Quantitative estimates on the normal form around a non semi-simple 1:-1 resonant periodic orbit
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hdl:2117/912
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Data publicació2003
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Abstract
The purpose of thiswork is to give precise estimates for the size of the remainder
of the normalized Hamiltonian around a non-semi-simple 1 : −1 resonant
periodic orbit, as a function of the distance to the orbit.
We consider a periodic orbit of a real analytic three-degrees of freedom
Hamiltonian system having a pairwise collision of its non-trivial characteristic
multipliers on the unit circle. Under generic hypotheses of non-resonance
and non-degeneracy of the collision, we present a constructive methodology
to reduce the Hamiltonian around the orbit to its (integrable) normal form, up
to any given order. This constructive process allows to obtain quantitative
estimates for the size of the remainder of the normal form, as a function
of the normalizing order. By selecting appropriately this order in terms of
the distance R to the resonant orbit (measured using suitable coordinates),
r = ropt(R) := 2 + ?exp(W(log(1/R1/(τ+1+ε))))?, we have proved that the size
of the remainder can be bounded (for small R) by Rropt(R)/2. Here, W(·) stands
for Lambert’s W function and verifies that W(z) exp(W(z)) = z, τ ? 1 is
the exponent of the required Diophantine condition and ε > 0 is any small
quantity. The reasons leading to this bound instead of classical exponentially
small estimates are also discussed.
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