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Preface |
Chapter 1. |
Systems of Differential Equations |
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1.1 |
A Simple Mass-Spring System |
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1.2 |
Coupled Mass-Spring Systems |
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1.3 |
Systems of First-Order Equations |
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1.4 |
Vector-Matrix Notation for Systems |
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1.5 |
The Need for a Theory |
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1.6 |
Existence, Uniqueness, and Continuity |
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1.7 |
The Gronwall Inequality |
Chapter 2. |
Linear Systems, with an Introduction to Phase Space Analysis |
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2.1 |
Introduction |
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2.2 |
Existence and Uniqueness for Linear Systems |
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2.3 |
Linear Homogeneous Systems |
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2.4 |
Linear Nonhomogeneous Systems |
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2.5 |
Linear Systems with Constant Coefficients |
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2.6 |
Similarity of Matrices and the Jordan Canonical Form |
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2.7 |
Asymptotic Behavior of Solutions of Linear Systems with Constant Coefficients |
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2.8 |
Autonomous Systems--Phase Space--Two-Dimensional Systems |
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2.9 |
Linear Systems with Periodic Coefficients; Miscellaneous Exercises |
Chapter 3. |
Existence Theory |
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3.1 |
Existence in the Scalar Case |
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3.2 |
Existence Theory for Systems of First-Order Equations |
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3.3 |
Uniqueness of Solutions |
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3.4 |
Continuation of Solutions |
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3.5 |
Dependence on Initial Conditions and Parameters; Miscellaneous Exercises |
Chapter 4. |
Stability of Linear and Almost Linear Systems |
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4.1 |
Introduction |
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4.2 |
Definitions of Stability |
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4.3 |
Linear Systems |
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4.4 |
Almost Linear Systems |
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4.5 |
Conditional Stability |
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4.6 |
Asymptotic Equivalence |
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4.7 |
Stability of Periodic Solutions |
Chapter 5. |
Lyapunov's Second Method |
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5.1 |
Introductory Remarks |
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5.2 |
Lyapunov's Theorems |
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5.3 |
Proofs of Lyapunov's Theorems |
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5.4 |
Invariant Sets and Stability |
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5.5 |
The Extent of Asymptotic Stability--Global Asymptotic Stability |
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5.6 |
Nonautonomous Systems |
Chapter 6. |
Some Applications |
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6.1 |
Introduction |
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6.2 |
The Undamped Oscillator |
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6.3 |
The Pendulum |
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6.4 |
Self-Excited Oscillations--Periodic Solutions of the Liénard Equation |
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6.5 |
The Regulator Problem |
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6.6 |
Absolute Stability of the Regulator System |
Appendix 1. |
Generalized Eigenvectors, Invariant Subspaces, and Canonical Forms of Matrices |
Appendix 2. |
Canonical Forms of 2 x 2 Matrices |
Appendix 3. |
The Logarithm of a Matrix |
Appendix 4. |
Some Results from Matrix Theory |
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Bibliography; Index |
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