Averaging Methods in Nonlinear Dynamical Systems

Preface

Preface to the Revised 2nd Edition

Preface to the First Edition

List of Figures

List of Tables

List of Algorithms

Contents

Map of the book

1 Basic Material and Asymptotics

1.1 Introduction

1.2 The Initial Value Problem: Existence, Uniqueness and Continuation

1.3 The Gronwall Lemma

1.4 Concepts of Asymptotic Approximation

1.5 Naive Formulation of Perturbation Problems

1.6 Reformulation in the Standard Form

1.7 The Standard Form in the Quasilinear Case

2 Averaging: the Periodic Case

2.1 Introduction

2.2 Van der Pol Equation

2.3 A Linear Oscillator with Frequency Modulation

2.4 One Degree of Freedom Hamiltonian System

2.5 The Necessity of Restricting the Interval of Time

2.6 Bounded Solutions and a Restricted Time Scale of Validity

2.7 Counter Example of Crude Averaging

2.8 Two Proofs of First-Order Periodic Averaging

2.9 Higher-Order Periodic Averaging and Trade-Off

3 Methodology of Averaging

3.1 Introduction

3.2 Handling the Averaging Process

3.3 Averaging Periodic Systems with Slow Time Dependence

3.4 Unique Averaging

3.5 Averaging and Multiple Time Scale Methods

4 Averaging: the General Case

4.1 Introduction

4.2 Basic Lemmas; the Periodic Case

4.3 General Averaging

4.4 Linear Oscillator with Increasing Damping

4.5 Second-Order Approximations in General Averaging; Improved First- Order Estimate Assuming Differentiability

4.6 Application of General Averaging to Almost- Periodic Vector Fields

5 Attraction

5.1 Introduction

5.2 Equations with Linear Attraction

5.3 Examples of Regular Perturbations with Attraction

5.4 Examples of Averaging with Attraction

5.5 Theory of Averaging with Attraction

y

( t)

x(0) y(t)

y

( t)

y

( t) x( t)

5.6 An Attractor in the Original Equation

5.7 Contracting Maps

5.8 Attracting Limit-Cycles

5.9 Additional Examples

6 Periodic Averaging and Hyperbolicity

6.1 Introduction

6.2 Coupled Duffing Equations, An Example

6.3 Rest Points and Periodic Solutions

6.4 Local Conjugacy and Shadowing

6.5 Extended Error Estimate for Solutions Approaching an Attractor

6.6 Conjugacy and Shadowing in a Dumbbell-Shaped Neighborhood

6.7 Extension to Larger Compact Sets

6.8 Extensions and Degenerate Cases

7 Averaging over Angles

7.1 Introduction

7.2 The Case of Constant Frequencies

7.3 Total Resonances

7.4 The Case of Variable Frequencies

7.5 Examples

7.6 Secondary (Not Second Order) Averaging

7.7 Formal Theory

7.8 Systems with Slowly Varying Frequency in the Regular Case; the Einstein Pendulum

7.9 Higher Order Approximation in the Regular Case

7.10 Generalization of the Regular Case; an Example from Celestial Mechanics

8 Passage Through Resonance

8.1 Introduction

8.2 The Inner Expansion

8.3 The Outer Expansion

8.4 The Composite Expansion

8.5 Remarks on Higher-Dimensional Problems

8.6 Analysis of the Inner and Outer Expansion; Passage through Resonance

8.7 Two Examples

9 From Averaging to Normal Forms

9.1 Classical, or First-Level, Normal Forms

9.2 Higher Level Normal Forms

10 Hamiltonian Normal Form Theory

10.1 Introduction

10.2 Normalization of Hamiltonians around Equilibria

10.3 Canonical Variables at Resonance

10.4 Periodic Solutions and Integrals

10.5 Two Degrees of Freedom, General Theory

10.6 Two Degrees of Freedom, Examples

10.7 Three Degrees of Freedom, General Theory

10.8 Three Degrees of Freedom, Examples

11 Classical (First–Level) Normal Form Theory

11.1 Introduction

11.2 Leibniz Algebras and Representations

11.3 Cohomology

11.4 A Matter of Style

11.5 Induced Linear Algebra

11.6 The Form of the Normal Form, the Description Problem

12 Nilpotent (Classical) Normal Form

12.1 Introduction

12.2 Classical Invariant Theory

12.3 Transvectants

12.4 A Remark on Generating Functions

12.5 The Jacobson–Morozov Lemma

12.6 A GL

-Invariant Description of the First Level

Normal Forms for n < 6

12.7 A GL

-Invariant Description of the Ring of

Seminvariants for n 6

13 Higher–Level Normal Form Theory

13.1 Introduction

13.2 Abstract Formulation of Normal Form Theory

13.3 The Hilbert–Poincar e Series of a Spectral Sequence

13.4 The Anharmonic Oscillator

13.5 The Hamiltonian 1 : 2-Resonance

13.6 Averaging over Angles

13.7 Definition of Normal Form

13.8 Linear Convergence, Using the Newton Method

13.9 Quadratic Convergence, Using the Dynkin Formula

A The History of the Theory of Averaging

A.1 Early Calculations and Ideas

A.2 Formal Perturbation Theory and Averaging

A.3 Proofs of Asymptotic Validity

B A 4-Dimensional Example of Hopf Bifurcation

B.1 Introduction

B.2 The Model Problem

B.3 The Linear Equation

B.4 Linear Perturbation Theory

B.5 The Nonlinear Problem and the Averaged Equations

C Invariant Manifolds by Averaging

C.1 Introduction

C.2 Deforming a Normally Hyperbolic Manifold

C.3 Tori by Bogoliubov-Mitropolsky-Hale Continuation

C.4 The Case of Parallel Flow

C.5 Tori Created by Neimark–Sacker Bifurcation

D Some Elementary Exercises in Celestial Mechanics

D.1 Introduction

D.2 The Unperturbed Kepler Problem

D.3 Perturbations

D.4 Motion Around an ‘Oblate Planet’

D.5 Harmonic Oscillator Formulation for Motion Around an ‘ Oblate Planet’

D.6 First Order Averaging for Motion Around an ‘ Oblate Planet’

D.7 A Dissipative Force: Atmospheric Drag

D.8 Systems with Mass Loss or Variable G

D.9 Two-body System with Increasing Mass

E On Averaging Methods for Partial Differential Equations

E.1 Introduction

E.2 Averaging of Operators

E.3 Hyperbolic Operators with a Discrete Spectrum

E.4 Discussion

References

Index of Definitions & Descriptions

General Index