Control of the dynamics of complex systems with memory

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CONTROL OF THE DYNAMICS OF COMPLEX SYSTEMS WITH MEMORY

Ismailov B.I., Azerbaijan State Oil and Industry University, PhD, docent, Department of “Instrumentation Engineering”, Baku, Azerbaijan Republic.

Abstract

The article is devoted to the problems of controlling the behavior of complex coupled systems functioning within the framework of the Open System, the influence of control actions, as well as noise against the background of the corrective manifestation of the memory phenomenon inherent in fractional chaotic processes.

To evaluate, analyze and visualize the processes occurring when controlling the dynamics of a complex system, it is proposed to use the Poincare return times, Tsallis entropy, Lyapunov exponents, and fractal dimension.

The structure of studies of the dynamics of multidimensional fractional chaotic coupled systems and the structure of the study of adaptive coherent behavior of coupled systems are given.

Keywords: open system, fractional dynamics, memory, Poincare recurrence, Tsallis entropy.

The process of interaction of physical systems within the framework of the Open System is accompanied by structural changes in their dynamics caused by mixing processes, as well as the transport of information flows. In addition, in the process of controlling the dynamics of complex systems, one can observe the manifestation of effects in time, the cause of which is the system memory. Mathematical modeling of the reaction of complex systems consisting of coupled nonlinear physical systems requires controlling the behavior of information flows, which are manifested in the stability and instability of the manifold of fixed points of their periodic orbits [1]. Studies of nonlocality degree and memory degree, as an integral part of fractional dynamics, are applicable in transport systems - transfer of information flows [1,2].

In addition to the well-known characteristics in the behavior of chaotic processes about their excessive sensitivity to initial conditions and influencing factors, they also have hereditary character and its manifestations in the form of nonlocal effects in time. [1.3-6].

The relevance of the problem is related to the need of predicting the behavior of a complex system when controlling its dynamics in order to obtain satisfactory characteristics. The development of dynamic processes is accompanied by transient processes with the transformation of complex processes within the framework of the Open System into such stages as: quasi-regular, chaotic, hyper-chaotic, and other states [1,6-10].

Along with this, the article analyzes the influence of control actions and the course of processes in the space of influence of various nature of noise. An important place in the study is given to the manifestations of memory as a corrective condition that affects the stability of the system. In the structure of studies of memory manifestations, there are also stages of determining the local dimension of the process under study and its global value, the difference of which will indicate memory loss [6-8]. The structure of the study is presented in Figure 1 and covers the main stages of analysis, interactions and calculations.

Figure 1. Structure of studies of the dynamics of multidimensional fractional-order chaotic coupled systems.

Studies on the use of Levy flight-type super diffusion processes for mixing and transferring information flows in terms of fractional dynamics have shown their effective role in enhancing the transport-mixing-transport fractional structure effect, estimated by such parameters as the transport exponent, Poincare's return times et al. [1,10-13].

According to the above scheme, the control of the main informative parameters is carried out, among which an important place is occupied by the return times of Poincare. These indicators are essentially indicators and characteristics, showing how certain states of complex systems are repeated over time [10, 14-17]. The diagrams constructed as a result of the nonlinear recurrence analysis have important visual analysis features in the form of the texture and topology of points fixed on the secant plane from the paths of the orbits of processes in phase space, as well as their color palette. [1,8,10,15,16,18].

In the control process, synchronization issues in fractional systems are updated, which is implemented by an adaptive system implemented on the principle of minimum losses - minimum return time and will be accompanied by visual images in terms of nonlinear recurrence analysis [1,19]. The Lyapunov exponents and fractal dimensions are used as indicators of the chaotic behavior of the system.

The structure of the study, which allows adaptively investigating the coherent behavior of the system based on the analysis of the process trajectory in the context of the selected area of interest, is shown in Figure 2. The result of the structure algorithm is the search for satisfactory coherence, the Tsallis entropy, the stability and Poincare diagram [1,19].

Figure 2. The structure of the study of the behavior of related systems.

In the presented structure, Chimera states is represented as a noise resonant exciter [21,22]. By its nature, the state of the chimera is a chaotic transition process, in connection with which its use as an exciting effect in the context of the organization of new structures is proposed. In the research algorithm for the presented structure, the Tsallis entropy characterizes the interaction of complex fractional chaotic systems and is used as a measure of the coherence of coupled multidimensional chaotic heterogeneous systems with non-extensive topology. [23.24].

Conclusion

The article analyzes methods for controlling the dynamics of coupled multidimensional fractional chaotic systems in the framework of the Open System. Structures for studying the behavior of interacting systems in the field of corrective actions and resonant excitation are presented. The importance of using Poincare's recurrency, of transitional systems under study, Tsallis entropy, characterizing the topology and physics of interconnected systems, as well as Lyapunov's indicators of randomness of systems.

visualize exponent chimera fractal

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