The topology associated with cusp singular points

K. Efstathiou and A. Giacobbe
Nonlinearity, 25(12), pp. 3409-3422, 2012.
DOI: 10.1088/0951-7715/25/12/3409

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In this paper we investigate the global geometry associated with cusp singular points of two-degree of freedom completely integrable systems. It typically happens that such singular points appear in couples, connected by a curve of hyperbolic singular points. We show that such a couple gives rise to two possible topological types as base of the integrable torus bundle, that we call pleat and flap. When the topological type is a flap, the system can have non-trivial monodromy, and this is equivalent to the existence in phase space of a lens space compatible with the singular Lagrangian foliation associated to the completely integrable system.