Stability of nonautonomous differential equations / Luis Barreira, Claudia Valls.

Format:

Book

Author/Creator:

Barreira, Luís, 1968-

Contributor:

Valls, Claudia.

Series:

Lecture notes in mathematics (Springer-Verlag) ; 1926.

Lecture notes in mathematics, 0075-8434 ; 1926

Language:

English

Subjects (All):

Differential equations.

Stability.

Manifolds (Mathematics).

Physical Description:

xiv, 285 pages : illustrations ; 24 cm.

Place of Publication:

Berlin : New York : Springer, [2008]

Summary:

Main theme of this volume is the stability of nonautonomous differential equations in Banach spaces in the presence of nonuniform hyperbolicity. In particular, the linear variational equations are always assumed to possess a nonuniform exponential behavior, given either by the existence of a nonuniform exponential contraction or a nonuniform exponential dichotomy. Topics under discussion include the Lyapunov stability of solutions, the existence and smoothness of invariant manifolds, and the construction and regularity of topological conjugacies. The exposition is directed to researchers as well as graduate students interested in differential equations and dynamical systems, particularly in stability theory.

Contents:

1.1 Exponential contractions 2

1.2 Exponential dichotomies and stable manifolds 4

1.3 Topological conjugacies 7

1.4 Center manifolds, symmetry and reversibility 10

1.5 Lyapunov regularity and stability theory 13

Part I Exponential dichotomies

2 Exponential dichotomies and basic properties 19

2.1 Nonuniform exponential dichotomies 19

2.2 Stable and unstable subspaces 22

2.3 Existence of dichotomies and ergodic theory 24

3 Robustness of nonuniform exponential dichotomies 27

3.1 Robustness in semi-infinite intervals 27

3.1.1 Formulation of the results 27

3.1.2 Proofs 29

3.2 Stable and unstable subspaces 40

3.3 Robustness in the line 42

3.4 The case of strong dichotomies 49

Part II Stable manifolds and topological conjugacies

4 Lipschitz stable manifolds 55

4.1 Setup and standing assumptions 55

4.2 Existence of Lipschitz stable manifolds 56

4.3 Nonuniformly hyperbolic trajectories 59

4.4 Proof of the existence of stable manifolds 61

4.4.2 Solution on the stable direction 62

4.4.3 Behavior under perturbations of the data 64

4.4.4 Reduction to an equivalent problem 67

4.4.5 Construction of the stable manifolds 69

4.5 Existence of Lipschitz unstable manifolds 71

5 Smooth stable manifolds in R[superscript n] 75

5.1 C[superscript 1] stable manifolds 75

5.2 Nonuniformly hyperbolic trajectories 77

5.3 Example of a C[superscript 1] flow with stable manifolds 78

5.4 Proof of the C[superscript 1] regularity 79

5.4.1 A priori control of derivatives and auxiliary estimates 79

5.4.2 Lyapunov norms 83

5.4.3 Existence of an invariant family of cones 86

5.4.4 Construction and continuity of the stable spaces 92

5.4.5 Behavior of the tangent sets 95

5.4.6 C[superscript 1] regularity of the stable manifolds 100

5.5 C[superscript k] stable manifolds 102

5.6 Proof of the C[superscript k] regularity 106

5.6.1 Method of proof 106

5.6.2 Linear extension of the vector field 107

5.6.3 Characterization of the stable spaces 110

5.6.4 Tangential component of the extension 111

5.6.5 C[superscript k] regularity of the stable manifolds 116

6 Smooth stable manifolds in Banach spaces 119

6.1 Existence of smooth stable manifolds 120

6.2 Nonuniformly hyperbolic trajectories 121

6.3 Proof of the existence of smooth stable manifolds 123

6.3.1 Functional spaces 123

6.3.2 Derivatives of compositions 125

6.3.3 A priori control of the derivatives 127

6.3.4 Holder regularity of the top derivatives 129

6.3.5 Solution on the stable direction 133

6.3.6 Behavior under perturbations of the data 136

6.3.7 Construction of the stable manifolds 138

7 A nonautonomous Grobman-Hartman theorem 145

7.1 Conjugacies for flows 145

7.2 Conjugacies for maps 146

7.2.1 Setup 147

7.2.2 Existence of topological conjugacies 149

7.3 Holder regularity of the conjugacies 155

7.3.1 Main statement 155

7.3.2 Lyapunov norms 156

7.3.3 Proof of the Holder regularity 157

7.4 Proofs of the results for flows 162

7.4.1 Reduction to discrete time 162

7.4.2 Proofs 164

Part III Center manifolds, symmetry and reversibility

8 Center manifolds in Banach spaces 171

8.1 Standing assumptions 171

8.2 Existence of center manifolds 173

8.3 Proof of the existence of center manifolds 176

8.3.1 Functional spaces 176

8.3.2 Lipschitz property of the derivatives 179

8.3.3 Solution on the central direction 184

8.3.4 Reduction to an equivalent problem 187

8.3.5 Construction of the center manifolds 191

9 Reversibility and equivariance in center manifolds 197

9.1 Reversibility for nonautonomous equations 197

9.1.1 The notion of reversibility 197

9.1.2 Relation with the autonomous case 199

9.1.3 Nonautonomous reversible equations 201

9.2 Reversibility in center manifolds 202

9.2.1 Formulation of the main result 202

9.2.2 Auxiliary results 206

9.2.3 Proof of the reversibility 212

9.3 Equivariance for nonautonomous equations 213

9.4 Equivariance in center manifolds 214

Part IV Lyapunov regularity and stability theory

10 Lyapunov regularity and exponential dichotomies 219

10.1 Lyapunov exponents and regularity 219

10.2 Existence of nonuniform exponential dichotomies 222

10.3 Bounds for the regularity coefficient 226

10.3.1 Lower bound 226

10.3.2 Upper bound in the triangular case 227

10.3.3 Reduction to the triangular case 232

10.4 Characterizations of regularity 235

10.5 Equations with negative Lyapunov exponents 241

10.5.1 Lipschitz stable manifolds 242

10.5.2 Smooth stable manifolds 243

10.6 Measure-preserving flows 244

11 Lyapunov regularity in Hilbert spaces 249

11.1 The notion of regularity 249

11.2 Upper triangular reduction 252

11.3 Regularity coefficient and Perron coefficient 254

11.4 Characterizations of regularity 256

11.5 Lower and upper bounds for the coefficients 259

12 Stability of nonautonomous equations in Hilbert spaces 265

12.1 Setup 265

12.2 Stability results 267

12.3 Smallness of the perturbation 268

12.4 Norm estimates for the evolution operators 270

12.5 Proofs of the stability results 273.

Notes:

Includes bibliographical references (pages [277]-281) and index.

ISBN:

9783540747741

3540747745

OCLC:

174167876