Weakly Nonlinear Approach: Bifurcations and Amplitude Equations

This lecture introduces the weakly nonlinear approach through the gravitational pendulum, discussing multiple time scale expansions and classical bifurcations. It covers the Hopf bifurcation, Stuart-Landau equation, and non-resonance conditions. The instructor explains the importance of critical slowing, linearized equations, normal modes, and dispersion relations. The lecture also delves into nonlinearities, multiscale expansions, and compatibility conditions, emphasizing the impact of nonlinear terms on the system's behavior.