Nonlinear Dynamics: Bifurcations and Stability

This lecture covers the analysis of nonlinear dynamics, focusing on bifurcations and stability. It discusses the existence and stability of steady solutions in nonlinear systems, the concept of bifurcation branches, and the impact of nonlinearities on system behavior. The lecture also delves into the theory of weakly nonlinear systems, multiscale expansions, and the implications of nonlinear saturation. Additionally, it explores the canonical example of a Hopf bifurcation and the Stuart-Landau amplitude equation. The presentation concludes with discussions on global stability analysis, resonance phenomena, and mean flow corrections in nonlinearly damped oscillators.