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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)
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Location: MSRI: Baker Board Room
Speaker(s) Amadeu Delshams (Polytechnical University of Cataluña (Barcelona))
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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.

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