Averaging methods in nonlinear dynamical systems /

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Bibliographic Details
Author / Creator:	Sanders, J. A. (Jan A.)
Edition:	2nd ed.
Imprint:	New York : Springer, c2007.
Description:	1 online resource (xxi, 431 p.) : ill.
Language:	English
Series:	Applied mathematical sciences (Springer-Verlag New York Inc.) ; v. 59 Applied mathematical sciences (Springer-Verlag New York Inc.) ; v. 59.
Subject:	Differential equations, Nonlinear -- Numerical solutions. Differentiable dynamical systems. Averaging method (Differential equations) Averaging method (Differential equations) Differentiable dynamical systems. Differential equations, Nonlinear -- Numerical solutions.
Format:	E-Resource Book
URL for this record:	http://pi.lib.uchicago.edu/1001/cat/bib/8883107

Hidden Bibliographic Details
Other authors / contributors:	Verhulst, F. (Ferdinand), 1939- Murdock, James A.
ISBN:	9780387489162 0387489169 9780387489186 (e-ISBN) 0387489185 (e-ISBN) 9786611066208 6611066209
Notes:	Includes bibliographical references (p. [394]-411) and indexes.
Other form:	Print version: Sanders, J.A. (Jan A.). Averaging methods in nonlinear dynamical systems. 2nd ed. New York : Springer, c2007 9780387489162 0387489169

Table of Contents:

1. Basic Material and Asymptotics
1.1. Introduction
1.2. Existence and Uniqueness
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
2.9.1. Higher-Order Periodic Averaging
2.9.2. Estimates on Longer Time Intervals
2.9.3. Modified Van der Pol Equation
2.9.4. Periodic Orbit of the Van der Pol Equation
3. Methodology of Averaging
3.1. Introduction
3.2. Handling the Averaging Process
3.2.1. Lie Theory for Matrices
3.2.2. Lie Theory for Autonomous Vector Fields
3.2.3. Lie Theory for Periodic Vector Fields
3.2.4. Solving the Averaged Equations
3.3. Averaging Periodic Systems with Slow Time Dependence
3.3.1. Pendulum with Slowly Varying Length
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 Averaging
4.5.1. Example of Second-Order Averaging
4.6. Almost-Periodic Vector Fields
4.6.1. Example
5. Attraction
5.1. Introduction
5.2. Equations with Linear Attraction
5.3. Examples of Regular Perturbations with Attraction
5.3.1. Two Species
5.3.2. A perturbation theorem
5.3.3. Two Species, Continued
5.4. Examples of Averaging with Attraction
5.4.1. Anharmonic Oscillator with Linear Damping
5.4.2. Duffing's Equation with Damping and Forcing
5.5. Theory of Averaging with Attraction
5.6. An Attractor in the Original Equation
5.7. Contracting Maps
5.8. Attracting Limit-Cycles
5.9. Additional Examples
5.9.1. Perturbation of the Linear Terms
5.9.2. Damping on Various Time Scales
6. Periodic Averaging and Hyperbolicity
6.1. Introduction
6.2. Coupled Duffing Equations, An Example
6.3. Rest Points and Periodic Solutions
6.3.1. The Regular Case
6.3.2. The Averaging Case
6.4. Local Conjugacy and Shadowing
6.4.1. The Regular Case
6.4.2. The Averaging Case
6.5. Extended Error Estimate for Solutions Approaching an Attractor
6.6. Conjugacy and Shadowing in a Dumbbell-Shaped Neighborhood
6.6.1. The Regular Case
6.6.2. The Averaging Case
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
7.5.1. Einstein Pendulum
7.5.2. Nonlinear Oscillator
7.5.3. Oscillator Attached to a Flywheel
7.6. Secondary (Not Second Order) Averaging
7.7. Formal Theory
7.8. Slowly Varying Frequency
7.8.1. Einstein Pendulum
7.9. Higher Order Approximation in the Regular Case
7.10. Generalization of the Regular Case
7.10.1. Two-Body Problem with Variable Mass
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
8.5.1. Introduction
8.5.2. The Case of More Than One Angle
8.5.3. Example of Resonance Locking
8.5.4. Example of Forced Passage through Resonance
8.6. Inner and Outer Expansion
8.7. Two Examples
8.7.1. The Forced Mathematical Pendulum
8.7.2. An Oscillator Attached to a Fly-Wheel
9. From Averaging to Normal Forms
9.1. Classical, or First-Level, Normal Forms
9.1.1. Differential Operators Associated with a Vector Field
9.1.2. Lie Theory
9.1.3. Normal Form Styles
9.1.4. The Semisimple Case
9.1.5. The Nonsemisimple Case
9.1.6. The Transpose or Inner Product Normal Form Style
9.1.7. The sl[subscript 2] Normal Form
9.2. Higher Level Normal Forms
10. Hamiltonian Normal Form Theory
10.1. Introduction
10.1.1. The Hamiltonian Formalism
10.1.2. Local Expansions and Rescaling
10.1.3. Basic Ingredients of the Flow
10.2. Normalization of Hamiltonians around Equilibria
10.2.1. The Generating Function
10.2.2. Normal Form Polynomials
10.3. Canonical Variables at Resonance
10.4. Periodic Solutions and Integrals
10.5. Two Degrees of Freedom, General Theory
10.5.1. Introduction
10.5.2. The Linear Flow
10.5.3. Description of the w[subscript 1]: w[subscript 2]-Resonance in Normal Form
10.5.4. General Aspects of the k : l-Resonance, k [not equal] l
10.6. Two Degrees of Freedom, Examples
10.6.1. The 1 : 2-Resonance
10.6.2. The Symmetric 1 : 1-Resonance
10.6.3. The 1 : 3-Resonance
10.6.4. Higher-order Resonances
10.7. Three Degrees of Freedom, General Theory
10.7.1. Introduction
10.7.2. The Order of Resonance
10.7.3. Periodic Orbits and Integrals
10.7.4. The w[subscript 1]: w[subscript 2]: w[subscript 3]-Resonance
10.7.5. The Kernel of ad(H[superscript 0])
10.8. Three Degrees of Freedom, Examples
10.8.1. The 1 : 2 : 1-Resonance
10.8.2. Integrability of the 1 : 2 : 1 Normal Form
10.8.3. The 1 : 2 : 2-Resonance
10.8.4. Integrability of the 1 : 2 : 2 Normal Form
10.8.5. The 1 : 2 : 3-Resonance
10.8.6. Integrability of the 1 : 2 : 3 Normal Form
10.8.7. The 1 : 2 : 4-Resonance
10.8.8. Integrability of the 1 : 2 : 4 Normal Form
10.8.9. Summary of Integrability of Normalized Systems
10.8.10. Genuine Second-Order Resonances
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
11.4.1. Example: Nilpotent Linear Part in R[superscript 2]
11.5. Induced Linear Algebra
11.5.1. The Nilpotent Case
11.5.2. Nilpotent Example Revisited
11.5.3. The Nonsemisimple Case
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. Description of the First Level Normal Forms
12.6.1. The N[subscript 2] Case
12.6.2. The N[subscript 3] Case
12.6.3. The N[subscript 4] Case
12.6.4. Intermezzo: How Free?
12.6.5. The N[subscript 2,2] Case
12.6.6. The N[subscript 5] Case
12.6.7. The N[subscript 2,3] Case
12.7. Description of the First Level Normal Forms
12.7.1. The N[subscript 2,2,2] Case
12.7.2. The N[subscript 3,3] Case
12.7.3. The N[subscript 3,4] Case
12.7.4. Concluding Remark
13. Higher-Level Normal Form Theory
13.1. Introduction
13.1.1. Some Standard Results
13.2. Abstract Formulation of Normal Form Theory
13.3. The Hilbert-Poincare Series of a Spectral Sequence
13.4. The Anharmonic Oscillator
13.4.1. Case A[superscript r]: [Characters not reproducible] Is Invertible
13.4.2. Case A[superscript r]: [Characters not reproducible] Is Not Invertible, but [Characters not reproducible] Is
13.4.3. The m-adic Approach
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.2.1. Jacobi
A.2.2. Poincare
A.2.3. Van der Pol
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
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. 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
D.6. First Order Averaging
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.2.1. Averaging in a Banach Space
E.2.2. Averaging a Time-Dependent Operator
E.2.3. A Time-Periodic Advection-Diffusion Problem
E.2.4. Nonlinearities, Boundary Conditions and Sources
E.3. Hyperbolic Operators with a Discrete Spectrum
E.3.1. Averaging Results by Buitelaar
E.3.2. Galerkin Averaging Results
E.3.3. Example: the Cubic Klein-Gordon Equation
E.3.4. Example: Wave Equation with Many Resonances
E.3.5. Example: the Keller-Kogelman Problem
E.4. Discussion
References
Index of Definitions & Descriptions
General Index

Averaging methods in nonlinear dynamical systems /

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