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Mathematics
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Nonlinear Dynamics Systems with Hysteresis
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Nonlinear Dynamics Systems with Hysteresis

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "E2: Control Theory and Mechanics".

Deadline for manuscript submissions: closed (31 March 2023) | Viewed by 13342

Special Issue Editor

Prof. Dr. Mikhail E. Semenov

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Guest Editor
Department of Digital Technologies, Voronezh State University, Voronezh, Russia
Interests: nonlinear dynamics; control theory; non-smooth systems; systems with hysteresis; regular and chaotic dynamics

Special Issue Information

Dear Colleagues,

In recent years, there have been many fields where hysteresis manifests itself and finds very promising applications. Despite the interest in the hysteresis phenomenon as is on the fundamental stage, there are many areas where hysteresis plays an important role from the point of view of adequate modeling of various phenomena in such fields as physics, engineering, chemistry, biology, medicine, economics, etc. As is well known, a wide class of systems in the aforementioned fields exhibit hysteretic behavior, which is caused either by their internal structure or by dynamical features of the processes occurring in these systems.

On the modeling stage, there are two main approaches to the description of the hysteresis phenomena: design models, e.g., a backlash (or play-operator), a non-ideal relay, the Preisach model, the Ishlinskii–Prandtl model, and phenomenological models, such as the Bouc–Wen model, the Duhem model, the Iwan model, and others. All these approaches are well investigated and have found a wide range of applications. However, there are many interesting problems in the field of modern nonlinear dynamics where hysteresis exhibits unexpected features and leads to some interesting results. 

This Special Issue collects papers with the aim to uncover and exploit the hysteresis phenomenon in the field of nonlinear dynamics and control theory from both fundamental and applied points of view. Special attention is paid to the modeling of nonlinear dynamical systems that show both regular and chaotic behavior under hysteresis.

Prof. Dr. Mikhail E. Semenov
Guest Editor

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Keywords

  • Nonlinear dynamics
  • Hysteresis
  • Control
  • Regular and chaotic dynamics
  • Simulation

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Published Papers (7 papers)

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Research

9 pages, 277 KiB  
Article
Some Properties of Motion of Atoms near a Charged Wire
by Božena Dorociaková, Rudolf Olach, Radoslav Chupáč and Pavol Oršanský
Mathematics 2023, 11(3), 634; https://doi.org/10.3390/math11030634 - 27 Jan 2023
Viewed by 1172
Abstract
We are concerned with the classical approach to atomic motions in the vicinity of a wire charged by an oscillating voltage. This mechanism has been proposed to serve for trapping cold neutral atoms. We prove that, under some conditions, the atom must either [...] Read more.
We are concerned with the classical approach to atomic motions in the vicinity of a wire charged by an oscillating voltage. This mechanism has been proposed to serve for trapping cold neutral atoms. We prove that, under some conditions, the atom must either collide with the wire or escape to infinity in the radial direction. The presented results for the atom–wire problem extend and complement some earlier ones in the literature. Full article
(This article belongs to the Special Issue Nonlinear Dynamics Systems with Hysteresis)
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12 pages, 922 KiB  
Article
A Combinatorial Model for Determining Information Loss in Organizational and Technical Systems
by Alexey V. Yakovlev, Vladimir V. Alekseev, Maria V. Volchikhina and Sergey V. Petrenko
Mathematics 2022, 10(19), 3448; https://doi.org/10.3390/math10193448 - 22 Sep 2022
Viewed by 1209
Abstract
A combinatorial model is proposed for determining the probability and information losses in an organizational and technical system (OTS) under destructive external influences. Mathematical expressions are obtained to determine the loss of information in the clusters of the control system. It is shown [...] Read more.
A combinatorial model is proposed for determining the probability and information losses in an organizational and technical system (OTS) under destructive external influences. Mathematical expressions are obtained to determine the loss of information in the clusters of the control system. It is shown that the use of this model for a quantitative analysis of the probability of occurrence of events and information losses in the control system, under varying external influences on the dynamic OTS, makes it possible to carry out a quantitative analysis and synthesis of the structure of the control system that is resistant to destructive external influences. A decomposition of the probabilities of occurrence of events and the corresponding loss of information by the levels of the hierarchy of the analyzed air traffic control system is presented. The achieved result is due to the sensitivity of the model for determining information losses relative to changes in the structure of the system and destructive external influences, as well as the use of the mathematical apparatus in combinatorial analyses. Full article
(This article belongs to the Special Issue Nonlinear Dynamics Systems with Hysteresis)
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16 pages, 1112 KiB  
Article
A Model of Optimal Production Planning Based on the Hysteretic Demand Curve
by Mikhail E. Semenov, Sergei V. Borzunov, Peter A. Meleshenko and Alexey V. Lapin
Mathematics 2022, 10(18), 3262; https://doi.org/10.3390/math10183262 - 8 Sep 2022
Cited by 5 | Viewed by 1868
Abstract
The article considers a hysteretic model of consumer behaviour in mono-product markets. Demand generation with regard to an individual consumer is modeled using a non-ideal relay with inverted thresholds. Therefore, the sales rate is defined as an analogue of the Preisach converter. The [...] Read more.
The article considers a hysteretic model of consumer behaviour in mono-product markets. Demand generation with regard to an individual consumer is modeled using a non-ideal relay with inverted thresholds. Therefore, the sales rate is defined as an analogue of the Preisach converter. The article considers the problem of the optimal production, storage, and distribution of goods, taking into account the hysteretic nature of the demand curve. The problem is reduced to a non-classical optimal control problem with hysteretic non-linearities. The latter is solved using Pontryagin’s maximum principle. The adopted economic model is based on the binary relationship of consumers to the product: the product is bought or the product is not bought. Transitions between these states are determined within the framework of our model only by the price of the goods; therefore, only the operator of a non-ideal relay can accurately describe such a dependence. The article presents the results of computational experiments illustrating the theoretical assumptions. Full article
(This article belongs to the Special Issue Nonlinear Dynamics Systems with Hysteresis)
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14 pages, 575 KiB  
Article
A Model of a Universal Neural Computer with Hysteresis Dynamics for Avionics Problems
by Andrey M. Solovyov, Nikolay I. Selvesyuk, Vladislav V. Kosyanchuk and Evgeniy Y. Zybin
Mathematics 2022, 10(14), 2390; https://doi.org/10.3390/math10142390 - 7 Jul 2022
Cited by 3 | Viewed by 1518
Abstract
The paper proposes a method for implementing a universal neural network processor with hysteresis dynamics. This processor allows a wide range of heterogeneous tasks in real time to be performed without reprogramming and changing their internal structure. Adding hysteresis behavior to the system [...] Read more.
The paper proposes a method for implementing a universal neural network processor with hysteresis dynamics. This processor allows a wide range of heterogeneous tasks in real time to be performed without reprogramming and changing their internal structure. Adding hysteresis behavior to the system makes it possible to increase resistance to external influences, the complexity as well as non-linearity of intelligent output. The paper discusses the use of this processor as part of an on-board intelligent avionics system. Full article
(This article belongs to the Special Issue Nonlinear Dynamics Systems with Hysteresis)
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26 pages, 9818 KiB  
Article
Effect of the Impact Coefficient of Restitution on the Nonlinear Dynamics Phenomenon of Flat-Faced Follower Mechanism with Clearance
by Louay S. Yousuf
Mathematics 2022, 10(11), 1922; https://doi.org/10.3390/math10111922 - 3 Jun 2022
Cited by 2 | Viewed by 1741
Abstract
The effect of impact coefficient of restitution on the nonlinear response of the follower is investigated at different follower guides’ clearances, different cam speeds and different followers’ offsets. The impact between the cam and the follower and between the follower and its guide [...] Read more.
The effect of impact coefficient of restitution on the nonlinear response of the follower is investigated at different follower guides’ clearances, different cam speeds and different followers’ offsets. The impact between the cam and the follower and between the follower and its guide is considered in the presence of coefficient of restitution and follower offset. The nonlinear dynamics phenomenon of the follower due to the impact coefficient of restitution is detected using the approach of largest Lyapunov exponent. Moreover, the chaotic phenomenon is detected using a phase-plane diagram. The numerical simulation of the nonlinear response of the follower is calculated using the SolidWorks program. The chaotic phenomenon in the cam follower system is increased with the increase of the impact coefficient of restitution value in which the potential energy of the follower has been decreased in the presence of follower offset. The chaotic motion of the follower response occurs due to the increase in cam speeds, follower’s offsets, follower guides’ clearances and impact coefficient of restitution. Full article
(This article belongs to the Special Issue Nonlinear Dynamics Systems with Hysteresis)
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12 pages, 365 KiB  
Article
On a High-Precision Method for Studying Attractors of Dynamical Systems and Systems of Explosive Type
by Alexander N. Pchelintsev
Mathematics 2022, 10(8), 1207; https://doi.org/10.3390/math10081207 - 7 Apr 2022
Cited by 2 | Viewed by 2463
Abstract
The author of this article considers a numerical method that uses high-precision calculations to construct approximations to attractors of dynamical systems of chaotic type with a quadratic right-hand side, as well as to find the vertical asymptotes of solutions of systems of explosive [...] Read more.
The author of this article considers a numerical method that uses high-precision calculations to construct approximations to attractors of dynamical systems of chaotic type with a quadratic right-hand side, as well as to find the vertical asymptotes of solutions of systems of explosive type. A special case of such systems is the population explosion model. A theorem on the existence of asymptotes is proved. The extension of the numerical method for piecewise smooth systems is described using the Chua system as an example, as well as systems with hysteresis. Full article
(This article belongs to the Special Issue Nonlinear Dynamics Systems with Hysteresis)
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17 pages, 309 KiB  
Article
On the Periodic Solutions for the Perturbed Spatial Quantized Hill Problem
by Elbaz I. Abouelmagd, Sawsan Alhowaity, Zouhair Diab, Juan L. G. Guirao and Mahmoud H. Shehata
Mathematics 2022, 10(4), 614; https://doi.org/10.3390/math10040614 - 17 Feb 2022
Cited by 12 | Viewed by 2059
Abstract
In this work, we investigated the differences and similarities among some perturbation approaches, such as the classical perturbation theory, Poincaré–Lindstedt technique, multiple scales method, the KB averaging method, and averaging theory. The necessary conditions to construct the periodic solutions for the spatial quantized [...] Read more.
In this work, we investigated the differences and similarities among some perturbation approaches, such as the classical perturbation theory, Poincaré–Lindstedt technique, multiple scales method, the KB averaging method, and averaging theory. The necessary conditions to construct the periodic solutions for the spatial quantized Hill problem—in this context, the periodic solutions emerging from the equilibrium points for the spatial Hill problem—were evaluated by using the averaging theory, under the perturbation effect of quantum corrections. This model can be used to develop a Lunar theory and the families of periodic orbits in the frame work for the spatial quantized Hill problem. Thereby, these applications serve to reinforce the obtained results on these periodic solutions and gain its own significance. Full article
(This article belongs to the Special Issue Nonlinear Dynamics Systems with Hysteresis)
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