**[Cr:4, Lc:3, Tt:1, Lb:0]**

- System of Linear equations, Exponential of matrices, Existence of solutions for linear systems, Phase portrait, Stability theory for linear systems.
- Nonlinear systems, Lipschitz functions, Existence and uniqueness theorem for nonlinear systems, dependence on initial conditions, the maximal interval of existence, Flow defined by differential equations.
- Linearization of a nonlinear system, equilibrium points, Stable manifold theorem, Centre manifold theorem, The Hartman-Grobman theorem (statement), Stability and Liapunov functions.
- Sturmâ€™s Separation and Comparison theorem, regular Sturm-Liouville systems, eigenvalues and eigenfunctions.
- Global existence theorem for nonlinear systems, Periodic orbits, Limit cycles,
separatrix cycles. Statement and discussion of the Poincare-Bendixson theorem.

Additional Topics The Poincare map, proof of the Poincare-Bendixson theorem,
discussion of the Poincare-Hopf index theorem.

- Lawrence Perko: Differential Equations and Dynamical systems, Springer
- E. Coddington and Levinson: Theory of ODE
- Anton Zettl: Sturm-Liouville theory, AMS