The Lagrangian view of geophysical fluid dynamics relies heavily on nonlinear methods familiar to most geoscientists. These include identification of hyperbolic and elliptic regions in flow fields along with finite time and scale Lyapunov exponents and particle dispersion statistics. Here we identify elliptic and hyperbolic regions to study the life cycle of a large anticyclonic eddy in the Gulf of Mexico. Hyperbolic regions develop simultaneously down to 200m when the Loop Current sheds the ring. The ring’s migration across the Gulf is monitored by tracking the movement of its elliptic point. Its abrupt disappearance is the result of interaction with a nearby hyperbolic region, which exists to 150m. Some broader implications of this approach are discussed in the last section.