Degenerated singular cycles of inclination-flip type. (English) Zbl 0903.34024

The authors consider two-parameter families of three-dimensional vector fields, unfolding a degenerate singular cycle. The singular cycle contains a hyperbolic singularity with a two-dimensional stable manifold; the linearized vector field at the singularity possesses three distinct real eigenvalues. It further contains a hyperbolic periodic orbit of saddle type, a heteroclinic connection from the singularity to the periodic orbit and a heteroclinic connection from the periodic orbit to the singularity. The degeneracy condition involves a quadratic contact between the stable manifold of the singularity and the unstable manifold of the periodic orbit, along the heteroclinic connection from the periodic orbit to the singularity.
Different cases occur, depending on eigenvalue conditions and geometric configurations. The authors state bifurcation theorems, in which the main thrust is on the prevalence of hyperbolic dynamics.
Bifurcations from singular cycles of codimension one had been studied by R. Bamón, R. Labarca, R. Mañe and M. J. Pacífico [Publ. Math., Inst. Hautes Étud. Sci. 78, 207-232 (1993; Zbl 0801.58010)] and by M. J. Pacifico and A. Rovella [Ann. Sci. Ec. Norm. Supér., IV. Sér. 26, No. 6, 691-700 (1994; Zbl 0802.58036)].


34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
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