Matricial formalism of transient dynamics: II. The presence of time-dependent external forces

In this work the formalism of the preceding reference, which we will refer to as paper I, is now extended to the case of Gaussian stochastic fluctuations of time-dependent external forces. For those general nonlinear systems of two variables the formalism is developed explicitly when the external forces are taken as complex functions with time-dependent phase diffusion model. As an example in two variables, we describe the switch-on process of a laser model when the injected external signal is weak and has a time-dependent fluctuating phase.