Anholonomy and geometrical localization in dynamical systems

We characterize the geometrical and topological aspects of a dynamical system by associating a geometric phase with a phase space trajectory. Unlike Berry's phase, the path underlying this “anholonomy” is not in parameter space, but in a projective space of the phase space, which directly characterizes the geometry of the trajectories. Using the example of a non-linear driven damped oscillator, we show that this phase is resilient to fluctuations, responds to all bifurcations and finds new geometric transitions. It also provides a new characterization of chaotic trajectories. 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.