This manuscript provides an introduction to ordinary differential equations and dynamical systems. We start with some simple examples of explicitly solvable equations. Then we prove the fundamental results concerning the initial value problem: existence, uniqueness, extensibility, dependence on initial conditions. Furthermore we consider linear equations, the Floquet theorem, and the autonomous linear flow.
Then we establish the Frobenius method for linear equations in the complex domain and investigates Sturm–Liouville type boundary value problems including oscillation theory.
Next we introduce the concept of a dynamical system and discuss stability including the stable manifold and the Hartman–Grobman theorem for both continuous and discrete systems.
We prove the Poincar´e–Bendixson theorem and investigate several examples of planar systems from classical mechanics, ecology, and electrical engineering. Moreover, attractors, Hamiltonian systems, the KAM theorem, and periodic solutions are discussed as well.
Finally, there is an introduction to chaos. Beginning with the basics for iterated interval maps and ending with the Smale–Birkhoff theorem and the Melnikov method for homoclinic orbits.