Dynamical Systems I

This course deals with basic properties of solutions of ordinary Differential equations and Dynamical systems such as stability of equilibrium point, periodic orbits, etc. Course outlines are as follows:

Outline of the course

1.,Linear systems, Geometry of solutions, Bifurcations in linear systems, Linearized stability, Hartman-Grobman theorem, Liapunov Functions, Lassalle invariance principle

2. Autonomous systems in n dimension, Existence and uniqueness, Geometry of flows, Equivalence of flows, Elementary bifurcations

 3. Periodic orbits, Limit cycles and separatrix cycles, Poincare map, Flouque Theory, The Poincare-Bendixson theorem, Stability of periodic orbits, Local bifurcations of periodic orbits.

4. Bifurcation of equilibria and periodic orbits in n-dimensional systems, Center manifold theorems, Center manifolds in parameter-dependent systems. Birkhoff normal forms.

5. Second order Hamiltonian and gradient systems, and their Bifurcations.

References:

1. Hirsch, W.M., Smale, S., Devaney, R., Differential Equations, Dynamical Systems and an Introduction to Chaos, Elsevior, 2004.

2. Hale, J., Kocak, H.; Dynamics and Bifurcations, Springer-Verlag, New York, 1991.

3. Perko, L., Differntial Equations and Dynamical systems, Third Edition, Springer-Verlag , 2001.

4. Guckenheimer, J.; Holmes, P.; Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag, New York, 1988.

5. Wiggins, S.; Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, NewYork, 1990.