Duistermaat, J. J. Non-integrability of the 1:1:2-resonance. (English) Zbl 0537.58026 Ergodic Theory Dyn. Syst. 4, 553-568 (1984). 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. Cited in 16 Documents MSC: 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems Keywords:equilibrium points; resonance; formal integrability; homoclinic orbits PDFBibTeX XMLCite \textit{J. J. Duistermaat}, Ergodic Theory Dyn. Syst. 4, 553--568 (1984; Zbl 0537.58026) Full Text: DOI References: [1] DOI: 10.1007/BF01081586 · Zbl 0524.58015 · doi:10.1007/BF01081586 [2] Verhulst, Asymptotic Analysis II, Springer Lecture Notes in Math. 985 pp 137– (1983) · Zbl 0515.70014 · doi:10.1007/BFb0062366 [3] DOI: 10.1007/BF02591529 · Zbl 0144.36302 · doi:10.1007/BF02591529 [4] van der Aa, Asymptotic Analysis, Springer Lecture Notes in Math. 711 (1979) [5] DOI: 10.1086/109234 · doi:10.1086/109234 [6] DOI: 10.1016/0022-0396(76)90130-3 · Zbl 0343.58005 · doi:10.1016/0022-0396(76)90130-3 [7] Abraham, Foundations of Mechanics (1978) [8] Melnikov, Trans. Moscow Math. Soc. 12 pp 3– (1963) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.