Stabilization of Stochastic Dynamical Systems of a Random Structure with Markov Switches and Poisson Perturbations

Lukashiv, T. O.; Yuliia, Litvinchuk,; Malyk, Igor V.; Golebiewska, Anna; Nazarov, Petr V.

doi:10.3390/math11030582

Cited by 3 publications

(1 citation statement)

References 21 publications

(38 reference statements)

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“…Random perturbation of signals has been an active field of research in several mathematics and computer science areas, for example, for the linear tracking of signals [29], for global optimization metaheuristics [30,31], for neural network synchronization [32], for optimal control of a stochastic dynamical system [33], for analyzing the transient dynamics of a predator-prey system [34], for the asymptotic covariance estimation [35], and for the study of the numerical approximation of partial differential equations [36], to mention some representative works. However, to our knowledge, this type of analysis has yet to be applied to detect complex dynamics in CAs.…”

Section: Introductionmentioning

confidence: 99%

Mean-Field Analysis with Random Perturbations to Detect Gliders in Cellular Automata

Seck-Tuoh-Mora,

Medina-Marin,

Hernández-Romero

et al. 2023

Mathematics

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Cellular automata are mathematical models that represent systems with complex behavior through simple interactions between their individual elements. These models can be used to study unconventional computational systems and complexity. One notable aspect of cellular automata is their ability to create structures known as gliders, which move in a regular pattern to represent the manipulation of information. This paper introduces the modification of mean-field theory applied to cellular automata, using random perturbations based on the system’s evolution rule. The original aspect of this approach is that the perturbation factor is tailored to the nature of the rule, altering the behavior of the mean-field polynomials. By combining the properties of both the original and perturbed polynomials, it is possible to detect when a cellular automaton is more likely to generate gliders without having to run evolutions of the system. This methodology is a useful approach to finding more examples of cellular automata that exhibit complex behavior. We start by examining elementary cellular automata, then move on to examples of automata that can generate gliders with more states. To illustrate the results of this methodology, we provide evolution examples of the detected automata.

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Section: Introductionmentioning

confidence: 99%

Mean-Field Analysis with Random Perturbations to Detect Gliders in Cellular Automata

Seck-Tuoh-Mora,

Medina-Marin,

Hernández-Romero

et al. 2023

Mathematics

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Mean-Field Analysis with Random Perturbations to Detect Gliders in Cellular Automata

Seck-Tuoh-Mora,

Medina-Marin,

Hernández-Romero

et al. 2023

Preprint

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Cellular automata are mathematical models that represent systems with complex behavior through simple interactions between their individual elements. These models can be used to study unconventional computational systems and complexity. One notable aspect of cellular automata is their ability to create structures known as gliders, which move in a regular pattern to represent the manipulation of information. This paper introduces the modification of mean-field theory applied to cellular automata, using random perturbations based on the system’s evolution rule. The original aspect of this approach is that the perturbation factor is tailored to the nature of the rule, altering the behavior of the mean-field polynomials. By combining the properties of both the original and perturbed polynomials, it is possible to detect when a cellular automaton is more likely to generate gliders without having to run evolutions of the system.This methodology is a useful approach to finding more examples of cellular automata that exhibit complex behavior. We start by examining elementary cellular automata, then move on to examples of automata that can generate gliders with more states. To illustrate the results of this methodology, we provide evolution examples of the detected automata.

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Stability of stochastic dynamic systems of a random structure with Markov switching in the presence of concentration points

Lukashiv

Malyk

Chepeleva

et al. 2023

MATH

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<abstract><p>This article aims to investigate sufficient conditions for the stability of the trivial solution of stochastic differential equations with a random structure, particularly in contexts involving the presence of concentration points. The proof of asymptotic stability leverages the use of Lyapunov functions, supplemented by additional constraints on the magnitudes of jumps and jump times, as well as the Markov property of the system solutions. The findings are elucidated with an example, demonstrating both stable and unstable conditions of the system. The novelty of this work is in the consideration of jump concentration points, which are not considered in classical works. The assumption of the existence of concentration points leads to additional constraints on jumps, jump times and relations between them.</p></abstract>

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Stabilization of Stochastic Dynamical Systems of a Random Structure with Markov Switches and Poisson Perturbations

Cited by 3 publications

References 21 publications

Mean-Field Analysis with Random Perturbations to Detect Gliders in Cellular Automata

Mean-Field Analysis with Random Perturbations to Detect Gliders in Cellular Automata

Mean-Field Analysis with Random Perturbations to Detect Gliders in Cellular Automata

Stability of stochastic dynamic systems of a random structure with Markov switching in the presence of concentration points

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