CC BY
2126_Sciences of Europe # 45, (2019)
НЕЛИНЕЙНЫЙ РЕКУРРЕНТНЫЙ АНАЛИЗ В ЗАДАЧАХ ОБРАБОТКИ СИГНАЛОВ
Исмайлов Б.И.
Доцент кафедры: "Приборостроительная Инженерия" Азербайджанский Государственный Университет Нефти и Промышленности
Баку, Азербайджанская Республика
NONLINEAR RECURRENT ANALYSIS IN SIGNAL PROCESSING PROBLEMS
Ismailov B.
Docent, Department of "Instrumentation Engineering" Azerbaijan State Oil and Industry University, Baku, Azerbaijan Republic
АННОТАЦИЯ
Основой взаимодействия сложных систем в рамках «Открытой системы» являются нелинейные процессы, изучение которых позволит нам понять их природу, моделировать, управлять и прогнозировать развитие. В предлагаемой статье представлена структура исследования измерительной информации с использованием нелинейного рекуррентного анализа. На примере взаимодействующих систем показаны результаты нелинейного рекуррентного анализа с построением рекуррентных диаграмм и расчетом информативных параметров. Приведен пример работы адаптивного алгоритма «измерение - распознавание - принятие решения» и сценарий рефлексивного выбора лица, принимающего решение.
ABSTRACT
The basis of the interactions between complex systems in the framework of the "Open System" are non-linear processes, the study of which will allow us to understand their nature, model, manage and predict development. The proposed article presents the structure of the study of measurement information using nonlinear recurrent analysis. On the example of interacting systems, the results of nonlinear recurrent analysis with the construction of recurrent diagrams and the calculation of informative parameters are shown. An example of the work of the adaptive algorithm "measurement - recognition - decision making" and the scenario of reflexive choice of the decision maker are presented.
Ключевые слова: хаотические системы, теорема Пуанкаре, рекуррентные диаграммы, визуальный анализ, рефлексивный выбор.
Keywords: chaotic systems, Poincare theorem, recurrent diagrams, visual analysis, reflexive choice.
I. Introduction
The analysis of the behavior of complex systems interacting in the framework of the "Open System" is an actual problem of signal processing. The study of the behavior of nonlinear dynamic systems shows that when they interact, there is a complication of the information component, which leads to an increase in chaotic processes. Evolutionary trajectories in the state space of the system become sensitive to small information influences (fluctuations). As a result of the cumulative effect at a certain moment, the system can move from one evolutionary trajectory to another. So comes the point of bifurcation - the branch point of development options. Branches, in turn, catalyze an innovative surge, causing the intensification of various forms of information interaction. In the complex phase space, which characterizes the behavior of the medium, states are highlighted that attract to themselves the possible trajectories of the evolution of information systems. These states are defined as informational attrac-tors, by analogy with the concept of a strange attractor in the theory of dynamic chaos [1, 2].
II. Problems of evaluating measurement results and solutions
The task of identifying and assessing the parameters of interaction between the sources of complex (chaotic, stochastic) oscillations by the observed ones is interdisciplinary. It occurs in physics, biology, geophysics, medicine, economics, etc. Research from the
standpoint of a synergistic concept revealed an interesting property of such systems - it turned out that an increase in the intensity of stochastic terms can lead not only to an increase in disorder, but also the formation of ordered structures, that is, self-organization of the system, a decrease in its entropy [3].
For the first time, the French mathematician Henri Poincare drew attention to the recurrence of dissipative dynamical systems in the theorem of recurrence he formulated: ... neglecting some exceptional trajectories, the occurrence of which is infinitely improbable, it can be shown, that the system recurs infinitely many times as close as one wishes to its initial state [4].
Traditional methods of analyzing nonlinear processes require a significant amount of information, which makes it difficult to implement numerical modeling and complicates making the correct decision (forecast). One of the advantages of nonlinear recurrent analysis is noncritically to the volume of measurement information.
To apply the method of recurrent analysis in the study of multidimensional dynamic systems by Eckmann and co-authors, a method was proposed for visualizing the phase trajectory of the dynamic system under investigation on a two-dimensional matrix, which was called recurrence plot, RP [5].
Recurrence points are mappings of the phase trajectory of the system in the form of points on a two-dimensional square matrix, in which both coordinate axes are the axes of time. RP are characterized by a
number of specific visual images composed of grouped and separate points - recurrence points of the system's trajectory. A set of recurrence points form structures of type typology and texture, which group together form the basic 4 types of different classes. There are homogeneous RP, periodic, reflecting the drift, as well as abrupt transitions in the form of areas of contrasting clusters of points. According to their visual images, one can judge the dynamics and character of the process under
study, identify random and rarely repeated system states. The accumulated experience of studying the behavior of complex dynamic systems allows us to interpret the concrete situation that has been created, which helps in decision making with control problems [6, 7].
Informative visualization structures of the recurrent diagram on the example of the chaotic Lorentz system are shown in Fig. 1 [7].
Fig.1. Example of a recurrent diagram of the chaotic Lorenz system with measures and other features of the plot.
III. Control of chaotic systems dynamic
The structure of the adaptive system for conducting studies of the dynamics and control of transient processes of interacting multidimensional fractional-order chaotic systems based on the principles of synergetic,
the Poincare return theorem, nonlinear recurrent analysis and calculation of the parameters of the system under study is presented in Fig. 2 [8, 9].
Fig.2. The structure of the adaptive system.
The structure of the system consists of:
• SChM - stochastic and chaotic mappings;
• F - filter;
• RFS - phase space reconstruction;
• RP - recurrent diagramming;
• RQA - recurrent quantitative analysis;
• ChV - characteristic vector generation;
• ARP - analysis of recurrent diagrams;
• AChV - analysis of characteristic vector;
• N - norm setting (Lj, Lq_, Lœ ) ;
• C - calculation of: correlation integral C(s), new fractional dimension of Kaplan-Yorke DKY and implementation filtering chaotic information, recurrent analysis of controlled processes.
The proposed adaptive research structure works Consider as an example the work of the proposed
according to the "measurement-recognition-decision- structure for the case of the impact of fractional Color making" algorithm. noise on the Chen fractional chaotic system in order to
find the system behavior that meets the requirements.
Algorithm implementation
• Let's transfer the observed time series corresponding chaotic processes {U , {u into square matrices
mm ]e R2 :
• Define the disturbance as:
Mu }f ^[n x N ]
M2 fe}N ^[NxN].
(1) (2)
G : M j ^ M2.
(3)
The resulting perturbed mapping is defined as:
MG = M1 + M2, MG e R2
(4)
The recurrence diagram of the resulting mapping M is defined as:
M G ^ Rm; = Q( st -
x . x
),x e Rm,i,j = 1,N.
(5)
The correlation dimension is determined as [10]:
C (e) =T7T
N2 i
j=i i*j
x x
ij
The Kaplan York dimension is determined as [9]:
where k. and k - senior and junior exponent indicators.
1 2
• A chaotic XB vector is formed as:
DKY ~2- A1 / A2 ,
XB = ¥ (MG,C( s ),Dky)
def
(6)
(7)
(8)
If an unsatisfactory result is obtained in the context of decision-making, an adjustment is made according to the norms of H or disturbing influences [11].
The result of the work of the described algorithm for finding acceptable solutions to control the dynamics of the system under study is shown in Fig. 3 as an example of 4 selected areas of interest (AI), differing in both calculated and visual indicators.
The allocation of areas of interest measured by (x, y, z) is made according to the criterion:
FR( x, y, z) = supinf I / a3,
def
sem
v
(10)
where FR(x, y, z) - area of interests formed on the thesis of highlighting information in such a way that it is as meaningful as possible from a semantic point of view sup and minimal dimension of information inf
sem
; a3 - is adequate FR considering reflexivity.
v
)
Next, it is necessary to sublimate the results obtained by visual analysis, as a result of which satisfactory values of the system behavior in the area of interest (AI) are selected, such informative parameters as Lya-punov stability, fractal dimension, etc. are checked. In this case, consider the model of the subject, who must make a choice of one of the possible actions [9, 12].
Let the function of the readiness of the subject to the choice is:
A1 = (a3 ^ a2) ^ a1 . (11)
The variable a 1 describes the pressure of the medium to the choice of one of the alternatives. This is the real pressure of the environment, which is not realized by the subject, but is felt on a subconscious level. The subject's perception of environmental pressure is described by a variable a 2. The variable a 3 describes the subject's plans for choosing one of the alternatives, then his intentions or desires, which he would like to carry out. A1 - this is the alternative that the subject is ready to choose.
Let the variables of formula (11) take any values on the Boolean lattice of norms < L, <> independently of each other. Expression:
A2 = (a3 ^ a2)
in formula (11) is interpreted as "an image of oneself' for subject A, that is, his self-esteem. If the subject has no plans or intentions, his willingness to choose A1 does not depend on a3 and is described by a value a2 ^ a1, which is called a primitive choice.
If the subject's willingness to choose coincides with his intentions, that is A1 = a3 , then the choice of the subject is called realistic. In this case, the subject makes the choice of awareness, in accordance with their own desires.
The conditions of realistic choice are defined as:
(a3 ^ a2) ^ a1 = a3. (12)
In other words, the external environment and psychological set (the expected pressure of the world) of the subject form his intentions, which he is able to translate into action.
It is known that the subject has the opportunity to make a realistic choice if his intentions lie between the real pressure of the world and the primitive choice:
a1 < a3 < a2 ^ a1. (13)
Hence the conclusion: if a person strives for the best behavior for himself, this leads to a realistic choice.
Indeed, it can be seen from (13) that the best realistic choice a subject can make is determined by a primitive choice:
a3 = a2 ^ al ^ a2 v a1 = sup(> a2, a1) (14)
that is, as the strongest rate of > a2 and al.
The reflective selection shown carries some abstract tint. In addition, to "self-image", it is necessary to take into account such a concept as "flash of intuition" [10], which has the function of completion for realistic choice.
Conclusion
Difficulties in evaluating the behavior of interacting information flows of dynamic systems to make satisfactory decisions require the use of modern methods of processing and interpreting the results obtained. Studying the trajectories of complex dynamic systems, it seems that their development should be evaluated by traditional methods with the calculation of informative parameters, as well as with the help of additional, difficultly formalized experience of a professional expert or a decision maker based on reflexive estimates of visual images of signal dynamics.
References
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4. Poincaré H. (1890) Sur la problem des trois corps et les équations de la dynamique. Acta Mathemat-ica. 13: pp. 1-271.
5. Eckman J., Kamphorst S., Ruelle D., Recurrence Plots of Dynamical Systems, Europhysics Letters, 4 (9), 1987. Pp. 973-977.
6. Webber C.L., Zbilut J.P. (2005) Recurrence quantification analysis of nonlinear dynamical systems. Chapter 2. In: Riley MA, Van Orden G (eds) Tutorials in contemporary nonlinear methods for the behavioural sciences, pp. 26-92 https://www.nsf.gov/pubs/2005/nsf05057/nmbs/nmbs. pdf. Accessed 5 July 2018.
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Karaganda state technical university, Karaganda
АННОТАЦИЯ
Одним из приоритетов стратегического развития научно-технических возможностей Республики Казахстан является формирование космической отрасли. Главное направление отрасли ориентировано на
