Dynamical Systems
Autonomous systems of ODEs in the two-dimensional phase plane, equilibrium points and their stability properties, the importance of the nonlinear terms. Population dynamics of two competing species (Lotka-Volterra model) and other applications. Hamiltonian dynamical systems, gradient systems. Local vs. global stability, Lyapunov functions. Periodic solutions, limit cycles and the Poincaré-Bendixson theorem. The Van der Pol oscillator and other applications. The notion of structural stability/instability. Bifurcations of equilibrium points and periodic trajectories: saddle-node, transcritical, pitchfork and Hopf bifurcations. Systems of ODEs with a phase space of three or more dimensions, the appearance of chaotic behavior. The Lorenz attractor and other chaotic ("strange") attractors in phase space.