Assess the learner's ability to analyze stability and bifurcations of equilibria in nonlinear continuous and discrete dynamical systems. Tasks include: determine local stability via linearization (Jacobian) and eigenvalues, treat nonhyperbolic cases using center manifold and normal-form reductions, identify and distinguish saddle-node, transcritical, pitchfork, Hopf (continuous) and period-doubling, Neimark-Sacker (discrete) bifurcations, compute bifurcation conditions and leading normal-form coefficients, apply Lyapunov methods for stability conclusions, and interpret bifurcation diagrams and simple numerical continuation results. Questions may require short analytic derivations and concise numerical evidence.