# Mathematical Sciences Research Institute

Home » Lunch with Hamilton: Exponentially small splitting of separatrices associated to 3D whiskered tori with cubic frequencies

# Seminar

Lunch with Hamilton: Exponentially small splitting of separatrices associated to 3D whiskered tori with cubic frequencies November 07, 2018 (12:00 PM PST - 01:00 PM PST)
Parent Program: Hamiltonian systems, from topology to applications through analysis MSRI: Baker Board Room
Speaker(s) Amadeu Delshams (Polytechnical University of Cataluña (Barcelona))
Description No Description
Video
Abstract/Media

We measure the splitting of invariant manifolds of whiskered(hyperbolic)

tori with three frequencies in a nearly integrable Hamiltonian system,

whose hyperbolic part is given by a pendulum. This splitting depends

strongly on the arithmetic properties of the frequencies. For

3-dimensional frequency vectors, the standard theory of continued

fractions cannot be applied, so we develop a methodology for determining

the behavior of the small divisors for cubic frequencies. A paradigmatic

case is the cubic golden vector, generated by the (real) number

$\Omega=1-\Omega^3$. We show that the splitting is exponential small in

the perturbation parameter $\epsilon$ with an exponent which is a

quasiperiodic function of $\ln\epsilon$. This is a joint work with

Marina Gonchenko and Pere Gutierrez.