Ordinary Differential Equations
Linear systems of differential equations: uncoupled linear systems, diagonalization, exponentials of operators, the fundamental theorem for linear systems, planar linear systems, complex eigenvalues, multiple eigenvalues, stability theory, nonhomogeneous linear systems.
Local theory of nonlinear systems: initial value problem, hyperbolic equilibrium point, Stable Manifold Theorem. Hartman-Grobman Theorem. Stability and Liapunov functions. Saddles, nodes, foci and centers. Nonhyperbolic critical points. Center manifold theory.
Global theory of nonlinear systems: limit set, attractor, limit cycle, Poincaré map, stable manifold theorem for periodic orbits, Poincaré-Bendixson theory. Mathematical background: Fundaments of perturbation analysis. The Multiple Scale Method. Basic concepts of bifurcation analysis: Bifurcation points, Linear codimension of a bifurcation, Imperfections, Fundamental path, Center Manifold Theory.
Basic mechanisms of multiple bifurcations: divergence, Hopf, nonresonant or resonant double-Hopf, Divergence-Hopf, Double-zero bifurcation.
Lawrence Perko, Differential equations and dynamical systems, Springer-Verlag, 2001