Ordinary Differential Equations (ODEs) are used to describe various types of phenomena in nature. On one hand, they allow us to look at systems in a comprehensive and abstract way. However, finding general solutions even for linear problems can be challenging. Adding nonlinearities makes the situation even more complex, as, among other things, the principle of superposition no longer holds. In this short course, we will introduce simple yet powerful principles that can be used to get an overview of the entire family of solutions of nonlinear ODEs. Through examples (Lasers, Josephson Junctions, Ising Model, Electrical Circuits, Oscillators, Chemical Reactions, etc.), concepts such as phase space, fixed point, limit cycle, attracting, repelling, and chaotic orbits, and bifurcations (when a solution transforms into another through parameter changes) will be discussed.

Material

The material (slides, educational simulations, scripts) can be downloaded directly from my GitHub repository.

Objectives

• Understand the fundamentals of dynamical systems and their applications in modeling physical and biological systems.
• Analyze reduced models of ODEs in 1, 2, and 3 dimensions using phase space analysis and bifurcations.
• Overview of the periodicity of limit cycles and strange attractors.

Bibliography

1. Strogatz, S. (2015) Nonlinear Dynamics and Chaos: with Applications to Physics, Biology, Chemistry, and Engineering. Westview Press
2. Abraham & Shaw 1992. Dynamics: the geometry of behavior. 2nd ed. Addison Wesley
3. Monteiro LHA (2023) Sistemas Dinâmicos 4a ed. Livraria da Física
4. Izhikevich E.M. (2007) Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting. The MIT press
5. Y. Kuznetsov (2004) Elements of Applied Bifurcation Theory. Springer