Recent Trends in Nonlinear, Chaotic and Complex Systems

Topic Information

Dear Colleagues,

We live in a world dominated by nonlinearities, which play an important role in many natural phenomena. Furthermore, chaos is a common phenomenon in nature and exists in various nonlinear systems. Theoretical research on chaos has lasted for almost five decades. Various definitions and properties, theoretical bases and frameworks, correlations with fractals, and test methods of chaos have been presented and examined.

Therefore, the theoretical concept of chaos helps us to examine phenomena and behaviors not only in physical, chemical, and biological systems, but also in economic and sociological models. At the same time, chaos has also been used in many practical and useful applications, such as in cryptography, secure communications, optimization algorithms, and robotics. Furthermore, with the development of nonlinear dynamics, chaotic phenomena have been found in recent nonlinear systems, such as fractional differential systems, fractional discrete systems, time delays systems, and discontinuous dynamical systems. Especially in the last three decades, complex dynamics, including chaos, bifurcation, and other mechanisms, have also been found in these new systems. Nonlinear systems, bifurcation, chaos, and fractals are intertwined, which constitute several major topics in the study of nonlinear dynamics.

In this sense, research in this topic should contribute to presenting recent trends in theory, analysis, numerical simulation, and experimental realization, promising new results and novel practical applications on various topics of current interest on chaos and the related field of nonlinear dynamics, as well as on complex systems.

Dr. Christos Volos
Dr. Karthikeyan Rajagopal
Dr. Sajad Jafari
Dr. Jacques Kengne
Prof. Dr. Jesus M. Munoz-Pacheco
Topic Editors

Keywords

Participating Journals

Journal Name	Impact Factor	CiteScore	Launched Year	First Decision (median)	APC
Entropy entropy	2.1	4.9	1999	22.4 Days	CHF 2600	Submit
Fractal and Fractional fractalfract	3.6	4.6	2017	20.9 Days	CHF 2700	Submit
Dynamics dynamics	-	1.2	2021	14.4 Days	CHF 1000	Submit
Symmetry symmetry	2.2	5.4	2009	16.8 Days	CHF 2400	Submit
Algorithms algorithms	1.8	4.1	2008	15 Days	CHF 1600	Submit

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Published Papers (12 papers)

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All Entropy Fractal Fract Dynamics Symmetry Algorithms

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11 pages, 11842 KiB  

Open AccessArticle

Control of a New Financial Risk Contagion Dynamic Model Based on Finite-Time Disturbance

by Yifeng Wei, Chengrong Xie, Xia Qing and Yuhua Xu

Entropy 2024, 26(12), 999; https://doi.org/10.3390/e26120999 - 21 Nov 2024

Viewed by 241

Abstract

With the widespread application of chaotic systems in many fields, research on chaotic systems is becoming increasingly in-depth. This article first proposes a new dynamic model of financial risk contagion based on financial principles and discusses some basic dynamic characteristics of the new [...] Read more.

With the widespread application of chaotic systems in many fields, research on chaotic systems is becoming increasingly in-depth. This article first proposes a new dynamic model of financial risk contagion based on financial principles and discusses some basic dynamic characteristics of the new chaotic system, such as equilibrium points, dissipativity, Poincaré diagrams, bifurcation diagrams, etc. Secondly, with the consideration of privacy during data transmission, the method was designed to protect the privacy of controlled systems in finite time based on perturbation. A controller designed for finite time was developed based on Lyapunov stability principles, which achieves system synchronization within a finite time and protects the privacy of the controlled system. The effectiveness was also verified by numerical simulations. Full article

(This article belongs to the Topic Recent Trends in Nonlinear, Chaotic and Complex Systems)

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16 pages, 3080 KiB  

Open AccessArticle

Load Frequency Optimal Active Disturbance Rejection Control of Hybrid Power System

by Kuansheng Zou, Yue Wang, Baowei Liu and Zhaojun Zhang

Algorithms 2024, 17(9), 403; https://doi.org/10.3390/a17090403 - 9 Sep 2024

Viewed by 622

Abstract

The widespread adoption of the power grid has led to increased attention to load frequency control (LFC) in power systems. The LFC strategy of multi-source hybrid power systems, including hydroelectric generators, Wind Turbine Generators (WTGs), and Photovoltaic Generators (PVGs), with thermal generators is [...] Read more.

The widespread adoption of the power grid has led to increased attention to load frequency control (LFC) in power systems. The LFC strategy of multi-source hybrid power systems, including hydroelectric generators, Wind Turbine Generators (WTGs), and Photovoltaic Generators (PVGs), with thermal generators is more challenging. Existing methods for LFC tasks pose challenges in achieving satisfactory outcomes in hybrid power systems. In this paper, a novel method for the multi-source hybrid power system LFC task by using an optimal active disturbance rejection control (ADRC) strategy is proposed, which is based on the combination of the improved linear quadratic regulator (LQR) and the ADRC controller. Firstly, an established model of a hybrid power system is presented, which incorporates multiple regions and multiple sources. Secondly, utilizing the state space representation, a novel control strategy is developed by integrating improved LQR and ARDC. Finally, a series of comparative simulation experiments has been conducted using the Simulink model. Compared with the LQR with ESO, the maximum relative error of the maximum peaks of frequency deviation and tie-line exchanged power of the hybrid power system is reduced by 96% and 83%, respectively, by using the proposed strategy. The experimental results demonstrate that the strategy proposed in this paper exhibits a substantial enhancement in control performance. Full article

(This article belongs to the Topic Recent Trends in Nonlinear, Chaotic and Complex Systems)

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44 pages, 24290 KiB  

Open AccessArticle

A Piecewise Linear Approach for Implementing Fractional-Order Multi-Scroll Chaotic Systems on ARMs and FPGAs

by Daniel Clemente-López, Jesus M. Munoz-Pacheco, Ernesto Zambrano-Serrano, Olga G. Félix Beltrán and Jose de Jesus Rangel-Magdaleno

Fractal Fract. 2024, 8(7), 389; https://doi.org/10.3390/fractalfract8070389 - 29 Jun 2024

Viewed by 974

Abstract

This manuscript introduces a piecewise linear decomposition method devoted to a class of fractional-order dynamical systems composed of piecewise linear (PWL) functions. Inspired by the Adomian decomposition method, the proposed technique computes an approximated solution of fractional-order PWL systems using only linear operators [...] Read more.

This manuscript introduces a piecewise linear decomposition method devoted to a class of fractional-order dynamical systems composed of piecewise linear (PWL) functions. Inspired by the Adomian decomposition method, the proposed technique computes an approximated solution of fractional-order PWL systems using only linear operators and specific constants vectors for each sub-domain of the PWL functions, with no need for the Adomian polynomials. The proposed decomposition method can be applied to fractional-order PWL systems composed of nth PWL functions, where each PWL function may have any number of affine segments. In particular, we demonstrate various examples of how to solve fractional-order systems with 1D 2-scroll, 4-scroll, and $4\times 4$-grid scroll chaotic attractors by applying the proposed approach. From the theoretical and implementation results, we found the proposed approach eliminates the unneeded terms, has a low computational cost, and permits a straightforward physical implementation of multi-scroll chaotic attractors on ARMs and FPGAs digital platforms. Full article

(This article belongs to the Topic Recent Trends in Nonlinear, Chaotic and Complex Systems)

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17 pages, 53744 KiB  

Open AccessArticle

Fractal Tent Map with Application to Surrogate Testing

by Ekaterina Kopets, Vyacheslav Rybin, Oleg Vasilchenko, Denis Butusov, Petr Fedoseev and Artur Karimov

Fractal Fract. 2024, 8(6), 344; https://doi.org/10.3390/fractalfract8060344 - 7 Jun 2024

Cited by 3 | Viewed by 903

Abstract

Discrete chaotic maps are a mathematical basis for many useful applications. One of the most common is chaos-based pseudorandom number generators (PRNGs), which should be computationally cheap and controllable and possess necessary statistical properties, such as mixing and diffusion. However, chaotic PRNGs have [...] Read more.

Discrete chaotic maps are a mathematical basis for many useful applications. One of the most common is chaos-based pseudorandom number generators (PRNGs), which should be computationally cheap and controllable and possess necessary statistical properties, such as mixing and diffusion. However, chaotic PRNGs have several known shortcomings, e.g., being prone to chaos degeneration, falling in short periods, and having a relatively narrow parameter range. Therefore, it is reasonable to design novel simple chaotic maps to overcome these drawbacks. In this study, we propose a novel fractal chaotic tent map, which is a generalization of the well-known tent map with a fractal function introduced into the right-hand side. We construct and investigate a PRNG based on the proposed map, showing its high level of randomness by applying the NIST statistical test suite. The application of the proposed PRNG to the task of generating surrogate data and a surrogate testing procedure is shown. The experimental results demonstrate that our approach possesses superior accuracy in surrogate testing across three distinct signal types—linear, chaotic, and biological signals—compared to the MATLAB built-in randn() function and PRNGs based on the logistic map and the conventional tent map. Along with surrogate testing, the proposed fractal tent map can be efficiently used in chaos-based communications and data encryption tasks. Full article

(This article belongs to the Topic Recent Trends in Nonlinear, Chaotic and Complex Systems)

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21 pages, 761 KiB  

Open AccessArticle

Chaos, Fractionality, Nonlinear Contagion, and Causality Dynamics of the Metaverse, Energy Consumption, and Environmental Pollution: Markov-Switching Generalized Autoregressive Conditional Heteroskedasticity Copula and Causality Methods

by Melike Bildirici, Özgür Ömer Ersin and Blend Ibrahim

Fractal Fract. 2024, 8(2), 114; https://doi.org/10.3390/fractalfract8020114 - 14 Feb 2024

Cited by 4 | Viewed by 1586

Abstract

Metaverse (MV) technology introduces new tools for users each day. MV companies have a significant share in the total stock markets today, and their size is increasing. However, MV technologies are questioned as to whether they contribute to environmental pollution with their increasing [...] Read more.

Metaverse (MV) technology introduces new tools for users each day. MV companies have a significant share in the total stock markets today, and their size is increasing. However, MV technologies are questioned as to whether they contribute to environmental pollution with their increasing energy consumption (EC). This study explores complex nonlinear contagion with tail dependence and causality between MV stocks, EC, and environmental pollution proxied with carbon dioxide emissions (CO₂) with a decade-long daily dataset covering 18 May 2012–16 March 2023. The Mandelbrot–Wallis and Lo’s rescaled range (R/S) tests confirm long-term dependence and fractionality, and the largest Lyapunov exponents, Shannon and Havrda, Charvât, and Tsallis (HCT) entropy tests followed by the Kolmogorov–Sinai (KS) complexity measure confirm chaos, entropy, and complexity. The Brock, Dechert, and Scheinkman (BDS) test of independence test confirms nonlinearity, and White‘s test of heteroskedasticity of nonlinear forms and Engle’s autoregressive conditional heteroskedasticity test confirm heteroskedasticity, in addition to fractionality and chaos. In modeling, the marginal distributions are modeled with Markov-Switching Generalized Autoregressive Conditional Heteroskedasticity Copula (MS-GARCH–Copula) processes with two regimes for low and high volatility and asymmetric tail dependence between MV, EC, and CO₂ in all regimes. The findings indicate relatively higher contagion with larger copula parameters in high-volatility regimes. Nonlinear causality is modeled under regime-switching heteroskedasticity, and the results indicate unidirectional causality from MV to EC, from MV to CO_2, and from EC to CO₂, in addition to bidirectional causality among MV and EC, which amplifies the effects on air pollution. The findings of this paper offer vital insights into the MV, EC, and CO₂ nexus under chaos, fractionality, and nonlinearity. Important policy recommendations are generated. Full article

(This article belongs to the Topic Recent Trends in Nonlinear, Chaotic and Complex Systems)

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16 pages, 2096 KiB  

Open AccessArticle

The Caputo Nonlocal Structural Derivative Ultraslow Diffusion Model of Language Change and the Microscopic Mechanism

by Wei Xu, Hui Liu, Yingjie Liang and Shijun Zhao

Fractal Fract. 2024, 8(1), 66; https://doi.org/10.3390/fractalfract8010066 - 17 Jan 2024

Cited by 1 | Viewed by 1376

Abstract

Numerous studies have observed and analyzed the dynamics of language change from a diffusion perspective. As a complex and changeable system, the process of language change is characterized by a long memory that conforms to ultraslow diffusion. However, it is not perfectly suited [...] Read more.

Numerous studies have observed and analyzed the dynamics of language change from a diffusion perspective. As a complex and changeable system, the process of language change is characterized by a long memory that conforms to ultraslow diffusion. However, it is not perfectly suited for modeling with the traditional diffusion model. The Caputo nonlocal structural derivative is a further development of the classic Caputo fractional derivative. Its kernel function, characterized as an arbitrary function, proves highly effective in dealing with ultraslow diffusion. In this study, we utilized an extended logarithmic function to formulate a Caputo nonlocal structural derivative diffusion model for qualitatively analyzing the evolution process of language. The mean square displacement that grows logarithmically was derived through the Tauberian theorem and the Fourier–Laplace transform. Its effectiveness and credibility were verified by the appearance of already popular words on Japanese blogs. Compared to the random diffusion model, the Caputo nonlocal structural derivative diffusion model proves to be more precise in simulating the process of language change. The microscopic mechanism of ultraslow diffusion was explored using the continuous time random walk model, which involves a logarithmic function with a long tail. Both models incorporate memory effects, which can provide useful guidance for modeling diffusion behavior in other social phenomena. Full article

(This article belongs to the Topic Recent Trends in Nonlinear, Chaotic and Complex Systems)

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24 pages, 2472 KiB  

Open AccessArticle

News Waves: Hard News, Soft News, Fake News, Rumors, News Wavetrains

by Nikolay K. Vitanov, Zlatinka I. Dimitrova and Kaloyan N. Vitanov

Entropy 2024, 26(1), 5; https://doi.org/10.3390/e26010005 - 19 Dec 2023

Viewed by 5587

Abstract

We discuss the spread of a piece of news in a population. This is modeled by SIR model of epidemic spread. The model can be reduced to a nonlinear differential equation for the number of people affected by the news of interest. The [...] Read more.

We discuss the spread of a piece of news in a population. This is modeled by SIR model of epidemic spread. The model can be reduced to a nonlinear differential equation for the number of people affected by the news of interest. The differential equation has an exponential nonlinearity and it can be approximated by a sequence of nonlinear differential equations with polynomial nonlinearities. Exact solutions to these equations can be obtained by the Simple Equations Method (SEsM). Some of these exact solutions can be used to model a class of waves associated with the spread of the news in a population. The presence of exact solutions allow to study in detail the dependence of the amplitude and the time horizon of the news waves on the wave parameters, such as the size of the population, initial number of spreaders of the piece of the news, transmission rate, and recovery rate. This allows for recommendations about the change of wave parameters in order to achieve a large amplitude or appropriate time horizon of the news wave. We discuss five types of news waves on the basis of the values of the transmission rate and recovery rate—types A, B, C, D, and E of news waves. In addition, we discuss the possibility of building wavetrains by news waves. There are three possible kinds of wavetrains with respect of the amplitude of the wave: increasing wavetrain, decreasing wavetrain, and mixed wavetrain. The increasing wavetrain is especially interesting, as it is connected to an increasing amplitude of the news wave with respect to the amplitude of the previous wave of the wavetrain. It can find applications in advertising, propaganda, etc. Full article

(This article belongs to the Topic Recent Trends in Nonlinear, Chaotic and Complex Systems)

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13 pages, 2318 KiB  

Open AccessArticle

Research on the Dynamical Behavior of Public Opinion Triggered by Rumor Based on a Nonlinear Oscillator Model

by Wanglai Li, Huizhang Shen, Zhangxue Huang and Hanzhe Yang

Entropy 2023, 25(12), 1614; https://doi.org/10.3390/e25121614 - 1 Dec 2023

Cited by 2 | Viewed by 1022

Abstract

In public opinion triggered by rumors, the authenticity of the information remains uncertain, and the main topic oscillates between diverse opinions. In this paper, a nonlinear oscillator model is proposed to demonstrate the public opinion triggered by rumors. Based on the model and [...] Read more.

In public opinion triggered by rumors, the authenticity of the information remains uncertain, and the main topic oscillates between diverse opinions. In this paper, a nonlinear oscillator model is proposed to demonstrate the public opinion triggered by rumors. Based on the model and actual data of one case, it is found that a continuous flow of new information about rumors acts as external forces on the system, probably leading to the chaotic behavior of public opinion. Moreover, similar features are observed in three other cases, and the same model is also applicable to these cases. Based on these results, it is shown that our model possesses generality, revealing the evolutionary trends of a certain type of public opinion in real-world scenarios. Full article

(This article belongs to the Topic Recent Trends in Nonlinear, Chaotic and Complex Systems)

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23 pages, 14185 KiB  

Open AccessArticle

Bifurcation and Chaotic Behavior of Duffing System with Fractional-Order Derivative and Time Delay

by Cuiyan Wang, Meiqi Wang, Wuce Xing and Shaoxuan Shi

Fractal Fract. 2023, 7(8), 638; https://doi.org/10.3390/fractalfract7080638 - 21 Aug 2023

Cited by 4 | Viewed by 1627

Abstract

In this paper, the abundant nonlinear dynamical behaviors of a fractional-order time-delayed Duffing system under harmonic excitation are studied. By constructing Melnikov function, the necessary conditions of chaotic motion in horseshoe shape are detected, and the chaos threshold curve is obtained by comparing [...] Read more.

In this paper, the abundant nonlinear dynamical behaviors of a fractional-order time-delayed Duffing system under harmonic excitation are studied. By constructing Melnikov function, the necessary conditions of chaotic motion in horseshoe shape are detected, and the chaos threshold curve is obtained by comparing the results obtained through the Melnikov theory and numerical iterative algorithm. The results show that the trend of change is the same, which confirms the accuracy of the chaos threshold curve. It could be found that when the excitation frequency ω is larger than a certain value, the Melnikov theory is not valid for these values. Furthermore, by numerical simulation, some numerical results are obtained, including phase portraits, the largest Lyapunov exponents, and the bifurcation diagrams, Poincare maps, time histories, and frequency spectrograms at some typical points. These numerical simulation results show that the system exhibits some new complex dynamical behaviors, including entry into the state of chaotic motion from single period to period-doubling bifurcation and chaotic motion and periodic motion alternating under the necessary condition of chaotic occurrence. In addition, the effects of time delay, fractional-order coefficient, fractional order, linear viscous damping coefficient, and linear stiffness coefficient on the chaotic threshold curve are discussed, respectively. Those results reveal that there exist abundant nonlinear dynamic behaviors in this fractional-order system, and by adjusting these parameters reasonably, the system could be transformed from chaotic motion to non-chaotic motion. Full article

(This article belongs to the Topic Recent Trends in Nonlinear, Chaotic and Complex Systems)

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24 pages, 14255 KiB  

Open AccessArticle

Joint Encryption Model Based on a Randomized Autoencoder Neural Network and Coupled Chaos Mapping

by Anqi Hu, Xiaoxue Gong and Lei Guo

Entropy 2023, 25(8), 1153; https://doi.org/10.3390/e25081153 - 1 Aug 2023

Cited by 1 | Viewed by 1194

Abstract

Following an in-depth analysis of one-dimensional chaos, a randomized selective autoencoder neural network (AENN), and coupled chaotic mapping are proposed to address the short period and low complexity of one-dimensional chaos. An improved method is proposed for synchronizing keys during the transmission of [...] Read more.

Following an in-depth analysis of one-dimensional chaos, a randomized selective autoencoder neural network (AENN), and coupled chaotic mapping are proposed to address the short period and low complexity of one-dimensional chaos. An improved method is proposed for synchronizing keys during the transmission of one-time pad encryption, which can greatly reduce the usage of channel resources. Then, a joint encryption model based on randomized AENN and a new chaotic coupling mapping is proposed. The performance analysis concludes that the encryption model possesses a huge key space and high sensitivity, and achieves the effect of one-time pad encryption. Experimental results show that this model is a high-security joint encryption model that saves secure channel resources and has the ability to resist common attacks, such as exhaustive attacks, selective plaintext attacks, and statistical attacks. Full article

(This article belongs to the Topic Recent Trends in Nonlinear, Chaotic and Complex Systems)

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19 pages, 12076 KiB  

Open AccessArticle

Evaluation of Geometric Attractor Structure and Recurrence Analysis in Professional Dancers

by Michalina Błażkiewicz

Entropy 2022, 24(9), 1310; https://doi.org/10.3390/e24091310 - 16 Sep 2022

Cited by 5 | Viewed by 2101

Abstract

Background: Human motor systems contain nonlinear features. The purpose of this study was to evaluate the geometric structure of attractors and analyze recurrence in two different pirouettes (jazz and classic) performed by 15 professional dancers. Methods: The kinematics of the body’s center of [...] Read more.

Background: Human motor systems contain nonlinear features. The purpose of this study was to evaluate the geometric structure of attractors and analyze recurrence in two different pirouettes (jazz and classic) performed by 15 professional dancers. Methods: The kinematics of the body’s center of mass (CoM) and knee of the supporting leg (LKNE) during the pirouette were measured using the Vicon system. A time series of selected points were resampled, normalized, and randomly reordered. Then, every second time series was flipped to be combined with other time series and make a long time series out of the repetitions of a single task. The attractors were reconstructed, and the convex hull volumes (CHV) were counted for the CoM and LKNE for each pirouette in each direction. Recurrence quantification analysis (RQA) was used to extract additional information. Results: The CHVs calculated for the LKNE were significantly lower for the jazz pirouette. All RQA measures had the highest values for LKNE along the mediolateral axis for the jazz pirouette. This result underscores the high determinism, high motion recurrence, and complexity of this maneuver. Conclusions: The findings offer new insight into the evaluation of the approximation of homogeneity in motion control. A high determinism indicates a highly stable and predictive motion trajectory. Full article

(This article belongs to the Topic Recent Trends in Nonlinear, Chaotic and Complex Systems)

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14 pages, 5464 KiB  

Open AccessArticle

A Novel Hyperchaotic 2D-SFCF with Simple Structure and Its Application in Image Encryption

by Yongsheng Hu, Han Wu and Luoyu Zhou

Entropy 2022, 24(9), 1266; https://doi.org/10.3390/e24091266 - 9 Sep 2022

Cited by 3 | Viewed by 2140

Abstract

In this paper, a novel image encryption algorithm is proposed based on hyperchaotic two-dimensional sin-fractional-cos-fractional (2D-SFCF), called sin-fractional-cos-fractional image-encryption (SFCF-IE). The 2D-SFCF is constructed from two one-dimensional cosine fractional (1-DCFs), and it has a more complex chaotic behavior with a larger parameter space [...] Read more.

In this paper, a novel image encryption algorithm is proposed based on hyperchaotic two-dimensional sin-fractional-cos-fractional (2D-SFCF), called sin-fractional-cos-fractional image-encryption (SFCF-IE). The 2D-SFCF is constructed from two one-dimensional cosine fractional (1-DCFs), and it has a more complex chaotic behavior with a larger parameter space than one-dimensional chaotic systems. Compared with the two-dimensional (2D) chaotic system, the 2D-SFCF has a simple structure, and the parameter space in the chaotic state is continuous, which is beneficial to generating the keystream in the cryptosystem. Therefore, in the novel image encryption algorithm, we use the 2D-SFCF to generate the keystream of the cryptosystem. The encryption algorithm is a process of scrambling and diffusion. Different from common diffusion methods, the diffusion starting position of the SFCF-IE is randomly generated, enhancing the algorithm’s security. Simulation experiments show that the image encrypted by this algorithm has better distribution characteristics and can resist common attack methods. Full article

(This article belongs to the Topic Recent Trends in Nonlinear, Chaotic and Complex Systems)

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