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Non-integrability of the 1:1:2-resonance. (English) Zbl 0537.58026

A Hamiltonian system of n degrees of freedom, defined by the function F, with an equilibrium point at the origin, is called formally integrable if there exist formal power series \(\hat f{}_ 1,...,\hat f_ n\), functionally independent, in involution, and such that the Taylor expansion \(\hat F\) of F is a formal power series in the \(\hat f{}_ j\). Take \(n=3\), \(\hat F=\sum_{k\geq 2}F^{(k)}\), \(F^{(k)}\) homogeneous of degree k, \(F^{(2)}>0\) and the eigenfrequencies in ratio 1:1:2. If \(F^{(3)}\) avoids a certain hypersurface of ”symmetric” third order systems, then the F-system is not formally integrable. If \(F^{(3)}\) is symmetric but \(F^{(4)}\) is in a non-void open subset, then homoclinic intersection with Devaney spiralling occurs; the angle decays of order 1 when approaching the origin.

MSC:

37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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