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# Monodromy of the fibre with oscillatory singular point of type $1 : (−2)$

2016, Vol. 12, No. 3, pp.  413-541

Author(s): Nekhoroshev N. N.

In the present work, we prove the existence of fractional monodromy in a large class of compact Lagrangian fibrations of four-dimensional symplectic manifolds. These fibrations are considered in the neighbourhood of the singular fibre $\Lambda^0$, that has a single singular point corresponding to a nonlinear oscillator with frequencies in $1 : (−2)$ resonance. We compute the matrices of monodromy defined by going around the fibre $\Lambda^0$. For all fibrations in the class and for an appropriate choice of the basis in the one-dimensional homology group of the torus, these matrices are the same. The elements of the monodromy matrix are rational and there is a non-integer element among them. This work is a continuation of the analysis in [20, 21, 39] where the matrix of fractional monodromy was computed for most simple particular fibrations of the class.
Citation: Nekhoroshev N. N., Monodromy of the fibre with oscillatory singular point of type $1 : (−2)$, Rus. J. Nonlin. Dyn., 2016, Vol. 12, No. 3, pp.  413-541
DOI:10.20537/nd1603008