Geometric Phase in Dissipative System

In non-Hamiltonian dissipative systems that have attracting limit cycles there are phase shifts analogous to the geometric phase of Hamiltonian quantum systems. Here the phase variable is the equal-time parametrization of a limit cycle. The phase shift is calculable when the direrentia1 equations of the variables of the systems are known. We study such a system with both numerical and analytical methods and show that the phase shift sometimes, but not always, have a geometric meaning. The computation is obtain by using the IBM SP2 at NCHC. 1. Geometric Phase in a Dissipative System There is an analogous phenomena to geometric phase in classical dissipative systems [ 11. They are dissipative oscillatory systems which have attracting limit cycles. By definition an attracting limit cycle of a dynamical system is such that if the system comes to a point sufficiently near a limit cycle it will relax to the cycle. A limit cycle in a dissipative system is therefore plays the role of a state, or an orbit, in a quantum system. The fast motion of the system around a limit cycle is analogous to the motion of the quantum system caused by the action of the Hamiltonian. A phase variable that equal-time parameterizes the limit cycle corresponds to the phase of a quantum state. Now consider adiabatically moving the classical system through a parameter space. The adiabatic condition prevents large fluctuations from a limit cycle, and ensures that once a system enters a limit cycle, it will always stay close to that limit cycle. This is an exact analogy to the adiabatic condition in a quantum system, which is imposed to ensure that the system stays in one state. We start with a two-dimensional limit cycle whose first order derivative is given by 0-81 86-7901-8197 $10.00