The course will introduce students to the theory of dynamical systems. There will be many exercises given to students. The plan will be to go slowly and thoroughly over the year through the proofs, theory and applications of dynamical systems ? and to get to the bottom of things, so that students will get a real command of the field. Examples of applications will be given from a variety of disciplines, including nonlinear oscillators, diseases and epidemics, chemical reactions, electrical circuits, predator-prey systems, and more. Models relevant for neuroscience will be emphasized (Fitzhugh-Nagumo equations, Wilson-Cowan equations, Excitatory-Inhibitory networks, and other neural networks, etc).
1. Geometric approach to differential equations.
2. Linear systems: Solutions and phase portraits; nonhomogeneous systems: time dependent forcing.
3. The flow: Solutions of nonlinear equations. Solutions in multiple dimensions. Numerical solutions.
4. Phase portraits with emphasis on fixed points: Stability; nullclines; competitive populations.
5. Phase portraits using energy and other cost functions: Lyapunov functions; limit sets; gradient systems.
6. Periodic orbits: Poincare-Bendixson theorem; oscillators; Andronov-Hopf bifurcation; homoclinic bifurcation; change of area or volume by the flow; stability of periodic orbits and the Poincare map.
7. Chaotic attractors: Chaos; Lorenz system; Rossler attractor; Lyapunov exponents; tests for chaotic attractors.
8. Iterations of functions.