THE GEOMETRIC PHASE IN NONLINEAR DISSIPATIVE SYSTEMS

We review the recent progress made in generalizing the concept of the geometric phase to nonlinear dissipative systems. We first illustrate the usual form of the parallel transport law with an elementary example of the parallel shift of a line on the complex plane. Important results about the non-adiabatical geometric (Aharonov and Anandan or AA) phase18 for the Schrodinger equations are reviewed in order to make a comparison with results for dissipative systems. Following the line of Ref. 21 we show that a geometric phase can be defined for dissipative systems with the cyclic attractors introduced in Ref. 21. Systems undergoing the Hopf bifurcation with a continuous symmetry are shown to possess such cyclic attractors. Examples from laser physics are discussed to exhibit the applicability of our formalism and the widespread existence of the geometric phase in dissipative systems defined in Ref. 21.