Fractal Approach, Bifurcation Poker and Suc-Logic for Nonlinear Dynamics Forecasting

Abstract

The paper is devoted to the novel logic (SUC-logic) of the nonlinear dynamics forecasting. The SUC-logic is based on three main points: the special sections (S) to build the 2D projections of multidimensional spaces without the loss of useful information; the special units (U) of measurement to estimate the nonlinear dynamics evolution; the special consecutions (C) to realize the nonlinear dynamics forecasting step-by-step. The fractal approach to forecasting the nonlinear dynamics in real-time together with the approach to build the modified bifurcation diagrams to research the regularities and the uncertainties of the evolution scenarios are developed with the SUC-logic. The physical meaning of the uncertainty zone, the stability margin, the risk estimation, the farthest forecasting and the earliest forecasting are considered from the viewpoint of the nonlinear dynamics aspect. Reasonings and discussions are based on experimental and computational results.

Experimental Identification of Uncertainties in Dynamics of PWM Buck Converter

“…As a result, the range [ <1> max ,  <2> min ] is identified, where  <1> max is the maximum -value when 1-process is identified,  <2> min is the minimum -value when 2-process is identified, Kolokolov and Monovskaya (2013a). If i-samplings are acquired with zero initial conditions, then the hysteresis can be at least minimized, Kolokolov and Monovskaya (2013b). To accelerate building a bifurcation boundary it is advisable to use the preliminary solutions of a mathematical model and to use the regularities of shifting the operating process bifurcation boundary with multidimensional parametrical variation, Kolokolov et al (2005).…”

“…In particular, the corresponding 2D-region was denoted as an uncertainty zone ( UZ -zone). It was shown that the structure of each  UZ -zone is heterogeneous and consists of several parts, Kolokolov and Monovskaya (2013b): the uncertainty zones of the I st , II d and III d kinds ( I -zone,  II -zone and  III -zone correspondingly). Here the presence of the  I -zone (the narrow part of the (,R 3 ) -space between the solid lines of the  I -zone bounadary, fig.3a) is caused, first of all, by the hysteresis effects, and can be minimized by the use of startup (zero) initial conditions.…”

“…But for systems with a variable structure, there are several qualitative thresholds, through which the successive evolution of the system behavior occurs within the operating process domain, Kolokolov and Monovskaya (2013a). So, the researches follow the approach to the formalized experimental bifurcation analysis presented in Kolokolov and Monovskaya (2013b) to identify such thresholds. And we try to reveal which of the thresholds corresponds in the best way to the reference boundary used in the algorithms.…”

Implementation of the real-time nonlinear dynamics prediction: Experimental research on location

“…In the past decade, the general state of affairs in the field of nonlinear dynamics prediction has not changed much and remains at the initial stage [1][2][3][4][5][6]. Bifurcation analysis remains the only tool to explain the physical meaning and explore the patterns of the process of development in any nonlinear system.…”

“…Reasoning from this statement, a sequence of production is considered in which particular solu tions of prediction tasks are synthesized in the context of the basic [11,19,20], shadow [6,17], and parallel [11,17,20] scenarios of development of nonlinear dynamics. This point of view is based on the findings of research obtained in the framework of the fractal approach.…”