1 Preliminaries
1.1 Vectors and Matrices
1.2 MATLAB
1.3 Special Kinds of Matrices
1.4 The Geometry of Vector Operations
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2 Solving Linear Equations
2.1 Systems of Linear Equations and Matrices
2.2 The Geometry of Low-Dimensional Solutions
2.3 Gaussian Elimination
2.4 Reduction to Echelon Form
2.5 Linear Equations with Special Coefficients
2.6 *Uniqueness of Reduced Echelon Form
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3 Matrices and Linearity
3.1 Matrix Multiplication of Vectors
3.2 Matrix Mappings
3.3 Linearity
3.4 The Principle of Superposition
3.5 Composition and Multiplication of Matrices
3.6 Properties of Matrix Multiplication
3.7 Solving Linear Systems and Inverses
3.8 Determinants of 2 X 2 Matrices
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4 Solving Ordinary Differential Equations
4.1 A Single Differential Equation
4.2 Graphing Solutions to Differential Equations
4.3 Phase Space Pictures and Equilibria
4.4 *Separation of Variables
4.5 Uncoupled Linear Systems of Two Equations
4.6 Coupled Linear Systems
4.7 The Initial Value Problem and Eigenvectors
4.8 Eigenvalues of 2 X 2 Matrices
4.9 Initial Value Problems Revisited
4.10 *Markov Chains
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5 Vector Spaces
5.1 Vector Spaces and Subspaces
5.2 Construction of Subspaces
5.3 Spanning Sets and MATLAB
5.4 Linear Dependence and Linear Independence
5.5 Dimension and Bases
5.6 The Proof of the Main Theorem
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6 Closed Form Solutions for Planar ODEs
6.1 The Initial Value Problem
6.2 Closed Form Solutions by the Direct Method
6.3 Solutions Using Matrix Exponentials
6.4 Linear Normal Form Planar Systems
6.5 Similar Matrices
6.6 *Formulas for Matrix Exponentials
6.7 Second Order Equations
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7 Qualitative Theory of Planar ODEs
7.1 Sinks, Saddles, and Sources
7.2 Phase Portraits of Sinks
7.3 Phase Portraits of Nonhyperbolic Systems
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8 Determinants and Eigenvalues
8.1 Determinants
8.2 Eigenvalues
8.3 *Appendix: Existence of Determinants
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9 Linear Maps and Changes of Coordinates
9.1 Linear Mappings and Bases
9.2 Row Rank Equals Column Rank
9.3 Vectors and Matrices in Coordinates
9.4 Matrices of Linear Maps on a Vector Space
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10 Orthogonality
10.1 Orthonormal Bases
10.2 Least Squares Approximations
10.3 Least Squares Fitting of Data
10.4 Symmetric Matrices
10.5 Orthogonal Matrices and QR Decompositions
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11 Autonomous Planar Nonlinear Systems
11.1 Introduction
11.2 Equilibria and Linearization
11.3 Periodic Solutions
11.4 Stylized Phase Portraits
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12 Bifurcation Theory
12.1 Two Species Population Models
12.2 Examples of Bifurcations
12.3 The Continuous Flow Stirred Tank Reactor
12.4 The Remaining Global Bifurcations
12.5 *Saddle-Node Bifurcations Revisited
12.6 *Hopf Bifurcations Revisited
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13 Matrix Normal Forms
13.1 Real Diagonalizable Matrices
13.2 Simple Complex Eigenvalues
13.3 Multiplicity and Generalized Eigenvectors
13.4 The Jordan Normal Form Theorem
13.5 *Appendix: Markov Matrix Theory
13.6 *Appendix: Proof of Jordan Normal Form
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14 Higher Dimensional Systems
14.1 Linear Systems in Jordan Normal Form
14.2 Qualitative Theory Near Equilibria
14.3 MATLAB \tt ode45 in One Dimension
14.4 Higher Dimensional Systems Using ode45
14.5 Quasiperiodic Motions and Tori
14.6 Chaos and the Lorenz Equation
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15 Linear Differential Equations
15.1 Solving Systems in Original Coordinates
15.2 Higher Order Equations
15.3 Linear Differential Operators
15.4 Undetermined Coefficients
15.5 Periodic Forcing and Resonance
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16 Laplace Transforms
16.1 The Method of Laplace Transforms
16.2 Laplace Transforms and Their Computation
16.3 Partial Fractions
16.4 Discontinuous Forcing
16.5 RLC Circuits
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17 Additional Techniques for Solving ODEs
17.1 Nonconstant Coefficient Linear Equations
17.2 Variation of Parameters for Systems
17.3 The Wronskian
17.4 Higher Order Equations
17.5 Simplification by Substitution
17.6 Exact Differential Equations
17.7 Hamiltonian Systems
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18 Numerical Solutions of ODEs
18.1 A Description of Numerical Methods
18.2 Error Bounds for Euler's Method
18.3 Local and Global Error Bounds
18.4 Appendix: Variable Step Methods
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