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We study the dynamics of the five-parameter quadratic family ofvolume-preserving diffeomorphisms of R^3. This family is the unfolded normalform for a bifurcation of a fixed point with a triple-one multiplier and alsois the general form of a quadratic three-dimensional map with a quadraticinverse. Much of the nontrivial dynamics of this map occurs when its two fixedpoints are saddle-foci with intersecting two-dimensional stable and unstablemanifolds that bound a spherical ``vortex-bubble''. We show that this occursnear a saddle-center-Neimark-Sacker (SCNS) bifurcation that also creates, atleast in its normal form, an elliptic invariant circle. We develop a simplealgorithm to accurately compute these elliptic invariant circles and theirlongitudinal and transverse rotation numbers and use it to study theirbifurcations, classifying them by the resonances between the rotation numbers.In particular, rational values of the longitudinal rotation number are shown togive rise to a string of pearls that creates multiple copies of the originalspherical structure for an iterate of the map.

Quadratic Volume-Preserving Maps: Invariant Circles and Bifurcations