The matrix formalism of quasi-deterministic (QD) approach is developed to characterize, through the statistical properties of the passage time distributions, the rotating unstable Langevin-type dynamics under the influence of weak external fluctuations. The theory is developed for those two variable systems by assuming a phase diffusion model for the fluctuating external force. The proposal is applied to the same laser system studied by Dellunde et al. [Opt. Commun. 109 (1994) 435] and shows the conditions, in the appropriate dynamical representation, under which the rotational effects of the laser system are not exhibited in the theoretical description of this reference.

In Physica A 237 (1997) 113, we have proposed in terms of Gaussian white noise (GWN) a generalized method, in terms of eigenfunctions and eigenvalues, of the quasideterministic (QD) approach and its connection with the nonlinear relaxation times (NLRT) to describe the transient stochastic dynamics of multivariate systems. Contrary to what happens with the standard formulation of QD approach, the generalized theory is focused on those unstable systems, which are not necessarily derived from a potential function. In the present work, we extend the generalized method to the case of Gaussian colored noise (GCN). The coupling between the noise and the initial state of the system, as a natural effect in colored noise problem, is also addressed. To justify the theory, we study the same two analytical models, in two and three variables, of the preceding reference and calculate the time scales associated with those models.

The objective of this paper is to generalize the formulation, reported in previous works, which we term as standard formulation of quasideterministic approach, to multivariate unstable systems. As usual, the new formalism is connected with the nonlinear relaxation times, to characterize the transient stochastic dynamics of those unstable systems which are governed by a multivariate Langevin-type equation. The generalized method is applied to study two analytical models in two and three variables in order to obtain explicit results and calculate the time scale associated with the decay of the unstable state of such systems. The theoretical formalism constitutes an alternative method to study some physical systems in two and three varibles. The laser systems represent one possibility in this theoretical context.

On a recent work we studied the transient stochastic dynamics driven by Gaussian white noise (GWN), of linear systems with time-dependent control parameters, by means of the connection between the nonlinear relaxation times (NLRT) and the quasideterministic (QD) approach. In the present study we make an extension of the analysis to the Gaussian colored noise (GCN) problem. Here, we first calculate the characteristic time associated with the decay of the unstable state of such linear systems, when the control parameter is a linear function of time of the form a(t) = bt − a0, with b > a0 > 0, which is continuously swept from below to above a threshold t̄ = (a0/b). This type of linear modulation is known as the ramp model. Then, going further, we consider a general case where the control parameter is modulated by a family of functions a(t) = btδ ∮ a0, with δ > 0. The effects of the coupling between the initial state, at time t = 0, of the system with the noise are specially emphasized. The results of NLRT for the ramp model and the general case are compared.

A systematic method is developed for the calculation of characteristic times, called nonlinear relaxation times (NLRT), to describe the dynamical relaxation of the linear transient stochastic systems whose control parameters are time-dependent functions. For those control parameters, which are modulated by a family of functions of the form a(t) = btδ −a0, with δ > 0, the method is applied to calculate the NLRT associated with the decay of unstable states of the linear stochastic systems when these parameters are continuously swept from below to above threshold (t- = (a0b)1δ). The ramp modulation is a model for which δ = 1, it is studied and formulated in terms of the time differences s = t − t_ with t- = (a0b). The time scales of both models are compared under certain requirements of the involved parameters.

We characterize the transient behavior of the time-dependent linear stochastic systems in presence of an external force, by means of the integration of the nonlinear relaxation time (NLRT), supported by the quasi-deterministic (QD) theory. The formalism is applied to study the transient dynamics of the switch-on times of a single mode semiconductor laser, in presence of an external optical signal, as a prototype model. The time scale, for this laser model is essentially given in terms of a relevant scaling parameter.

We present the relationship between nonlinear-relaxation-time (NLRT) and quasideterministic approaches to characterize the decay of an unstable state. The universal character of the NLRT is established. The theoretical results are applied to study the dynamical relaxation of the Landau model in one and n variables and also a laser model

We present a general theory of nonlinear relaxation times (NLRT) for stochastic transient dynamics of systems driven by an asymmetric dichotomous Markov noise (DMN). Two limiting cases of this general result are studied: the Poissonian white shot noise (WSN) and the Gaussian white noise (GWN).

The decay of an unstable state under the influence of external colored noise has been studied by means of analog experiments and digital simulations. For both fixed and random initial conditions, the time evolution of the second moment 〈${x}^{2}$(t)〉 of the system variable was determined and then used to evaluate the nonlinear relaxation time. The results obtained are found to be in excellent agreement with the theoretical predictions of the immediately preceding paper [Casademunt, Jim\'enez-Aquino, and Sancho, Phys. Rev. A 40, 5905 (1989)].

The Non-Linear Relaxation Time (NLRT) technique is presented in the framework of a general and systematic formalism for the characterization of the transient evolution of systems described by Langevin equations. Several examples are studied. The theoretical results are compared with numerical and simulation data and also with those obtained via Mean First Passage Time techniques.

A novel approach to evaluate characteristic times of general relaxation processes is presented. The decay of the unstable state of a prototype model is studied. Our results are compared with those given by mean first passage time techniques. The advantages and possibilities of the new approach are briefly discussed.