  We characterize the geometrical and topological aspects of a dynamical system
by associating a geometric phase with a phase space trajectory. Using the
example of a nonlinear driven damped oscillator, we show that this phase is
resilient to fluctuations, responds to all bifurcations in the system, and also
finds new geometric transitions. Enriching the phase space description is a
novel phenomenon of ``geometrical localization'' which manifests itself as a
significant deviation from planar dynamics over a short time interval.
