1 Introduction
1.1 Definitions and notation
1.2 Existence and uniqueness
1.3 Gronwall's inequality
2 Autonomous equations
2.1 Phase-space, orbits
2.2 Critical points and linearisation
2.3 Periodic solutions
2.4 First integrals and integral manifolds
2.5 Evolution of a volume element, Liouville's theorem
2.6 Exercises
3 Critical points
3.1 Two-dimensional linear systems
3.2 Remarks on three-dimensional linear systems
3.3 Critical points of nonlinear equations
3.4 Exercises
4 Periodic solutions
4.1 Bendixson's criterion
4.2 Geometric auxiliaries, preparation for the Poincaré-Bendixson theorem
4.3 The Poincaré-Bendixson theorem
4.4 Applications of the Poincaré-Bendixson theorem
4.5 Periodic solutions in?n
4.6 Exercises
5 Introduction to the theory of stability
5.1 Simple examples
5.2 Stability of equilibrium solutions
5.3 Stability of periodic solutions
5.4 Linearisation
5.5 Exercises
6 Linear Equations
6.1 Equations with constant coefficients
6.2 Equations with coefficients which have a limit
6.3 Equations with periodic coefficients
6.4 Exercises
7 Stability by linearisation
7.1 Asymptotic stability of the trivial solution
7.2 Instability of the trivial solution
7.3 Stability of periodic solutions of autonomous equations
7.4 Exercises
8 Stability analysis by the direct method
8.1 Introduction
8.2 Lyapunov functions
8.3 Hamiltonian systems and systems with first integrals
8.4 Applications and examples
8.5 Exercises
9 Introduction to perturbation theory
9.1 Background and elementary examples
9.2 Basic material
9.3 Naïve expansion
9.4 The Poincaré expansion theorem
9.5 Exercises
10 The Poincaré-Lindstedt method
10.1 Periodic solutions of autonomous second-order equations
10.2 Approximation of periodic solutions on arbitrary long time-scales
10.3 Periodic solutions of equations with forcing terms
10.4 The existence of periodic solutions
10.5 Exercises
11 The method of averaging
11.1 Introduction
11.2 The Lagrange standard form
11.3 Averaging in the periodic case
11.4 Averaging in the general case
11.5 Adiabatic invariants
11.6 Averaging over one angle, resonance manifolds
11.7 Averaging over more than one angle, an introduction
11.8 Periodic solutions
11.9 Exercises
12 Relaxation Oscillations
12.1 Introduction
12.2 Mechanical systems with large friction
12.3 The van der Pol-equation
12.4 The Volterra-Lotka equations
12.5 Exercises
13 Bifurcation Theory
13.1 Introduction
13.2 Normalisation
13.3 Averaging and normalisation
13.4 Centre manifolds
13.5 Bifurcation of equilibrium solutions and Hopf bifurcation
13.6 Exercises
14 Chaos
14.1 Introduction and historical context
14.2 The Lorenz-equations
14.3 Maps associated with the Lorenz-equations
14.4 One-dimensional dynamics
14.5 One-dimensional chaos: the quadratic map
14.6 One-dimensional chaos: the tent map
14.7 Fractal sets
14.8 Dynamical characterisations of fractal sets
14.9 Lyapunov exponents
14.10 Ideas and references to the literature
15 Hamiltonian systems
15.1 Introduction
15.2 A nonlinear example with two degrees of freedom
15.3 Birkhoff-normalisation
15.4 The phenomenon of recurrence
15.5 Periodic solutions
15.6 Invariant tori and chaos
15.7 The KAM theorem
15.8 Exercises
Appendix 1: The Morse lemma
Appendix 2: Linear periodic equations with a small parameter
Appendix 3: Trigonometric formulas and averages
Appendix 4: A sketch of Cotton's proof of the stable and unstable manifold theorem 3.3
Appendix 5: Bifurcations of self-excited oscillations
Appendix 6: Normal forms of Hamiltonian systems near equilibria
Answers and hints to the exercises
References

On the subject of differential equations many elementary books have been written. This book bridges the gap between elementary courses and research literature. The basic concepts necessary to study differential equations - critical points and equilibrium, periodic solutions, invariant sets and invariant manifolds - are discussed first. Stability theory is then developed starting with linearisation methods going back to Lyapunov and Poincaré. In the last four chapters more advanced topics like relaxation oscillations, bifurcation theory, chaos in mappings and differential equations, Hamiltonian systems are introduced, leading up to the frontiers of current research: thus the reader can start to work on open research problems, after studying this book. This new edition contains an extensive analysis of fractal sets with dynamical aspects like the correlation- and information dimension. In Hamiltonian systems, topics like Birkhoff normal forms and the Poincaré-Birkhoff theorem on periodic solutions have been added. There are now 6 appendices with new material on invariant manifolds, bifurcation of strongly nonlinear self-excited systems and normal forms of Hamiltonian systems. The subject material is presented from both the qualitative and the quantitative point of view, and is illustrated by many examples.