MTH402: (Ordinary Differential Equations)

        [Cr:4, Lc:3, Tt:1, Lb:0]

Course Outline

• 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.
• Sturm’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.

Reference Texts:

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