Course information

ntroduction to Bifurcation Theory: Trajectories, equilibrium points, periodic oscillatory trajectories, phase diagrams. Attractors and bifurcations. The Duffing oscillator (without external forcing). The Van der Pol oscillator. Introduction to the theory of invariant manifolds: stable, unstable, central. Overview of nonlinear phenomena. Local bifurcation of equilibrium solutions and stability in one dimension and two dimensions: Turning points, transcritical and pitchfork bifurcation. Buckling and oscillation of a rod. Local Bifurcation of Periodic solutions from equilibrium solutions in two and more dimensions: Hopf-Andronov bifurcation. Fluid convection dynamics described by Lorentz equations. Poincaré maps: The periodically forced Duffing oscillator. Stability of periodic solutions of autonomous systems. The monodromy matrix. Bifurcations of equilibrium points of maps. Mechanisms of stability loss. Bifurcation points of periodic solutions, period-doubling, torus bifurcation. The example of FitzHugh neuroexcitation equations. Phase-locking. The example of the forced Van der Pol oscillator. Global bifurcations: Homoclinic and Heteroclinic bifurcation. Andronov-Leontovich Theorem. Melnikov method for homoclinic orbits. Introduction to chaotic dynamics: Strange Attractors. The Lorenz convection equations. The Duffing oscillator. Routes to Chaos. Route via torus, via period-doubling, via intermittency. Lyapunov exponents. Characterization of attractors. Calculation of Lyapunov exponents from time series. Power spectra.