems that have applications to functional differential equations as well as classes of dissipative partial differential equations. However, as in the earlier edition, the major emphasis is on retarded functional differential equations. This updated version also contains much material on neutral functional differential equations. The results in the earlier edition on Morse-Smale systems for maps are extended to a class of semiflows, which include retarded functional differential equations and parabolic partial differential equations.

Table of Contents

Preface	v
Introduction	1	(6)
Invariant Sets and Attractors	7	(12)
Functional Differential Equations on Manifolds	19	(38)
RFDE on manifolds	19	(10)
Examples of RFDE on manifolds	29	(15)
NFDE on manifolds	44	(1)
NFDE on Rn	45	(6)
General properties	46	(4)
Equivalence of point and compact dissipative	50	(1)
An example of NFDE on S1	51	(3)
A canonical ODE in the Frechet category	54	(3)
The Dimension of the Attractor	57	(8)
Stability and Bifurcation	65	(16)
Stability of Morse-Smale Maps and Semiflows	81	(28)
Morse-Smale maps	81	(17)
Morse-Smale semiflows	98	(6)
An example	104	(5)
One-to-Oneness, Persistence, and Hyperbolicity	109	(20)
The semiflow of an RFDE on a compact manifold M	110	(1)
Hyperbolic invariant sets	111	(1)
Hyperbolic sets as hyperbolic fixed points	111	(7)
Persistence of hyperbolicity and perturbations with one-to-oneness	118	(5)
Nonuniform hyperbolicity and invariant manifolds	123	(6)
Nonuniform hyperbolicity	123	(1)
Regular points	124	(1)
Hyperbolic measures and nonuniform hyperbolicity	125	(2)
The infinite dimensional case	127	(2)
Realization of Vector Fields and Normal Forms	129	(66)
Realization of vector fields on center manifolds	129	(11)
Normal forms for RFDE in finite dimensional spaces	140	(9)
Applications to Hopf bifurcation	149	(12)
Hopf Bifurcation for Scalar RFDE: The General Case	149	(7)
Hopf bifurcation for a delayed predator-prey system	156	(5)
Applications to Bogdanov-Takens bifurcation	161	(6)
Bogdanov-Takens bifurcation for scalar RFDE: The general case	161	(3)
Square and pulse waves	164	(3)
Singularity with a pure imaginary pair and a zero as simple eigenvalues	167	(3)
Normal forms for RFDE in infinite dimensional spaces	170	(16)
Periodic RFDE on Rn and autonomous RFDE on Banach spaces	186	(3)
A Viscoelastic model	189	(6)
Attractor Sets as C1-Manifolds	195	(14)
Monotonicity	209	(20)
Usual cones	209	(1)
Cones of rank k	210	(2)
Monotonicity in finite dimensions	212	(4)
Monotonicity in infinite dimensions	216	(5)
The Chafee-Infante problem	217	(2)
An infinite dimensional Morse-Smale map	219	(2)
Negative feedback: Morse decomposition	221	(8)
The Kupka-Smale Theorem	229	(12)
A Conley Index Theory in Noncompact Spaces	241	(26)
References	267	(10)
Index	277