Introduction to real dynamical systems with applications

Jorge Mozo Fernández

Summary: The idea of this course is to give an introduction to the theory of real dynamical systems from scratch, focusing on the main elements of qualitative theory: equilibria, periodic orbits, limit cycles, … An outline of the contents is as follows:

1. Systems of differential equations: existence and uniqueness theorems.
2. Dynamical systems. Phase spaces. Classical examples: Logistic equations, predator – prey systems, Lotka- Volterra, …
3. Linear dynamical systems. Types of equilibria.
4. Non-linear dynamical systems. Topological and analytic equivalence of singular points.
5. Stability. Lyapunov functions.
6. Periodic solutions. Poincaré-Bendixon Theorem. Stability of periodic orbits. Limit cycles.
7. Applications.