J.I. Jiménez-Aquino

Multivariate formulation of transient stochastic dynamics

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.

Matricial formalism of transient dynamics

An alternative method in terms of a matricial formulation of quasideterministic (QD) approach and its connection with the nonlinear relaxation times (NLRT) has recently been proposed by (Jimenez-Aquino, 1997), to characterize the decay of unstable states of nonlinear systems, in the context of Gaussian white noise (GWN). The objective of this paper is to extend the method to the study of transient stochastic dynamics in the presence of a constant external force. In this matricial scheme, besides the influence of internal fluctuations and of the external force in the dynamical relaxation of the system, we also study the effect of fluctuations of initial conditions. The theory is explicitly applied to general systems of the two physical variables, and particularly to the study of a laser system in the presence of weak optical signals.

MATRICIAL FORMULATION OF TRANSIENT STOCHASTIC DYNAMICS DRIVEN BY GAUSSIAN COLORED NOISE

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.

On the transient stochastic dynamics driven by Gaussian colored noise of systems with time-dependent control parameters. The effect of initial conditions

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 t = (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.

The characteristic times of the transient stochastic dynamics with time-dependent control parameters distributed initial conditions

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_witht- = (a0b). The time scales of both models are compared under certain requirements of the involved parameters.

Three variable models in multivariate transient stochastic dynamics. The presence of constant external force

In this work we show that the generalized formalism of the Quasideterministic (QD) approach and its connection with the Nonlinear Relaxation Times (NLRT), is absolutely consistent with 3×3 matricial models whose elements of matrix are all different from zero, in the context of both Gaussian white and colored noise. In the absence of external force, more general results than those reported in Refs. [1,2] are given. The consistency of the formalism with such models in the presence of constant external force is also verified for Gaussian white noise.

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

Abstract 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.

The quasideterministic approach in the dynamical characterization of rotating unstable systems

In this work, we report a new theoretical scheme to deal with the dynamical characterization of rotating unstable systems driven by both internal noise and constant external force. We use the quasideterministic (QD) approach and the passage time (PT) distribution to characterize the dynamical relaxation of systems of two and three variables and show that, in this new scheme, the validity regime of QD approach is satisfied only if the intensity of external force is smaller than or of the same order as the internal noise. For the systems of three variables, a special mathematical approach is given for a better understanding of underlying physics. Numerical simulation results are given to support our proposal.