Lecturer:
Course Type:
PhD Course
Academic Year:
2024-2025
Period:
October - December
Duration:
50 h
Description:
The main assumption of the classical implict function theorem in Banach spaces is that the linearized operator has a bounded inverse. This is sufficient for constructing bifurcation theory of periodic solutions of finite dimensional dynamical systems. On the other hand, there are several problems where this assumption is not satisfied, i.e. the linearized operator is unbounded, for example for the search of quasi-periodic solutions. To overcome this challenge, the Nash-Moser theory was developed.
In this course we shall prove a Nash-Moser-Zehnder implicit function theorem for the search of zeros of a nonlinear operator acting on scales of Banach spaces, of either analityic or differentiable functions. As applications we shall prove the Siegel linearization theorem for an analytic map close to an elliptic fixed point and the classical KAM (Kolmogorov–Arnold–Moser) theorem concerning the persistence of quasi-periodic solutions for sufficiently small and smooth perturbations of completely integrable, non-degenerate, finite dimensional Hamiltonian systems.
Other applications shall be considered time permitting.
Next lectures:
Wednesday 16 October, 11-13, room 133
Thursday 17 October, 14-16, room 133
Tuesday 22 October, 11-13, room 136
Thursday 24 October, 14-16, room 133
Tuesday 5 November, 11-13, room 136
Thursday 7 November, 11-13, room 134
Tuesday 12 November, 11-13, room 136
Thursday 14 November, 14-16, room 133
Tuesday 19 November, 11-13, room 136
Thursday 21 November, 14-16, room 133
Tuesday 26 November, 11-13, room 136
Thursday 28 November, 14-16, room 133
Tuesday 10 December, 11-13, room 136
Thursday 12 December 14-16, room 133
Tuesday 17 December, 11-13, room 136
Thursday 19 December 14-16, room 133
Research Group:
Location:
A-133