Modern symbolic computation methods: Lyapunov quantities and 16th Hilbert problem

Леонов, Геннадий Алексеевич; Kuznetsov, Nikolay V.; Кудряшова, Е. В.

doi:10.15622/sp.16.1

Cited by 2 publications

(1 citation statement)

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“…True periodic orbits of this system can be visualized from the obtained initial data. For this system approximations of the nonlinearity ẋ cos(x) by its expansion in the Taylor series leads one to the case of the 16th Hilbert problem on the maximum number of coexisting periodic attractors and repellers and their disposition in two-dimensional polynomial systems (which was formulated in 1900 [26] and is still unsolved even for quadratic polynomials [40,46,92,93]). Remark that for this system the similar straightforward application of the XPPAUT package, as above for the Chua system, does not provide initial data for the visualization of all hidden periodic attractors.…”

Section: The Advanced Harmonic Balance Methodsmentioning

confidence: 99%

Hidden attractors in Chua circuit: mathematical theory meets physical experiments

et al. 2022

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After the discovery in early 1960s by E. Lorenz and Y. Ueda of the first example of a chaotic attractor in numerical simulation of a real physical process, a new scientific direction of analysis of chaotic behavior in dynamical systems arose. Despite the key role of this first discovery, later on a number of works have appeared supposing that chaotic attractors of the considered dynamical models are rather artificial, computer-induced objects, i.e., they are generated not due to the physical nature of the process, but only by errors arising from the application of approximate numerical methods and finite-precision computations. Further justification for the possibility of a real existence of chaos in the study of a physical system developed in two directions. Within the first direction, effective analytic-numerical methods were invented providing the so-called computer-assisted proof of the existence of a chaotic attractor. In the framework of the second direction, attempts were made to detect chaotic behavior directly in a physical experiment, by designing a proper experimental setup. The first remarkable result in this direction is the experiment of L. Chua, in which he designed a simple RLC circuit (Chua circuit) containing a nonlinear element (Chua diode), and managed to demonstrate the real evidence of chaotic behavior in this circuit on the screen of oscilloscope. The mathematical model of the Chua circuit (further, Chua system) is also known to be the first example of a system in which the existence of a chaotic hidden attractor was discovered and the bifurcation scenario of its birth was described. Despite the nontriviality of this discovery and cogency of the procedure for hidden attractor localization, the question of detecting this type of attractor in a physical experiment remained open. This article aims to give an exhaustive answer to this question, demonstrating both a detailed formulation of a radiophysical experiment on the localization of a hidden attractor in the Chua circuit, as well as a thorough description of the relationship between a physical experiment, mathematical modeling, and computer simulation.

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Section: The Advanced Harmonic Balance Methodsmentioning

confidence: 99%

Hidden attractors in Chua circuit: mathematical theory meets physical experiments

et al. 2022

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Use of a Numerical Analytical Model for Assessing the Functional Effectiveness of an Information Protection System by Analysing Its Probabilistic-Temporal Characteristics

Alferov

Butskikh

Krisilov

et al. 2020

Vestn. Dagest. gos. teh. univ., Teh. nauki

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Abstract. Aim Currently, the implementation of computational experiments to determine the probabilistic-temporal characteristics of protection functions for automated information systems is a complex and costly task. In order to study the dynamics of transitions between the states of this type of system, it is necessary to develop a mathematical model and an algorithm for computing the corresponding characteristics.Method. To achieve this goal, a mathematical model of the information security system was developed in the MATLAB software environment. The main advantages of this software environment consist in a high level of visualisation, the ability to modify models to analyse other systems of this type and the availability of integration tools with other software products.Results. The article presents a numerical and analytical model of a system for protecting information from unauthorised access. The functional dynamics of the system are described using a stochastic Petri net. In order to solve the integral equations and determine the probabilities of reaching the final state in a given time, the Laplace transform is used. The solution is carried out in an analytical mode to obtain an explicit form of the dependences of the probability-time characteristics of the system on the probabilities of transitions between states and the average times the system stays in each state. The paper presents the results of calculating the probability-time characteristics of the “Turning on the personal computer and user identification” subsystem of the “Guard NT” system for preventing unauthorised access to information.Conclusion. The developed model, which can be used to study the dynamics of transitions between states of an information protection system against unauthorised access in an automated system, as well as to optimise the time it takes to complete functional tasks, can also be used to improve the operational efficiency of these systems.

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Modern symbolic computation methods: Lyapunov quantities and 16th Hilbert problem

Cited by 2 publications

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Hidden attractors in Chua circuit: mathematical theory meets physical experiments

Hidden attractors in Chua circuit: mathematical theory meets physical experiments

Use of a Numerical Analytical Model for Assessing the Functional Effectiveness of an Information Protection System by Analysing Its Probabilistic-Temporal Characteristics

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