Geometric phases in dissipative systems

It is shown that a phenomenon analogous to the geometric phase shifts of Berry and Hannay occurs for dissipative oscillatory systems and can be detected in numerical simulations of chemical oscillators. The approach herein to the theory of geometric phases begins with a study of simple first-order differential equations on the circle (circle dynamics). It is shown how more complicated systems exhibit geometric phases through reduction to a circle dynamics. In this way, the various manifestations of the phenomenon are seen from a single unified perspective. The results are illustrated in numerical experiments on several model systems ranging from analytically solvable, but contrived, to realistic models of chemical oscillators.