Nonlinear Dynamics and Chaos Theory

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Computational and Applied Mathematics".

Deadline for manuscript submissions: closed (31 December 2023) | Viewed by 20325

Special Issue Editors

School of Mathematics and Statistics, Northwestern Polytechnical University, Xi'an 710072, China
Interests: chaos control and synchronization; random dynamical systems; complex system; molecular dynamics simulation
School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001, China
Interests: delay differential equations; nonlinear time series analysis and predication; bifurcation; stability; chaos; synchronization; nonlinear dynamics; reservoir computing
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Nonlinear dynamics and chaos theory plays an ever-important role in the exploration of science and technology. It aims to study the mechanisms and applications of rich dynamical phenomena in the real world. Applications and examples of nonlinear dynamical systems are ubiquitous over a diverse set of areas. The growing demand for determining the laws and mechanisms of rich nonlinear dynamical phenomena as well as their applications to diverse fields is inducing growth in the demand for nonlinear dynamics and chaos theory.  

In this Special Issue, we aim to present the recent developments in the theory and applications of nonlinear dynamics and chaos, with a special emphasis on stability, bifurcation, chaos, hidden and multi-scroll attractors, image encryption, chaotic circuits, high-speed trains, hydraulic cylinders, neural networks, and environmental protection.

This Special Issue will accept high-quality papers containing original research results and review articles of exceptional merit in the following fields:

  • Stability, bifurcation and chaos;
  • Extreme multi-stability, and hidden and multi-scroll attractors;
  • Image encryption and chaotic circuits;
  • Neural networks;
  • High-speed trains;
  • Hydraulic cylinders;
  • Mathematical biology.

Prof. Dr. Youming Lei
Prof. Dr. Lijun Pei
Guest Editors

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Keywords

  • stability
  • bifurcation
  • chaos
  • hidden and multi-scroll attractors
  • singularities

Published Papers (12 papers)

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Research

19 pages, 5229 KiB  
Article
An Unconditionally Stable Integration Method for Structural Nonlinear Dynamic Problems
by Chuanguo Jia, Hongchen Su, Weinan Guo, Yutao Li, Biying Wu and Yingqi Gou
Mathematics 2023, 11(13), 2987; https://doi.org/10.3390/math11132987 - 04 Jul 2023
Viewed by 932
Abstract
This paper presents an unconditionally stable integration method, which introduces a linearly implicit algorithm featuring an explicit displacement expression. The technique that is being considered integrates one Newton iteration into the mean acceleration method. The stability of the proposed algorithm in solving equations [...] Read more.
This paper presents an unconditionally stable integration method, which introduces a linearly implicit algorithm featuring an explicit displacement expression. The technique that is being considered integrates one Newton iteration into the mean acceleration method. The stability of the proposed algorithm in solving equations of motion containing nonlinear restoring force and nonlinear damping force is analyzed using the root locus method. The objective of this investigation was to assess the accuracy and consistency of the proposed approach in contrast to the Chang method and the CR method. This is achieved by analyzing the dynamic response of three distinct structures: a three-layer shear structure model outfitted with viscous dampers, a three-layer shear structure model featuring metal dampers, and an eight-story planar frame structure. Empirical evidence indicates that the algorithm in question exhibits a notable degree of precision and robustness when applied to nonlinear dynamic problem-solving. Full article
(This article belongs to the Special Issue Nonlinear Dynamics and Chaos Theory)
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22 pages, 73567 KiB  
Article
Bifurcation Analysis, Synchronization and FPGA Implementation of a New 3-D Jerk System with a Stable Equilibrium
by Sundarapandian Vaidyanathan, Ahmad Taher Azar, Ibrahim A. Hameed, Khaled Benkouider, Esteban Tlelo-Cuautle, Brisbane Ovilla-Martinez, Chang-Hua Lien and Aceng Sambas
Mathematics 2023, 11(12), 2623; https://doi.org/10.3390/math11122623 - 08 Jun 2023
Cited by 4 | Viewed by 1104
Abstract
This research paper addresses the modelling of a new 3-D chaotic jerk system with a stable equilibrium. Such chaotic systems are known to exhibit hidden attractors. After the modelling of the new jerk system, a detailed bifurcation analysis has been performed for the [...] Read more.
This research paper addresses the modelling of a new 3-D chaotic jerk system with a stable equilibrium. Such chaotic systems are known to exhibit hidden attractors. After the modelling of the new jerk system, a detailed bifurcation analysis has been performed for the new chaotic jerk system with a stable equilibrium. It is shown that the new jerk system has multistability with coexisting attractors. Next, we apply backstepping control for the synchronization design of a pair of new jerk systems with a stable equilibrium taken as the master-slave chaotic systems. Lyapunov stability theory is used to establish the synchronization results for the new jerk system with a stable equilibrium. Finally, we show that the FPGA design of the new jerk system with a stable equilibrium can be implemented using the FPGA Zybo Z7-20 development board. The design of the new jerk system consists of multipliers, adders and subtractors. It is observed that the experimental attractors are in good agreement with simulation results. Full article
(This article belongs to the Special Issue Nonlinear Dynamics and Chaos Theory)
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14 pages, 7111 KiB  
Article
Chaos Synchronization of Two Györgyi–Field Systems for the Belousov–Zhabotinsky Chemical Reaction
by Andrei Victor Oancea and Ilie Bodale
Mathematics 2022, 10(21), 3947; https://doi.org/10.3390/math10213947 - 24 Oct 2022
Cited by 1 | Viewed by 946
Abstract
Chemical reactions with oscillating behavior can present a chaos state in specific conditions. In this study, we analyzed the dynamic of the chaotic Belousov–Zhabotinsky (BZ) reaction using the Györgyi–Field model in order to identify the conditions of the chaos behavior. We studied the [...] Read more.
Chemical reactions with oscillating behavior can present a chaos state in specific conditions. In this study, we analyzed the dynamic of the chaotic Belousov–Zhabotinsky (BZ) reaction using the Györgyi–Field model in order to identify the conditions of the chaos behavior. We studied the behavior of the reaction under different parameters that included both a low and high flux of chemical species. We performed our analysis of the flow regime in the conditions of an open reaction system, as this provides information about the behavior of the reaction over time. The proposed method for determining the favorable conditions for obtaining the state of chaos is based on the time evolution of the intermediate species and phase portraits. The synchronization of two Györgyi–Field systems based on the adaptive feedback method of control is presented in this work. The transient time until synchronization depends on the initial conditions of the two systems and on the strength of the controllers. Among the areas of interest for possible applications of the control method described in this paper, we can include identification of the reaction parameters and the extension to the other chaotic systems. Full article
(This article belongs to the Special Issue Nonlinear Dynamics and Chaos Theory)
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22 pages, 4644 KiB  
Article
Global Dynamics of the Vibrating System of a Tristable Piezoelectric Energy Harvester
by Yijun Zhu and Huilin Shang
Mathematics 2022, 10(16), 2894; https://doi.org/10.3390/math10162894 - 12 Aug 2022
Cited by 4 | Viewed by 1118
Abstract
Global dynamics of a piezoelectric energy harvester with tristable potential is investigated. The dynamical model of a cantilever beam energy harvester is considered; its static bifurcation is also discussed. Multiple intra-well attractors and their basins of attraction are presented to discuss the mechanism [...] Read more.
Global dynamics of a piezoelectric energy harvester with tristable potential is investigated. The dynamical model of a cantilever beam energy harvester is considered; its static bifurcation is also discussed. Multiple intra-well attractors and their basins of attraction are presented to discuss the mechanism of multistability and its initial sensitivity. Moreover, the Melnikov method is applied to present the conditions for global bifurcations and the induced complex dynamics. The results show that the variation of coefficients of the polynomial may affect the number and shapes of potential wells, while the increase of the excitation amplitude may trigger multistability around one equilibrium, initial-sensitive jump, inter-well attractor and chaos. The results may provide some theoretical reference for increasing the working performance of energy harvesters. Full article
(This article belongs to the Special Issue Nonlinear Dynamics and Chaos Theory)
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20 pages, 393 KiB  
Article
Asymptotic Hyperstability and Input–Output Energy Positivity of a Single-Input Single-Output System Which Incorporates a Memoryless Non-Linear Device in the Feed-Forward Loop
by Manuel De la Sen
Mathematics 2022, 10(12), 2051; https://doi.org/10.3390/math10122051 - 13 Jun 2022
Viewed by 1068
Abstract
This paper visualizes the role of hyperstable controllers in the closed-loop asymptotic stability of a single-input single-output system subject to any nonlinear and eventually time-varying controller within the hyperstable class. The feed-forward controlled loop (or controlled plant) contains a strongly strictly positive real [...] Read more.
This paper visualizes the role of hyperstable controllers in the closed-loop asymptotic stability of a single-input single-output system subject to any nonlinear and eventually time-varying controller within the hyperstable class. The feed-forward controlled loop (or controlled plant) contains a strongly strictly positive real transfer function in parallel with a non-linear and memory-free device. The properties of positivity and boundedness of the input–output energy are examined based on the “ad hoc” use of the Rayleigh energy theorem on the truncated relevant signals for finite time intervals. The cases of minimal and non-minimal state-space realizations of the linear part are characterized from a global asymptotic stability (asymptotic hyperstability) point of view. Some related extended results are obtained for the case when the linear part is both positive real and externally positive and for the case of incorporation of other linear components which are stable but not necessarily positive real. Full article
(This article belongs to the Special Issue Nonlinear Dynamics and Chaos Theory)
20 pages, 47952 KiB  
Article
Chaotic and Hyperchaotic Dynamics of a Clapp Oscillator
by Jiri Petrzela
Mathematics 2022, 10(11), 1868; https://doi.org/10.3390/math10111868 - 30 May 2022
Cited by 8 | Viewed by 1614
Abstract
This paper describes recent findings achieved during a numerical investigation of the circuit known as the Clapp oscillator. By considering the generalized bipolar transistor as an active element and after applying the search-for-chaos optimization approach, parameter regions that lead to either chaotic or [...] Read more.
This paper describes recent findings achieved during a numerical investigation of the circuit known as the Clapp oscillator. By considering the generalized bipolar transistor as an active element and after applying the search-for-chaos optimization approach, parameter regions that lead to either chaotic or hyperchaotic dynamics were discovered. For starters, the two-port that represents the transistor was firstly assumed to have a polynomial-forward trans-conductance; then the shape of trans-conductance changes into the piecewise-linear characteristics. Both cases cause vector field symmetry and allow the coexistence of several different attractors. Chaotic and hyperchaotic behavior were deeply analyzed by using standard numerical tools such as Lyapunov exponents, basins of attraction, bifurcation diagrams, and solution sensitivity. The structural stability of strange attractors observed numerically was finally proved via a real practical experiment: a flow-equivalent chaotic oscillator was constructed as the lumped electronic circuit, and desired attractors were captured and provided as oscilloscope screenshots. Full article
(This article belongs to the Special Issue Nonlinear Dynamics and Chaos Theory)
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13 pages, 4824 KiB  
Article
Research on Rumor-Spreading Model with Holling Type III Functional Response
by Yanhui Wei, Liang’an Huo and Hongguang He
Mathematics 2022, 10(4), 632; https://doi.org/10.3390/math10040632 - 18 Feb 2022
Cited by 1 | Viewed by 1212
Abstract
In this paper, a rumor-spreading model with Holling type III functional response was established. The existence of the equilibrium points was discussed. According to the Routh–Hurwitz criteria, the locally asymptotic stability of the equilibrium points was analyzed. The global stability of the equilibrium [...] Read more.
In this paper, a rumor-spreading model with Holling type III functional response was established. The existence of the equilibrium points was discussed. According to the Routh–Hurwitz criteria, the locally asymptotic stability of the equilibrium points was analyzed. The global stability of the equilibrium points was proven based on Lasalle’s invariance principle and generalized Bendixson–Dulac theorem. Numerical simulations were carried out to illustrate the impact of different parameters on the spread of rumors. When the stifling rate λ increases, or the predation capacity β or the system coming rate k decreases, the number of rumor-spreaders is reduced to extinction. The results provide theory, method and decision support for effectively controlling the spread of rumors. Full article
(This article belongs to the Special Issue Nonlinear Dynamics and Chaos Theory)
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20 pages, 15188 KiB  
Article
Controlling Chaos in Van Der Pol Dynamics Using Signal-Encoded Deep Learning
by Hanfeng Zhai and Timothy Sands
Mathematics 2022, 10(3), 453; https://doi.org/10.3390/math10030453 - 30 Jan 2022
Cited by 6 | Viewed by 3911
Abstract
Controlling nonlinear dynamics is a long-standing problem in engineering. Harnessing known physical information to accelerate or constrain stochastic learning pursues a new paradigm of scientific machine learning. By linearizing nonlinear systems, traditional control methods cannot learn nonlinear features from chaotic data for use [...] Read more.
Controlling nonlinear dynamics is a long-standing problem in engineering. Harnessing known physical information to accelerate or constrain stochastic learning pursues a new paradigm of scientific machine learning. By linearizing nonlinear systems, traditional control methods cannot learn nonlinear features from chaotic data for use in control. Here, we introduce Physics-Informed Deep Operator Control (PIDOC), and by encoding the control signal and initial position into the losses of a physics-informed neural network (PINN), the nonlinear system is forced to exhibit the desired trajectory given the control signal. PIDOC receives signals as physics commands and learns from the chaotic data output from the nonlinear van der Pol system, where the output of the PINN is the control. Applied to a benchmark problem, PIDOC successfully implements control with a higher stochasticity for higher-order terms. PIDOC has also been proven to be capable of converging to different desired trajectories based on case studies. Initial positions slightly affect the control accuracy at the beginning stage yet do not change the overall control quality. For highly nonlinear systems, PIDOC is not able to execute control with a high accuracy compared with the benchmark problem. The depth and width of the neural network structure do not greatly change the convergence of PIDOC based on case studies of van der Pol systems with low and high nonlinearities. Surprisingly, enlarging the control signal does not help to improve the control quality. The proposed framework can potentially be applied to many nonlinear systems for nonlinear controls. Full article
(This article belongs to the Special Issue Nonlinear Dynamics and Chaos Theory)
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17 pages, 3107 KiB  
Article
Computational Analysis and Bifurcation of Regular and Chaotic Ca2+ Oscillations
by Xinxin Qie and Quanbao Ji
Mathematics 2021, 9(24), 3324; https://doi.org/10.3390/math9243324 - 20 Dec 2021
Cited by 3 | Viewed by 1906
Abstract
This study investigated the stability and bifurcation of a nonlinear system model developed by Marhl et al. based on the total Ca2+ concentration among three different Ca2+ stores. In this study, qualitative theories of center manifold and bifurcation were used to [...] Read more.
This study investigated the stability and bifurcation of a nonlinear system model developed by Marhl et al. based on the total Ca2+ concentration among three different Ca2+ stores. In this study, qualitative theories of center manifold and bifurcation were used to analyze the stability of equilibria. The bifurcation parameter drove the system to undergo two supercritical bifurcations. It was hypothesized that the appearance and disappearance of Ca2+ oscillations are driven by them. At the same time, saddle-node bifurcation and torus bifurcation were also found in the process of exploring bifurcation. Finally, numerical simulation was carried out to determine the validity of the proposed approach by drawing bifurcation diagrams, time series, phase portraits, etc. Full article
(This article belongs to the Special Issue Nonlinear Dynamics and Chaos Theory)
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12 pages, 2014 KiB  
Article
Quasi-Periodic Oscillations of Roll System in Corrugated Rolling Mill in Resonance
by Dongping He, Huidong Xu, Tao Wang and Zhihua Wang
Mathematics 2021, 9(24), 3201; https://doi.org/10.3390/math9243201 - 11 Dec 2021
Viewed by 1622
Abstract
This paper investigates quasi-periodic oscillations of roll system in corrugated rolling mill in resonance. The two-degree of freedom vertical nonlinear mathematical model of roller system is established by considering the nonlinear damping and nonlinear stiffness within corrugated interface of corrugated rolling mill. In [...] Read more.
This paper investigates quasi-periodic oscillations of roll system in corrugated rolling mill in resonance. The two-degree of freedom vertical nonlinear mathematical model of roller system is established by considering the nonlinear damping and nonlinear stiffness within corrugated interface of corrugated rolling mill. In order to investigate the quasi-periodic oscillations at the resonance points, the Poincaré map is established by solving the power series solution of dynamic equations. Based on the Poincaré map, the existence and stability of quasi-periodic oscillations from the Neimark-Sacker bifurcation in the case of resonance are analyzed. The numerical simulation further verifies the correctness of the theoretical analysis. Full article
(This article belongs to the Special Issue Nonlinear Dynamics and Chaos Theory)
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13 pages, 25297 KiB  
Article
Fast-Slow Coupling Dynamics Behavior of the van der Pol-Rayleigh System
by Danjin Zhang and Youhua Qian
Mathematics 2021, 9(23), 3004; https://doi.org/10.3390/math9233004 - 23 Nov 2021
Cited by 1 | Viewed by 1423
Abstract
In this paper, the dynamic behavior of the van der Pol-Rayleigh system is studied by using the fast–slow analysis method and the transformation phase portrait method. Firstly, the stability and bifurcation behavior of the equilibrium point of the system are analyzed. We find [...] Read more.
In this paper, the dynamic behavior of the van der Pol-Rayleigh system is studied by using the fast–slow analysis method and the transformation phase portrait method. Firstly, the stability and bifurcation behavior of the equilibrium point of the system are analyzed. We find that the system has no fold bifurcation, but has Hopf bifurcation. By calculating the first Lyapunov coefficient, the bifurcation direction and stability of the Hopf bifurcation are obtained. Moreover, the bifurcation diagram of the system with respect to the external excitation is drawn. Then, the fast subsystem is simulated numerically and analyzed with or without external excitation. Finally, the vibration behavior and its generation mechanism of the system in different modes are analyzed. The vibration mode of the system is affected by both the fast and slow varying processes. The mechanisms of different modes of vibration of the system are revealed by the transformation phase portrait method, because the system trajectory will encounter different types of attractors in the fast subsystem. Full article
(This article belongs to the Special Issue Nonlinear Dynamics and Chaos Theory)
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25 pages, 80707 KiB  
Article
A Novel Plain-Text Related Image Encryption Algorithm Based on LB Compound Chaotic Map
by Shijie Zhang, Lingfeng Liu and Hongyue Xiang
Mathematics 2021, 9(21), 2778; https://doi.org/10.3390/math9212778 - 02 Nov 2021
Cited by 10 | Viewed by 1522
Abstract
Chaos systems have been widely used in image encryption algorithms. In this article, we introduce an LB (Logistic-Baker) compound chaotic map that can greatly improve the complexity of original Logistic map and Baker map, as well as the generated sequences have pseudo-randomness. Furthermore, [...] Read more.
Chaos systems have been widely used in image encryption algorithms. In this article, we introduce an LB (Logistic-Baker) compound chaotic map that can greatly improve the complexity of original Logistic map and Baker map, as well as the generated sequences have pseudo-randomness. Furthermore, based on the LB compound chaotic map, an image encryption algorithm is proposed. To resist the differential attack, and enhance the sensitivity of plain-text, the parameters of this algorithm are plain-text related. In this algorithm, the compound chaotic function is influenced by the plain-text image; thus, the specific form of this chaotic map, and its dynamics will be different when encrypting different images. Numerical experiment results indicate that the effect of this novel plain-text related image encryption scheme is excellent, as well as can be competitive with other corresponding algorithms. Full article
(This article belongs to the Special Issue Nonlinear Dynamics and Chaos Theory)
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