Variation of the dependence of the transient process duration on the initial conditions in systems with discrete time

Abstract

In the past decades, considerable attention of researchers is devoted to the investigation of nonlinear dynamical systems, both with discrete and continuous time. As a rule, the effort is concentrated on studying established regimes and on determining how one dynamical regime is changed by another in response to variations of the control parameters. At the same time, there are many phenomena related to the transient processes that frequently remain unstudied despite the fact that such processes can provide information about the whole system and about attractors realized in the phase space [1,2].Previously [3], we have studied transient processes in a one-dimensional (1D) system with discrete time, representing a logistic map (1)In particular, it was demonstrated that dependence of the transient process duration on the initial conditions T ε ( x 0 ) for this logistic map qualitatively changes when the control parameter λ varies and one dynamical regime is changed by another. It was also shown that dependence of the transient process duration on the initial conditions obeys certain scaling laws.In this study, we will consider the mechanisms leading to a qualitative change in the type of dependence of the transient process duration on the initial conditions in the case when the control parameters of a system vary within the framework of the same dynamical regime. The investigation is performed for a logistic map (1), which is a standard object of nonlinear dynamics, and the Eno map [4,5] (2) which transforms into a logistic map for b = 0. The transient processes in systems (1) and (2) were studied in a simplest regime offered by a stable cycle of period 1. Inother words, the control parameters were varied in such a manner that all observations referred to this dynamic regime, in which the behavior of system (2) corresponds to an immobile stable point ( x 0 , y 0 ), x 0 = y 0 = ( λ + b -1)/ λ , and the behavior of map (1), to an immobile stable point x 0 = ( λ -1)/ λ .Dependence of the transient process duration on the initial conditions, T ε ( x 0 ) in the logistic map (1) and T ε ( x 0 , y 0 ) in the Eno map (2), was determined for the given initial conditions and preset accuracy ε using the same method as in [3]. First, an attractor realized in the system was determined for a fixed set of the control parameters by N = 6500 iterations of an arbitrary initial point, after which it was assumed that the imaging point attained the attractor. Then, the obtained sequence ({ x n and { x n , y n for maps (1) and (2), respectively) was analyzed beginning with n = N -1, N -2, … in order to determine a period of the regime (stable point, 2-cycle, 4-cycle, etc.). Finally, by sequentially trying all the possible initial conditions with a certain partition step, an interval of the discrete time necessary for the imaging point to attain the attractor with an accuracy ε was determined for each initial condition.The results of this investigation showed that dependence of the transient process duration on the initial conditions T ε...