Contents
1 First-Order Equations and the Qualitative View
1.1 Differential equations and models
1.2 Linear first-order equations
1.3 Separable equations
1.4 Exact equations and integrating factors
1.5 Equations reducible to linear form
1.6 Autonomous equations and the phase line
1.7 First-order equations in software
1.8 Practice Problems
2 Existence and Uniqueness
2.1 The integral equation
2.2 Lipschitz conditions
2.3 The Picard–Lindelöf theorem
2.4 Gronwall’s inequality and continuous dependence
2.5 Peano’s existence theorem
2.6 Existence and uniqueness in software
2.7 Practice Problems
3 Maximal Solutions, Comparison and Dependence on Data
3.1 Maximal solutions
3.2 Global existence
3.3 Blow-up
3.4 Comparison theorems
3.5 The flow and dependence on initial data
3.6 Maximal solutions in software
3.7 Practice Problems
4 Linear Systems
4.1 The space of solutions
4.2 Liouville’s formula
4.3 The inhomogeneous system
4.4 The adjoint system
4.5 When is the fundamental matrix an exponential?
4.6 Linear systems in software
4.7 Practice Problems
5 Constant-Coefficient Systems
5.1 The matrix exponential
5.2 Computing the exponential: the Jordan form
5.3 Stability
5.4 Planar systems
5.5 Forced oscillations and resonance
5.6 Constant-coefficient systems in software
5.7 Practice Problems
6 Higher-Order Linear Equations and the Laplace Transform
6.1 The homogeneous equation
6.2 Undetermined coefficients
6.3 The impulse response and Green’s function
6.4 The Laplace transform
6.5 Higher-order equations in software
6.6 Practice Problems
7 Periodic Coefficients and Floquet Theory
7.1 The monodromy matrix
7.2 Floquet’s theorem
7.3 Floquet multipliers and stability
7.4 Hill’s equation
7.5 Mathieu’s equation and parametric resonance
7.6 Floquet theory in software
7.7 Practice Problems
8 Series Solutions and Special Functions
8.1 Power series at an ordinary point
8.2 Regular singular points: the method of Frobenius
8.3 Bessel functions
8.4 Legendre polynomials
8.5 Series solutions in software
8.6 Practice Problems
9 Boundary Value Problems and Sturm–Liouville Theory
9.1 Green’s formula and self-adjointness
9.2 The Green’s function and the Fredholm alternative
9.3 The Sturm–Liouville eigenvalue problem
9.4 The Rayleigh quotient
9.5 Sturm’s comparison and oscillation theorems
9.6 Boundary value problems in software
9.7 Practice Problems
10 Non-linear Systems and the Phase Plane
10.1 Orbits, equilibria and invariant sets
10.2 Linearisation and stability
10.3 Hartman–Grobman and stable manifolds
10.4 Competing species
10.5 Predator and prey
10.6 The phase plane in software
10.7 Practice Problems
11 Lyapunov Functions, Gradient and Hamiltonian Systems
11.1 Lyapunov’s direct method
11.2 Limit sets and LaSalle’s invariance principle
11.3 Basins of attraction and the Lyapunov equation
11.4 Gradient systems
11.5 Hamiltonian systems
11.6 Lyapunov methods in software
11.7 Practice Problems
12 Periodic Orbits and Limit Cycles
12.1 Limit cycles
12.2 The Poincaré–Bendixson theorem
12.3 Ruling out periodic orbits
12.4 Liénard systems and the van der Pol oscillator
12.5 Stability of periodic orbits
12.6 Limit cycles in software
12.7 Practice Problems
13 Bifurcations and Chaos
13.1 When can a bifurcation occur?
13.2 Zero eigenvalue: saddle-node, transcritical and pitchfork
13.3 The Hopf bifurcation
13.4 Chaos in the Lorenz system
13.5 Bifurcations and chaos in software
13.6 Practice Problems
14 Perturbation Methods, Numerical Methods and Review
14.1 Regular perturbation and secular terms
14.2 The Poincaré–Lindstedt method
14.3 Averaging and multiple scales
14.4 Singular perturbation and boundary layers
14.5 Numerical methods
14.6 Review of the course
14.7 Perturbation and numerical methods in software
14.8 Practice Problems
References
References