Author(s): Youssef Raffoul
Edition: 1
Publisher: Elsevier Inc, Academic Press Inc
Year: 2022
1: Preliminaries and Banach spaces
1.1 Preliminaries
1.2 Escape velocity
1.3 Applications to epidemics
1.4 Metrics and Banach spaces
1.5 Variation of parameters
1.5.1 RC circuit
1.6 Special differential equations
1.7 Exercises
2: Existence and uniqueness
2.1 Existence and uniqueness of solutions
2.2 Existence on Banach spaces
2.3 Existence theorem for linear equations
2.4 Continuation of solutions
2.5 Dependence on initial conditions
2.6 Exercises
3: Systems of ordinary differential equations
3.1 Existence and uniqueness
3.2 x ′ = A(t)x
3.2.1 Fundamental matrix
3.2.2 x ′ = Ax
3.2.3 Exponential matrix e^At
3.3 x ′ = A(t)x + g(t)
3.4 Discussion
3.5 Exercises
4: Stability of linear systems
4.1 Definitions and examples
4.2 x ′ = A(t)x
4.3 Floquet theory
4.3.1 Mathieu’s equation
4.3.2 Applications to Mathieu’s equation
4.4 Exercises
5: Qualitative analysis of linear systems
5.1 Preliminary theorems
5.2 Near-constant systems
5.3 Perturbed linear systems
5.4 Autonomous systems in the plane
5.5 Hamiltonian and gradient systems
5.6 Exercises
6: Nonlinear systems
6.1 Bifurcations in scalar systems
6.2 Stability of systems by linearization
6.3 An SIR epidemic model
6.4 Limit cycle
6.5 Lotka–Volterra competition model
6.6 Bifurcation in planar systems
6.7 Manifolds and Hartman–Grobman theorem
6.7.1 The stable manifold theorem
6.7.2 Global manifolds
6.7.3 Center manifold
6.7.4 Center manifold and reduced systems
6.7.5 Hartman–Grobman theorem
6.8 Exercises
7: Lyapunov functions
7.1 Lyapunov method
7.1.1 Stability of autonomous systems
7.1.2 Time-varying systems; non-autonomous
7.2 Global asymptotic stability
7.3 Instability
7.4 ω-limit set
7.5 Connection between eigenvalues and Lyapunov functions
7.6 Exponential stability
7.7 Exercises
8: Delay differential equations
8.1 Introduction
8.2 Method of steps
8.3 Existence and uniqueness
8.4 Stability using Lyapunov functions
8.5 Stability using fixed point theory
8.5.1 Neutral differential equations
8.5.2 Neutral Volterra integro-differential equations
8.6 Exponential stability
8.7 Existence of positive periodic solutions
8.8 Exercises
9: New variation of parameters
9.1 Applications to ordinary differential equations
9.1.1 Periodic solutions
9.2 Applications to delay differential equations
9.2.1 The main inversion
9.2.2 Variable time delay
9.3 Exercises
Index
