DUT Dynamics Seminar

This talk will give a brief introduction to the regularization theory, with a particular focus on the problems from celestial mechanics, such as the Kepler problem and the restricted three-body problem. Starting with the Kepler problem, we will demonstrate how to manipulate its equations of motion to regularize the singularity. Next, we will show the regularization method from the Hamiltonian perspective. After that, the planar circular restricted three-body problem and its regularization will be introduced. We will understand why this is important for both theoretical and numerical studies. If time permits, the elliptic problem will also be mentioned.

This talk continues our discussion of ergodic theory on horseshoes. We begin with a brief review of the horseshoe map, a classic example in differentiable dynamical systems, and its relationship with symbolic systems. We then recall some basic concepts in ergodic theory, including invariant measures, ergodic measures, and entropy. Finally, we investigate intermediate value properties and multifractal analysis on horseshoes.

This talk begins with a brief overview of KAM theory for Hamiltonian PDEs and the challenges posed by derivative nonlinearities. We then discuss recent progress on full-dimensional invariant tori for a 1D derivative nonlinear wave equation. To control frequency shifts while preserving non-resonance conditions through KAM iteration, we introduce a modified quasi-Töplitz framework unifying Töplitz-Lipschitz and quasi-Töplitz techniques. This yields linearly stable invariant tori with sub-exponentially decaying amplitudes under mild parameter assumptions.

In this talk， we will discuss the existence of full dimensional tori for Hamiltonian PDEs by KAM theory for infinite dimensional Hamiltonian systems.