- 97-402 Chandre C., Govin M., Jauslin H.R.
- KAM-Renormalization Group Approach to the Break-up of Invariant Tori
in Hamiltonian Systems
Jul 15, 97
(auto. generated ps),
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Abstract. We analyse the break-up of invariant tori in Hamiltonian
systems with two degrees of freedom using a combination of KAM theory
and renormalization group techniques.
We consider a class of Hamiltonians quadratic in the
action variables that is invariant under the chosen KAM transformations,
following the approach of Thirring.
The numerical implementation of the transformation
shows that the KAM iteration converges up to the critical coupling at
which the torus breaks up. By combining this iteration with a renormalization,
consisting of a shift of resonances and rescalings of momentum and energy,
we obtain a much more
efficient method that allows to determine the critical coupling with high
accuracy. This transformation is based on the physical mechanism of the
break-up of invariant tori.
We show that the critical surface of the transformation is the
stable manifold of codimension one of a nontrivial fixed point, and we discuss
its universality properties.