Dynamical Systems II

Course outline

1. Birkhoff normal forms, Hopf bifurcations,  Periodic orbits, limit cycles and separatrix cycles, Poincare

map, Flouque theory, the Poincare-Bendixson theorem, stability

of periodic orbits, local bifurcations of periodic orbits. Index Theory,

2. Methods of averaging, Melnikov methods, Perturbation of

planar homoclinic and periodic orbits. Abelian Integrals. Local codimension two Bifurcations of flows

3. One-Parameter bifurcations of fixed points in discrete-time

systems, including saddle-node, Flip and Neimark-Sacker bifurcation.

4. Differential equations on torus, rotation number, quasiperiodicity, Bifurcations of periodic orbits into tori,

5. Smale horshoes, Hypebolic sets, Markov partitions and strange attractors.

6. Orbits homoclinic to hyperbolic fixed points in two and three-dimensional

autonomous vector Fields, Lorenz bifurcations, Silnikov example.

References

1. Guckenheimer, J.; Holmes, P.; Nonlinear Oscillations, Dynamical

Systems and Bifurcations of Vector Fields, Springer-

Verlag, New York, 1988.

2. Wiggins, S.; Introduction to Applied Nonlinear Dynamical

Systems and Chaos, Springer-Verlag, NewYork, 1990.

3. Kuznetsov, Y. A.; Elements of Applied Bifurcation Theory,

Springer-Verlag, NewYork, 1995.