Nonlinear Autonomous Oscillations

Name: Nonlinear Autonomous Oscillations | Analytical Theory
Brand: Academic Press
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Analytical Theory

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Academic Press

Erschienen am 14. Mai 2014

343 Seiten

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978-0-08-095541-4 (ISBN)

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Front Cover
Nonlinear Autonomous Oscillations
Copyright Page
Contents
Preface
Chapter 1. Analysis of Vectors and Matrices
1.1 Norms of Vectors and Matrices
1.2 Canonical Forms of Matrices
1.3 Sequences and Series of Matrices
1.4 Logarithms of Matrices
1.5 Differentiation and Integration of Matrices
Chapter 2. Basic Theorems Concerning Ordinary Differential Equations
2.1 Fundamental Existence Theorems
2.2 Dependence of the Solution on Initial Conditions and Parameters
Chapter 3. Linear Differential Systems
3.1 Linear Homogeneous Systems
3.2 Linear Nonhomogeneous Systems
Chapter 4. Orbits of Autonomous Systems
4.1 Introduction
4.2 Critical Points of Autonomous Systems
4.3 Periodic Solutions and Closed Orbits
4.4 Continuity of Orbits
Chapter 5. Moving Orthonormal Systems along a Closed Orbit
5.1 A Method to Construct a Moving Orthonormal System along a Closed Orbit
5.2 Equations of Orbits with Respect to the Moving Orthogonal System
5.3 Multipliers of Solutions of the Normal Variation Equation
Chapter 6. Stability
6.1 Definition of Stability
6.2 Fundamental Theorems Concerning Stability
6.3 Orbital Stability
6.4 Orbital Stability of Critical Points
6.5 Orbital Stability of Periodic Solutions
Chapter 7. Perturbation of Autonomous Systems
7.1 Fundamental Formulas
7.2 Periodic Solutions of the Perturbed System
7.3 Stability of the Periodic Solution of the Perturbed System
Chapter 8. Perturbation of Fully Oscillatory Systems
8.1 Universal Periods
8.2 Preliminary Theorem
8.3 Perturbation of a Fully Oscillatory System
8.4 An Example
Chapter 9. Perturbation of Partially Oscillatory Systems
9.1 The Reduced Form of the Partially Oscillatory System
9.2 Perturbation of a Partially Oscillatory System
9.3 Stability of the Periodic Solution of the Perturbed System
9.4 An Example
Chapter 10. Analysis of Two-Dimensional Autonomous Systems
10.1 Fundamental Formulas
10.2 Stability of a Periodic Solution of the Unperturbed System
10.3 Perturbation
10.4 Perturbation of a Fully Oscillatory System
10.5 Fundamental Formulas for Analytic Systems
10.6 Stability of a Periodic Solution of the Analytic Unperturbed System
10.7 Perturbation of Analytic Systems
10.8 Multiplicities of Closed Orbits
10.9 Perturbation of Analytic Fully Oscillatory Systems
Chapter 11. Numerical Computation of Periodic Solutions
11.1 A Method to Compute a Periodic Solution
11.2 The Two-Dimensional Case
11.3 Periodic Solutions of the Autonomous van der Pol Equation
11.4 Remarks
Chapter 12. Center of the Autonomous System
12.1 First Reduction
12.2 The Universal Period of the Orbit
12.3 The Canonical Form of the Autonomous System in the Neighborhood of the Center
Chapter 13. Inverse Problems Connected with Periods of Oscillations Described by x + g(x) = 0
13.1 Preliminaries
13.2 Lemmas
13.3 The Period Function Associated with the Maximum Velocity
13.4 The Period Functions Associated with the Amplitude and the Half- Amplitudes
Appendix. The Newton Method and Step-by-Step Methods for Ordinary Differential Equations
A.l The Iterative Method to Solve Equations Numerically
A.2 The Newton Method
A.3 A Numerical Illustration of the Newton Method
A.4 Step-by-step Methods for Ordinary Differential Equations
A 5 Convergence of the Runge-Kutta Method
A.6 Convergence of the Multi-Step Methods
Bibliography
Subject Index

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