Assess learners' ability to analyze stability of equilibria in nonlinear dynamical systems using Lyapunov functions and bifurcation theory: construct and verify local and global Lyapunov candidates (positive definiteness, decrescent properties, time-derivative sign), classify equilibria (stable, asymptotically stable, unstable), compare linearization vs nonlinear conclusions, detect and classify common bifurcations (saddle-node, transcritical, pitchfork, Hopf), determine parameter thresholds and resulting phase-portrait changes, and interpret limit cycles and stability near Hopf points through worked examples and short proofs.