Complex invariants in two-dimension for coupled oscillator systems

Abstract

In order to extract some insight into the features of a dynamical system, we present here the possibility of its complex dynamical invariant. There are many systems which admit complex invariants. To achieve this we use Lie algebraic method to study two dimensional complex systems (coupled oscillator system) on the extended complex phase plane characterized by \(x=x_{1}+ip_{3}, y=x_{2} + ip_{4} , p_x=p_{1} + ix_{3}, p_y=p_{2}+ix_{4}\). Such invariants play an important role in the analysis of complex trajectories with regard to the calculation of semi-classical coherent state propagator in the path integral method.