| Abstract: |
| This talk presents recent and ongoing work on invariant tori in Hamiltonian systems with quasi-periodic time dependence and in conformally symplectic systems. In joint work with Pedro Porras and Alex Haro, we developed an a posteriori KAM theorem for Lagrangian invariant tori in Hamiltonian flows with quasi-periodic forcing. The proof is based on the parameterization method and a Newton-like scheme using adapted symplectic frames and an intrinsic torsion matrix, leading to efficient quadratically convergent algorithms under standard Diophantine and nondegeneracy conditions.
I will also discuss current work with Alex Haro and Arturo Vieiro on secondary tori near elliptic points and on resonance capture in near-symplectic dynamics via Birkhoff normal forms. These problems illustrate how the same functional analytic and geometric ideas extend from conservative to weakly dissipative settings.
The parameterization method provides a common framework throughout, connecting rigorous existence theory, effective numerical computation, and the study of breakdown and bifurcation phenomena for quasi-periodic invariant objects. |
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