Elementary Theory of Ordinary Differential Equations

• Scalar Autonomous Equations, Existence and Uniqueness, Geometry of Flows, Stability of Equilibria
• Elementary Bifurcations, Dependence on Parameters, Local perturbations Near Equilibria, Equivalence of Flows
• Planar Autonomous Systems, Examples of Planar Systems, General Properties and Geomtery, Product Systems, First Integral and Conservative Systems.
• Linear Systems, Properties of Linear systems, Reduction to Cannnical Forms, Qualitative Equivalence in Linear systems, Bifurcations in Linear Systems.
• Nonlinear systems, Asymptotic Stability and Instability from Linearization. Liapunov Equations, Lassalle Invariance Principle. Stable and Unstable manifolds, Hartman-Grobman Theorem.

Stability and Bifurcations of equilibrium points in the presence of Zero Eigenvalue using Center Manifold Theorem.

Textbooks:

• J. Hale, H Kocak, Dynamics and Bifurcations, Springer- Verlag, 1996
• M.W. Hirsch, S. Smale, R. Devaney, Differential Equations, Dynamical Systems and an Introduction to Chaos, Elsevier, 2004.