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.