Editing Geometric phase (section)

=== Geometric phase defined on attractors ===

While Berry's formulation was originally defined for linear Hamiltonian systems, it was soon realized by Ning and Haken<ref name="Ning-Haken92">{{cite journal |title=Geometrical phase and amplitude accumulations in dissipative systems with cyclic attractors |author=C.&nbsp;Z. Ning, H. Haken |journal=Phys. Rev. Lett. |volume=68 |year=1992 |issue=14 |pages=2109–2122 |doi=10.1103/PhysRevLett.68.2109 |bibcode=1992PhRvL..68.2109N |pmid=10045311}}</ref> that similar geometric phase can be defined for entirely different systems such as nonlinear dissipative systems that possess certain cyclic attractors. They showed that such cyclic attractors exist in a class of nonlinear dissipative systems with certain symmetries.<ref name="Ning-HakenMPL">{{cite journal |title=The geometric phase in nonlinear dissipative systems |author=C.&nbsp;Z. Ning, H. Haken |journal=Mod. Phys. Lett. B |volume=6 |year=1992 |issue=25 |pages=1541–1568 |doi=10.1142/S0217984992001265 |bibcode=1992MPLB....6.1541N }}</ref> There are several important aspects of this generalization of Berry's phase: 1) Instead of the parameter space for the original Berry phase, this Ning-Haken generalization is defined in phase space; 2) Instead of the adiabatic evolution in quantum mechanical system, the evolution of the system in phase space needs not to be adiabatic. There is no restriction on the time scale of the temporal evolution; 3) Instead of a Hermitian system or non-hermitian system with linear damping, systems can be generally nonlinear and non-hermitian.