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Dynamical Systems and System Analysis
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Dynamical Systems and System Analysis

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "C2: Dynamical Systems".

Deadline for manuscript submissions: closed (31 July 2023) | Viewed by 26891

Special Issue Editor

Prof. Dr. Vasyl Martsenyuk

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Guest Editor
Department of Computer Science and Automatics, University of Bielsko-Biala, Willowa 2, 43-309 Bielsko-Biala, Poland
Interests: dynamical systemss; system analysis; medical informatics; data mining; cybernetics
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

This Special Issue, devoted to dynamical systems and system analysis, will bring together the newest research achievements of scholars studying the models describing time dependences of the state in appropriate system space, focusing on their qualitative behavior. The issue will cover all aspects of this topic, starting with the problem of parameter identification, stability investigation, optimal control, nonlinear dynamics, studies using the theory of deterministic chaos, or any other techniques which can be applied in the field of the qualitative analysis of dynamical systems and their applications.

The Editors of this Special Issue are pleased to invite authors to submit their original results related to dynamical systems which can be described with the help of differential or difference equations, partial and lattice differential equations, differential equations on graphs, delayed equations, and even studies related to dynamical systems on time scales. We await the latest results related to the identification of the parameters of dynamical systems, initial and boundary value problems, local and global asymptotic stability, Lyapunov theory, construction of an optimal controller, and numerical research of nonlinear dynamics. We believe that researchers are eager to see how differential, difference, and equations on time scales are used in a variety of applications in engineering, economics and finance, biology and medicine, and what their importance in the field is. This Special Issue will also publish contributions related exclusively to system analysis, and we hope to find, among the results, new approaches for the qualitative theory of dynamical systems that could inspire further achievements in the field.

Prof. Dr. Vasyl Martsenyuk
Guest Editor

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Keywords

  • Dynamical System
  • Qualitative Theory
  • Stability
  • Nonlinear Analysis
  • Optimal Control
  • Differential Equations
  • Difference Equations
  • Delayed Differential Equations
  • Time Scales
  • Spatial-Temporal System

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Published Papers (11 papers)

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23 pages, 816 KiB  
Article
Language-Based Opacity Verification in Partially Observed Petri Nets through Linear Constraints
by Ikram Saadaoui, Abdeldjalil Labed, Zhiwu Li, Ahmed M. El-Sherbeeny and Huiran Du
Mathematics 2023, 11(18), 3880; https://doi.org/10.3390/math11183880 - 11 Sep 2023
Cited by 3 | Viewed by 1533
Abstract
Information security is an important area of concern in modern computer-integrated systems. It involves implementing preventative measures to protect confidential data from potential vulnerabilities, such as unauthorized access, secret disclosure, modification, or destruction. Considering such threats, we investigate a particular confidentiality property called [...] Read more.
Information security is an important area of concern in modern computer-integrated systems. It involves implementing preventative measures to protect confidential data from potential vulnerabilities, such as unauthorized access, secret disclosure, modification, or destruction. Considering such threats, we investigate a particular confidentiality property called opacity, which specifies a system’s ability to cover its ‘secret’ data from being interfered with by outside observers, termed as intruders. This paper discusses language-based opacity formulation and verification in the context of discrete event systems represented by partially observed Petri nets. In this context, we identify two opacity properties, called consistency and non-secrecy; then, we exploit the mathematical characterization of a net system, to separately check each property, by specifying two feasibility problems. The proposed method is carried out for two distinct settings of a system. The first setting is centralized, where an intruder is granted complete information about the system structure but a partial observation of its behavior. The second setting is decentralized, where a group of intruders cooperates to reveal the secret language, by using a coordinator. Finally, experimental findings are given, to demonstrate the proficiency of the proposed approach. Full article
(This article belongs to the Special Issue Dynamical Systems and System Analysis)
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17 pages, 1066 KiB  
Article
Bacterial Competition in the Presence of a Virus in a Chemostat
by Amer Hassan Albargi and Miled El Hajji
Mathematics 2023, 11(16), 3530; https://doi.org/10.3390/math11163530 - 15 Aug 2023
Cited by 12 | Viewed by 1463
Abstract
We derive a mathematical model that describes the competition of two populations in a chemostat in the presence of a virus. We suppose that only one population is affected by the virus. We also suppose that the substrate is continuously added to the [...] Read more.
We derive a mathematical model that describes the competition of two populations in a chemostat in the presence of a virus. We suppose that only one population is affected by the virus. We also suppose that the substrate is continuously added to the bioreactor. We obtain a model taking the form of an “SI” epidemic model using general increasing growth rates of bacteria on the substrate and a general increasing incidence rate for the viral infection. The stability of the steady states was carried out. The system can have multiple steady states with which we can determine the necessary and sufficient conditions for both existence and local stability. We exclude the possibility of periodic orbits and we prove the uniform persistence of both species. Finally, we give some numerical simulations that validate the obtained results. Full article
(This article belongs to the Special Issue Dynamical Systems and System Analysis)
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14 pages, 483 KiB  
Article
(Iq)–Stability and Uniform Convergence of the Solutions of Singularly Perturbed Boundary Value Problems
by Robert Vrabel
Mathematics 2023, 11(12), 2717; https://doi.org/10.3390/math11122717 - 15 Jun 2023
Viewed by 966
Abstract
In this paper, using the notion of (Iq)–stability and the method of a priori estimates, known as the method of lower and upper solutions, the sufficient conditions guaranteeing uniform convergence of solutions to the solution of a reduced problem on [...] Read more.
In this paper, using the notion of (Iq)–stability and the method of a priori estimates, known as the method of lower and upper solutions, the sufficient conditions guaranteeing uniform convergence of solutions to the solution of a reduced problem on the entire interval [a,b] have been established for four different types of boundary conditions for a singularly perturbed differential equation εy″=f(x,y,y′), a≤x≤b. In the second part of the paper, by employing the Peano phenomenon, we analyzed the structure of the solutions of the reduced problem f(x,y,y′)=0. Full article
(This article belongs to the Special Issue Dynamical Systems and System Analysis)
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13 pages, 2862 KiB  
Article
Delay-Dependent Stability Region for the Distributed Coordination of Delayed Fractional-Order Multi-Agent Systems
by Abbasali Koochakzadeh, Mojtaba Naderi Soorki, Aydin Azizi, Kamran Mohammadsharifi and Mohammadreza Riazat
Mathematics 2023, 11(5), 1267; https://doi.org/10.3390/math11051267 - 6 Mar 2023
Cited by 6 | Viewed by 1654
Abstract
Delay and especially delay in the transmission of agents’ information, is one of the most important causes of disruption to achieving consensus in a multi-agent system. This paper deals with achieving consensus in delayed fractional-order multi-agent systems (FOMAS). The aim in the present [...] Read more.
Delay and especially delay in the transmission of agents’ information, is one of the most important causes of disruption to achieving consensus in a multi-agent system. This paper deals with achieving consensus in delayed fractional-order multi-agent systems (FOMAS). The aim in the present note is to find the exact maximum allowable delay in a FOMAS with non-uniform delay, i.e., the case in which the interactions between agents are subject to non-identical communication time-delays. By proving a stability theorem, the results available for non-delayed networked fractional-order systems are extended for the case in which interaction links have nonequal communication time-delays. In this extension by considering a time-delay coordination algorithm, necessary and sufficient conditions on the time delays and interaction graph are presented to guarantee the coordination. In addition, the delay-dependent stability region is also obtained. Finally, the dependency of the maximum allowable delay on two parameters, the agent fractional-order and the largest eigenvalue of the graph Laplacian matrix, is exactly determined. Numerical simulation results are given to confirm the proposed methodologies. Full article
(This article belongs to the Special Issue Dynamical Systems and System Analysis)
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27 pages, 437 KiB  
Article
Dynamics of Classical Solutions of a Two-Stage Structured Population Model with Nonlocal Dispersal
by Maria A. Onyido, Rachidi B. Salako, Markjoe O. Uba and Cyril I. Udeani
Mathematics 2023, 11(4), 925; https://doi.org/10.3390/math11040925 - 12 Feb 2023
Cited by 3 | Viewed by 1709
Abstract
We study the dynamics of classical solutions of a two-stage structured population model with nonlocal dispersal in a spatially heterogeneous environment and address the question of the persistence of the species. In particular, we show that the species’ persistence is completely determined by [...] Read more.
We study the dynamics of classical solutions of a two-stage structured population model with nonlocal dispersal in a spatially heterogeneous environment and address the question of the persistence of the species. In particular, we show that the species’ persistence is completely determined by the sign of the principal spectrum point, λp, of the linearized system at the trivial solution: the species goes extinct if λp≤0, while it persists uniformly in space if λp>0. Sufficient conditions are provided to guarantee the existence, uniqueness, and stability of a positive steady state when the parameters are spatially heterogeneous. Furthermore, when the model parameters are spatially homogeneous, we show that the unique positive equilibrium is globally stable with respect to positive perturbations. Full article
(This article belongs to the Special Issue Dynamical Systems and System Analysis)
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13 pages, 373 KiB  
Article
Local H∞ Control for Continuous-Time T-S Fuzzy Systems via Generalized Non-Quadratic Lyapunov Functions
by Guolin Hu, Jian Zhang and Zhiguo Yan
Mathematics 2022, 10(19), 3438; https://doi.org/10.3390/math10193438 - 21 Sep 2022
Cited by 2 | Viewed by 1581
Abstract
This paper further develops a relaxed method to reduce conservatism in H∞ feedback control for continuous-time T-S fuzzy systems via a generalized non-quadratic Lyapunov function. Different from the results of some exisiting works, the generalized H∞ state feedback controller is designed. [...] Read more.
This paper further develops a relaxed method to reduce conservatism in H∞ feedback control for continuous-time T-S fuzzy systems via a generalized non-quadratic Lyapunov function. Different from the results of some exisiting works, the generalized H∞ state feedback controller is designed. The relaxed stabilization conditions are obtained by applying Finsler’s lemma with the homogenous polynomial multipliers, and the H∞ performance is acquired by solving an optimization problem. In addition, the proposed method could be expanded to handle other control problems for fuzzy systems. Two examples are given to show the validity of the proposed results. Full article
(This article belongs to the Special Issue Dynamical Systems and System Analysis)
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27 pages, 836 KiB  
Article
The Mathematical Model of Cyclic Signals in Dynamic Systems as a Cyclically Correlated Random Process
by Serhii Lupenko
Mathematics 2022, 10(18), 3406; https://doi.org/10.3390/math10183406 - 19 Sep 2022
Cited by 7 | Viewed by 3556
Abstract
This work is devoted to the procedure for constructing of a cyclically correlated random process of a continuous argument as a mathematical model of cyclic signals in dynamic systems, which makes it possible to consistently describe cyclic stochastic signals, both with regular and [...] Read more.
This work is devoted to the procedure for constructing of a cyclically correlated random process of a continuous argument as a mathematical model of cyclic signals in dynamic systems, which makes it possible to consistently describe cyclic stochastic signals, both with regular and irregular rhythms, not separating them, but complementing them within the framework of a single integrated model. The class of cyclically correlated random processes includes the subclass of cyclostationary (periodically) correlated random processes, which enable the use of a set of powerful methods of analysis and the forecasting of cyclic signals with a stable rhythm. Mathematical structures that model the cyclic, phase and rhythmic structures of a cyclically correlated random process are presented. The sufficient and necessary conditions that the structural function and the rhythm function of the cyclically correlated random process must satisfy have been established. The advantages of the cyclically correlated random process in comparison with other mathematical models of cyclic signals with a variable rhythm are given. The obtained results contribute to the emergence of a more complete and rigorous theory of this class of random processes and increase the validity of the methods of their analysis and computer simulation. Full article
(This article belongs to the Special Issue Dynamical Systems and System Analysis)
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17 pages, 1839 KiB  
Article
Parameterization of a Novel Nonlinear Estimator for Uncertain SISO Systems with Noise Scenario
by Ahmad Taher Azar, Farah Ayad Abdul-Majeed, Hasan Sh. Majdi, Ibrahim A. Hameed, Nashwa Ahmad Kamal, Anwar Jaafar Mohamad Jawad, Ali Hashim Abbas, Wameedh Riyadh Abdul-Adheem and Ibraheem Kasim Ibraheem
Mathematics 2022, 10(13), 2261; https://doi.org/10.3390/math10132261 - 28 Jun 2022
Cited by 4 | Viewed by 1822
Abstract
Dynamic observers are commonly used in feedback loops to estimate the system’s states from available control inputs and measured outputs. The presence of measurement noise degrades the performance of the observer and consequently degrades the performance of the controlled system. This paper presents [...] Read more.
Dynamic observers are commonly used in feedback loops to estimate the system’s states from available control inputs and measured outputs. The presence of measurement noise degrades the performance of the observer and consequently degrades the performance of the controlled system. This paper presents a novel nonlinear higher-order extended state observer (NHOESO) for efficient state and disturbance estimation in presence of measurement noise for nonlinear single-input–single-output systems. The proposed nonlinear function allows a fast reconstruction of the system’s states and is robust against uncertainties and measurement noise. An analytical parameterization technique is proposed to parameterize the coefficients of the proposed nonlinear higher-order extended state observer in the case of measurement noise in the output signal. Several scenarios are simulated to demonstrate the effectiveness of the proposed observer. Full article
(This article belongs to the Special Issue Dynamical Systems and System Analysis)
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16 pages, 4676 KiB  
Article
Dynamic Modeling and Vibration Characteristics Analysis of Deep-Groove Ball Bearing, Considering Sliding Effect
by Fanjie Li, Xiaopeng Li and Dongyang Shang
Mathematics 2021, 9(19), 2408; https://doi.org/10.3390/math9192408 - 28 Sep 2021
Cited by 9 | Viewed by 3222
Abstract
To study the vibration characteristics of deep-groove ball bearing, considering the influence of sliding, the dynamic model of the DGB 6205 system is established in this paper. The DGB 6205 system model includes the movement of the bearing inner ring in the X [...] Read more.
To study the vibration characteristics of deep-groove ball bearing, considering the influence of sliding, the dynamic model of the DGB 6205 system is established in this paper. The DGB 6205 system model includes the movement of the bearing inner ring in the X and Y directions, the rotation of the cage, the rotation movement of each ball, the revolution movement of each ball and the movement along the radial direction of each ball. Based on the system model, the differential equations of motion of the system are established, and the correctness of the model is verified by experiment. The slip characteristics of the DGB 6205 system are studied by numerical simulation. At the same time, the influence of time-varying load on the vibration characteristics of the system is studied. Then, the sensitivity of system parameters is analyzed. The results show that the sliding speed between the ball and the inner raceway is greater than that between the ball and the outer raceway. The radial vibration response of DGB 6205 system under time-varying load is less than that under constant load. The increase of radial clearance will increase the vibration response of DGB 6205 system. Full article
(This article belongs to the Special Issue Dynamical Systems and System Analysis)
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20 pages, 4321 KiB  
Article
Dynamic Characteristics Analysis of ISD Suspension System under Different Working Conditions
by Xiaopeng Li, Fanjie Li and Dongyang Shang
Mathematics 2021, 9(12), 1345; https://doi.org/10.3390/math9121345 - 10 Jun 2021
Cited by 12 | Viewed by 3101
Abstract
The “inerter-spring-damper” (ISD) suspension system is a suspension system composed of an inerter, spring, and damper. To study the ride comfort and stability of the vehicle by using the ISD suspension system, a vehicle model with ISD suspension is established in this paper. [...] Read more.
The “inerter-spring-damper” (ISD) suspension system is a suspension system composed of an inerter, spring, and damper. To study the ride comfort and stability of the vehicle by using the ISD suspension system, a vehicle model with ISD suspension is established in this paper. The vehicle model including vertical, pitch, roll, and yaw motion of the vehicle body. Based on the vehicle model, the differential equation of motion with ISD suspension is obtained. The dynamic responses of the ISD suspension system are investigated by using different road excitations. At the same time, the influence of coupled excitation and single excitation on the vibration reduction performance of the ISD suspension system is studied. Then, the dynamic responses of ISD suspension and passive suspension are compared, and the improvement of comprehensive vibration reduction performance of ISD suspension system is quantitatively analyzed. The numerical results illustrate the ISD suspension has the optimal vehicle speed under different road excitations, and the comprehensive vibration reduction performance of the ISD suspension is the best when driving at the optimal vehicle speed. Under different types of road excitation, ISD suspension shows excellent comprehensive vibration reduction performance. ISD suspension is more suitable for vibration reduction of complex roads than that of a single road. Full article
(This article belongs to the Special Issue Dynamical Systems and System Analysis)
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21 pages, 1081 KiB  
Article
Research on Optimization of Array Honeypot Defense Strategies Based on Evolutionary Game Theory
by Leyi Shi, Xiran Wang and Huiwen Hou
Mathematics 2021, 9(8), 805; https://doi.org/10.3390/math9080805 - 8 Apr 2021
Cited by 21 | Viewed by 4121
Abstract
Honeypot has been regarded as an active defense technology that can deceive attackers by simulating real systems. However, honeypot is actually a static network trap with fixed disposition, which is easily identified by anti-honeypot technology. Thus, honeypot is a “passive” active defense technology. [...] Read more.
Honeypot has been regarded as an active defense technology that can deceive attackers by simulating real systems. However, honeypot is actually a static network trap with fixed disposition, which is easily identified by anti-honeypot technology. Thus, honeypot is a “passive” active defense technology. Dynamic honeypot makes up for the shortcomings of honeypot, which dynamically adjusts defense strategies with the attack of hackers. Therefore, the confrontation between defenders and attackers is a strategic game. This paper focuses on the non-cooperative evolutionary game mechanism of bounded rationality, aiming to improve the security of the array honeypot system through the evolutionarily stable strategies derived from the evolutionary game model. First, we construct a three-party evolutionary game model of array honeypot, which is composed of defenders, attackers and legitimate users. Secondly, we formally describe the strategies and revenues of players in the game, and build the three-party game payoff matrices. Then the evolutionarily stable strategy is obtained by analyzing the Replicator Dynamics of various parties. In addition, we discuss the equilibrium condition to get the influence of the number of servers N on the stability of strategy evolution. MATLAB and Gambit simulation experiment results show that deduced evolutionarily stable strategies are valid in resisting attackers. Full article
(This article belongs to the Special Issue Dynamical Systems and System Analysis)
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