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Topological Degree Approach to Bifurcation Problems

Publisher: 
Springer
Number of Pages: 
261
Price: 
129.00
ISBN: 
9781402087233
Date Received: 
Monday, November 24, 2008
Reviewable: 
No
Include In BLL Rating: 
No
Reviewer Email Address: 
Michal Fečkan
Series: 
Topological Fixed Point Theory and Its Applications 5
Publication Date: 
2008
Format: 
Hardcover
Category: 
Monograph

1. Introduction

1.1. Preface

1.2. An Illustrative Perturbed Problem

1.3. A Brief Summary of the Book

2. Theoretical Background

2.1. Linear Functional Analysis

2.2. Nonlinear Functional Analysis

2.2.1. Implicit Function Theorem

2.2.2. Lyapunov-Schmidt Method

2.2.3. Leray-Schauder Degree

2.3. Differential Topology

2.3.1. Differentiable Manifolds

2.3.2. Symplectic Surfaces

2.3.3. Intersection Numbers of Manifolds

2.3.4. Brouwer Degree on Manifolds

2.3.5. Vector Bundles

2.3.6. Euler Characteristic

2.4. Multivalued Mappings

2.4.1. Upper Semicontinuity

2.4.2. Measurable Selections

2.4.3. Degree Theory for Set-Valued Maps

2.5. Dynamical Systems

2.5.1. Exponential Dichotomies

2.5.2. Chaos in Discrete Dynamical Systems

2.5.3. Periodic O.D.Eqns

2.5.4. Vector Fields

2.6. Center Manifolds For Infinite Dimensions

3. Bifurcation of Periodic Solutions

3.1. Bifurcation of Periodics from Homoclinics I

3.1.1. Discontinuous O.D.Eqns

3.1.2. The Linearized Equation

3.1.3. Subharmonics for Regular Periodic Perturbations

3.1.4. Subharmonics for Singular Periodic Perturbations

3.1.5. Subharmonics for Regular Autonomous Perturbations

3.1.6. Applications to Discontinuous O.D.Eqns

3.1.7. Bounded Solutions Close to Homoclinics

3.2. Bifurcation of Periodics from Homoclinics II

3.2.1. Singular Discontinuous O.D.Eqns

3.2.2. Linearized Equations

3.2.3. Bifurcation of Subharmonics

3.2.4. Applications to Singular Discontinuous O.D.Eqns

3.3. Bifurcation of Periodics from Periodics

3.3.1. Discontinuous O.D.Eqns

3.3.2. Linearized Problem

3.3.3. Bifurcation of Periodics in Nonautonomous Systems

3.3.4. Bifurcation of Periodics in Autonomous Systems

3.3.5. Applications to Discontinuous O.D.Eqns

3.3.6. Concluding Remarks

3.4. Bifurcation of Periodics in Relay Systems

3.4.1. Systems with Relay Hysteresis

3.4.2. Bifurcation of Periodics

3.4.3. Third-Order O.D.Eqns with Small Relay Hysteresis

3.5. Nonlinear Oscillators with Weak Couplings

3.5.1. Weakly Coupled Systems

3.5.2. Forced Oscillations from Single Periodics

3.5.3. Forced Oscillations from Families of Periodics

3.5.4. Applications to Weakly Coupled Nonlinear Oscillators

4. Bifurcation of Chaotic Solutions

4.1. Chaotic Differential Inclusions

4.1.1. Nonautonomous Discontinuous O.D.Eqns

4.1.2. The Linearized equation

4.1.3. Bifurcation of Chaotic Solutions

4.1.4. Chaos from Homoclinic Manifolds

4.1.5. Almost and Quasi Periodic Discontinuous O.D.Eqns

4.2. Chaos in Periodic Differential Inclusions

4.2.1. Regular Periodic Perturbations

4.2.2. Singular Differential Inclusions

4.3. More about Homoclinic Bifurcations

4.3.1. Transversal Homoclinic Crossing Discontinuity

4.3.2. Homoclinic Sliding on Discontinuity

5. Topological Transversality

5.1. Topological Transversality and Chaos

5.1.1. Topologically Transversal Invariant Sets

5.1.2. Difference Boundary Value Problems

5.1.3. Chaotic Orbits

5.1.4. Periodic Points and Extensions on Invariant Compact Subsets

5.1.5. Perturbed Topological Transversality

5.2. Topological Transversality and Reversibility

5.2.1. Period Blow-up

5.2.2. Period Blow-up for Reversible Diffeomorphisms

5.2.3. Perturbed Period Blow-up

5.2.4. Perturbed Second Order O.D.Eqns

5.3. Chains of Reversible Oscillators

5.3.1. Homoclinic Period Blow-up for Breathers

5.3.2. Heteroclinic Period Blow-up for Non-Breathers

5.3.3. Period Blow-up for Traveling Waves

6. Traveling waves on lattices

6.1. Traveling Waves in Discretized P.D.Eqns

6.2. Center Manifold Reduction

6.3. A Class of Singularly Perturbed O.D.Eqns

6.4. Bifurcation of Periodic Solutions

6.5. Traveling Waves in Homoclinic Cases

6.6. Traveling Waves in Heteroclinic Cases

6.7. Traveling Waves in 2 Dimensions

7. Periodic Oscillations of Wave Equations

7.1. Periodics of Undamped Beam Equations

7.1.1. Undamped Forced Nonlinear Beam Equations

7.1.2. Existence Results on Periodics

7.1.3. Subharmonics from Homoclinics

7.1.4. Periodics from Periodics

7.1.5. Applications to Forced Nonlinear Beam Equations

7.2. Weakly Nonlinear Wave Equations

7.2.1. Excluding Small Divisors

7.2.2. Lebesgue Measures of Nonresonances

7.2.3. Forced Periodic Solutions

7.2.4. Theory of Numbers and Nonresonances

8. Topological Degree for Wave Equations

8.1. Discontinuous Undamped Wave Equations

8.2. Standard Classes of Multi-Mappings

8.3. M-Regular Multi-Functions

8.4. Classes of Admissible Mappings

8.5. Semilinear Wave Equations

8.6. Construction of Topological Degree

8.7. Local Bifurcations

8.8. Bifurcations from Infinity

8.9. Bifurcations for Semilinear Wave Equations

8.10. Chaos for Discontinuous Beam Equations

Bibliography

Index

Publish Book: 
Modify Date: 
Monday, November 24, 2008

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