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In this research work, the convective flow of Lorenz attractor in the fractional domain is modeled with a nonlinear flexible structure of radial basis neural network (RBNN). The physical parameters of the complex fractional-order system are initially computed with fractional order Runge–Kutta solver for different stochastic scenarios of control constraints and designed its parametric model with RBFN under various initial conditions of the chaotic Lorenz system. Multiple chaos of the Lorenz system with the dynamical neural network structure are computed for different fractional order solutions to analyze the sensitivity of chaotic behavior by means of Lyapunov exponents. Phase diagrams of the chaotic pattern show the fractional-order Lorenz system has compact dynamical behavior and has the potential for the design application of a real dynamical system. Embedded dimension and time delay of the fractional Lorenz system are computed using the average mutual information (AMI) technique to evaluate the suitability of the time delay chaotic pattern in the fractional domain, which is affected by the numerical algorithm and fluctuated on minute time step size. The proposed design of RBFN has been confirmed as an exceptionally high-performance tool for soft computing and dynamics analysis of chaotic systems in the fractional order domain. [ABSTRACT FROM AUTHOR]

In this paper, we have considered a three-dimensional Lotka–Volterra attractor in the frame of the Caputo fractional derivative to examine its dynamics. The theoretical concepts like existence and uniqueness and boundedness of the solution are analyzed. To regulate the chaos in this fractional-order system, we have developed a sliding mode controller and conditions for global stability of the controlled system with and without uncertainties and outside disruptions are derived. The ability of the designed controller is examined in terms of both commensurate and non-commensurate fractional order derivatives for all the aspects. The Lyapunov exponent is the novelty of this paper which is used to illustrate the behavior of the chaos and demonstrate the dissipativeness of the considered chaotic system. We have examined the effect of fractional order derivatives in this system. With the help of numerical simulations, the theoretical claims regarding the impact of the controller on the system are established. [ABSTRACT FROM AUTHOR]

Stochastic configuration networks (SCNs) have universal approximation capability and fast modeling properties, which have been successfully employed in large-scale data analytics. Based on SCNs, Stochastic configuration networks with block increments (BSC) use the node block increments mechanism to improve training speed but increase the complexity of the model. This paper presents a parallel configuration method (PCM), develops an extension of the original BSC with chaos theory and proposes stochastic configuration networks with chaotic maps (SCNCM), and establishes a hierarchical learning strategy (HLS) to enhance the compactness and construction speed of the model. Firstly, PCM randomly assigns the input weights w and biases b of hidden layer nodes by using uniform and normal distributions. In PCM, an iterative learning algorithm is intended to generate the scope control set and improve configuration efficiency. Secondly, the paper presents two kinds of stochastic configuration networks with chaotic maps, which are SCNCM-I and SCNCM-II. SCNCM-I adjusts block size by using multiple error values and chaotic maps to improve the training speed. Based on SCNCM-I, SCNCM-II utilizes node removal mechanism to enhance the compactness. Finally, HLS integrates with SCNCM-I, SCNCM-II, and the Harris-hawks optimization algorithm (HHO). The purpose of training is to enhance the training speed and compactness for three algorithms. The experiments are conducted on four benchmark data sets and an industrial application shows its effectiveness. [ABSTRACT FROM AUTHOR]

In this paper, we modify the methods of Zhou et al. (2004) and Shang and Han (2005) associated with proving the existence of a homoclinic orbit of Shilnikov type. We construct the series expressions of the solution based at a saddle-focus on stable and unstable manifolds, and give the sufficient conditions of the existence of homoclinic orbit and spiral chaos. Then, we consider the Rössler system with the typical parameters under which the system exhibits chaotic behavior. Using our modified method, we verify that there exists a homoclinic orbit of Shilnikov type in the Rössler system with a group of typical parameters, and prove the existence of spiral chaos by using the Shilnikov criterion, and we carry out numerical simulations to support the analytic results. [ABSTRACT FROM AUTHOR]

A recently developed metaheuristic optimization algorithm, Salp Swarm Algorithm (SSA), has manifested its capability in solving various optimization problems and many real-life applications. SSA is based on salps' swarming behaviour when finding their way and searching for food in the oceans. Nonetheless, like most metaheuristic algorithms, SSA experiences low convergence and stagnation in local optima and rate. There is a need to enhance SSA to speed its convergence and effectiveness to solve complex problems. In the present study, we will introduce chaos into SSA (CSSA) to increase its global search mobility for robust global optimization. Detailed studies are carried out on real-world nonlinear benchmark systems and CEC 2013 benchmark functions with chaotic map (Tent). Here, the algorithm utilizes a Tent map to tune the salp leaders' attractive movement around food sources. The experimental results, considering both convergence and accuracy simultaneously, demonstrate the effectiveness of CSSA for 12 nonlinear systems and 28 unconstrained optimization problems CEC 2013. Two nonparametric statistical tests, the Friedman test and Wilcoxon Signed-Rank Test, are conducted to show the superiority of CSCA over other states of the art algorithms and our results' significance. • This paper introduces chaos into Salp Swarm Algorithm. • Solve 28 unconstrained optimization problems, CEC and 12 nonlinear systems. • The effectiveness and efficiency of our algorithm are provided. • Experimental results prove superiority of our algorithm over the state-of-the-arts. [ABSTRACT FROM AUTHOR]

A two-dimensional cross-mode hyperchaotic map based on logistic and sine maps (2D-CLSS) is presented. The hyperchaotic map consists of a logistic map and two sine maps with cross structure. The chaotic behavior of the system is analyzed using bifurcation diagrams, Lyapunov exponential spectra, phase diagrams, etc. The outcomes demonstrate that the system has promising ergodicity and a wide range of hyperchaotic phenomena. Using the proposed 2D-CLSS, an image cryptography algorithm is developed. This image encryption system employs a strategy that simultaneously combines permutation and diffusion to alter the location and the value of the pixels. Experiments and security simulations indicate that the scheme is effective in encrypting images with excellent security against various attacks (differential, noise and data loss, etc.). [ABSTRACT FROM AUTHOR]

Double-circumferential-slot air bearing (DCSAB) systems provide multidirectional supporting forces and have high stiffness, increasing the stability of instruments at high rotational speeds. However, DCSAB systems may exhibit chaotic motion because of a nonlinear pressure distribution within the gas film, supplied gas imbalances, or an inappropriate design. This study investigated the occurrence of nonperiodic motion in a DCSAB system by analyzing the dynamic response of systems with different rotor masses and bearing numbers. The dynamic trajectory, spectral response, bifurcation, Poincaré map, and maximum Lyapunov exponent were analyzed to identify chaotic behavior. Behavior was found to be highly sensitive to rotor mass and bearing number; the system exhibits chaotic behavior when the rotor mass has values in three intervals within 0.1–6.0 kg given a fixed bearing number of Λ = 3.8. To reduce the computational cost of predicting chaotic behavior, the maximum Lyapunov exponent was predicted using various machine learning models; a voting model combining random forest with XGBoost has the highest performance. The results can be used as a guideline for designing of DCASB systems for use in industrial applications. • Effect of rotor mass and bearing number on slots air bearing system is analyzed. • Complex periodic, non-periodic and chaotic motion behavior is observed. • Chaos occurs at specific rotor mass and bearing number proven by Lyapunov exponent. • Voting model combining random forest and XGBoost obtains better prediction accuracy. [ABSTRACT FROM AUTHOR]

A new problem at the intersection of multi-agent systems, chaotic time series prediction, and flow map learning is formulated in this paper. The problem involves agents collaborating to track moving targets in chaotic dynamic systems by communicating. Inspired by the multi-scale hierarchical time-stepper (HiTS), a novel Distributed Prediction Network based on Multi-scale Attention (DPNMA) is proposed to fuse predictions from agents at different scales through an enhanced self-attention mechanism. The experimental evaluation demonstrates that DPNMA effectively mitigates cumulative errors and enhances the accuracy and robustness of the predictions, which has important implications for the scenarios where the agents have heterogeneous and constrained capabilities. [ABSTRACT FROM AUTHOR]

In this work, an autocatalytic chemical reactions model with mutation is investigated, and a nonlinear control is proposed to lead the system to a behavior that maintains the system with a desired behavior and with suppress the chaotic behavior. Numerical results considering a broad analysis of the system parameters confirmed the chaotic behavior using phase portraits, bifurcation diagrams and Lyapunov exponents. An analysis of the system behavior in fractional order considering the Atangana–Baleanu–Caputo operator was performed to include the memory effect in the system dynamics. The numerical results obtained demonstrated that the system is extremely sensitive to variations in the order of the derivative, taking the system from chaotic behavior to periodic behavior with small variations in the order of the derivative. The chaotic behavior of the system is proven in fractional order by the 0–1 test and a scale index based on the wavelets test. The proposed control demonstrated to be efficient in maintaining the main product (ethanol) output concentration at the desired level. • Observation of chaotic behavior in bioreactors • Influence of fractional order on the dynamics of nonlinear systems • Application of control in the optimization of biological processes [ABSTRACT FROM AUTHOR]

In this paper, a fractional-order hyperchaotic system based on dual memristors is proposed by introducing flux-controlled and charge-controlled memristors into a simple RLC circuit. Dynamics of the hyperchaotic system are investigated using bifurcation diagrams, Lyapunov exponents spectrum (LEs), phase diagrams, time-domain diagrams, spectral entropy (SE) and C 0 complexity. The results show that it has a plane of equilibria and exhibits rich dynamical characteristics, including hyperchaos, homogeneous and heterogeneous extreme multistability. Meanwhile, the transient transition phenomena as well as the effect of parameters on the complexity and chaotic behavior of the system are also studied. Furthermore, the practical implementation of the system is realized through analog and digital circuit. The experimental results validate the correctness of the theoretical analysis and help to make better use of the hyperchaotic system in applications such as secure communications. • A fractional-order hyperchaotic memristive system with infinite equilibrium point is proposed • Compared with integer-order system, the fractional-order system has stronger chaotic characteristics and higher complexity • The system exhibits homogeneous and heterogeneous extreme multistability behavior • The proposed system is implemented by analog circuit and digital circuit [ABSTRACT FROM AUTHOR]

Optimization techniques are required to find the best solutions to challenging problems in many engineering disciplines. Metaheuristic algorithms that effectively increase search performance using various evolutionary strategies have become increasingly popular in recent years. The Golden Sinus Algorithm (GoldSa) is a population-based optimization algorithm that uses the sine function and the golden ratio. In this study, a chaotically enhanced opposition-based Golden Sinus Algorithm (Co-GoldSa) has been proposed to improve the efficiency of the exploitation and exploration ability of the GoldSa method. While designing this approach, it is first necessary to analyze the chaotic behavior of the GoldSa parameters. For this purpose, three chaotic GoldSa methods have been developed using eight chaotic maps to determine the effect of the chaotic maps on the parameters with different behaviors. Secondly, the opposition-based learning strategy is adjusted to the cGoldSa to enhance the searching ability. To investigate the proficiency of the proposed Co-GoldSa method, it has been examined with well-known and newly introduced metaheuristic approaches on benchmark functions and classical engineering design problems. Besides, an efficient framework of the multilevel thresholding image segmentation has been presented based on the Co-GoldSa method since the efficient processing of pathological images is quite important in medical diagnostics. The experimental outcomes reveal the superiority of the proposed method in solving global optimization problems, image segmentation, and engineering problems. Thus, the outcomes of the benchmark functions, image segmentation, and classical engineering problems support that the proposed Co-GoldSa approach can be considered a promising method for resolving challenging optimization problems. • Different ergodic chaotic systems are used to generate chaotic values on the parameters with different behaviors. • The Co-GoldSa method is proposed to enhance the searching ability and initial population. • The efficiency of the Co-GoldSa method is tested on benchmark functions, classical engineering problems and multilevel thresholding. • An efficient image segmentation framework is developed for pathological images. [ABSTRACT FROM AUTHOR]

Dynamical systems can be analysed as computational devices capable of performing information processing. In coupled oscillators, enlarged capabilities are expected when the set of units is formed by subsets with collective behaviour within them but weak correlation between the subsets. A system of non-linear oscillators weakly coupled in the phase approximation is studied. The informational distance maps show different regimes of collective behaviour, ranging from independent, local, and global, as the control parameters of the system are changed. This rich set of behaviours happens despite the simple nature of the model used. Complex hierarchies between oscillators can be identified near the edge of chaos region in the informational distance dendrograms. We identify the emergence of local collective regimes, further corroborated by the spatiotemporal maps. Correlation diagrams also exhibit non-trivial dependence for the system at the edge of chaos. • Surge in complexity measures at EOC signals enhancement of information processing. • Informational distance matrix can be used to identify complex correlations at EOC. • Dendrograms at EOC shows emergence of local collective behaviour. • Fine tuning can be avoided for sub-optimal improvement of information processing. [ABSTRACT FROM AUTHOR]

The forced van der Pol oscillator has played a fundamental role in the development of nonlinear science. It is notable that the van der Pol oscillator in the absence of an AC forcing term explains the underlying mechanism that induces limit cycles and relaxation oscillations; the forced van der Pol oscillator was the first electric oscillator that, via measurements of the emitted sound, was inferred to exhibit chaotic behavior (van der Pol and van der Mark, 1927). This oscillator thus had a significant influence on the development of chaos theory. It was subsequently demonstrated, via a nonstandard analysis undertaken in the 1980s, that a canard explosion was present in the dynamics exhibited by this oscillator. In previous works (Inaba and Kousaka, (2020); Inaba et al., (2023)), we established the existence of bifurcation structures that are referred to as nested mixed-mode oscillations (MMOs); these structures are generated by a forced Bonhoeffer–van der Pol (BVP) oscillator. Nested MMOs, despite their complexity, are robust, repeatable, and composed of higher dimensional oscillations. In this study, we show that nested MMOs can also be produced by the forced van der Pol oscillator. This paper is of extreme importance as it demonstrates that nested MMOs are not a phenomenon that can only be observed in specific BVP systems. Based on the fact that nested MMO bifurcations also occur in the forced van der Pol oscillator, this study indicates that nested MMOs could represent a phenomenon that is more widely observed than previously believed. • The van der Pol equation has played significant roles in nonlinear science. • Nested mixed-mode oscillations (MMOs) occur in the forced van der Pol equation. • Nested MMOs are a complex phenomenon that evidently has strong orders. • MMOs are at least doubly nested in the forced van der Pol oscillator. • This work indicates that nested MMOs are a widespread phenomenon. [ABSTRACT FROM AUTHOR]

The exploration of the nonlinear dynamics related to the new coupled Konno–Oono equation, which determines the propagation of magnetic fields, is the focus of this work. Through the employing of Lie group analysis, the bifurcation phase portraits, and chaos theory, the project will investigate symmetry reductions in dynamical systems and examine the dynamic behavior of perturbed dynamical systems. The 3D phase portrait, 2D phase portrait, Lyapunov exponent, time series analysis, sensitivity analysis, and an examination of the existence of multistability in the autonomous system under various initial conditions constitute a few of the methods used for recognizing chaotic behavior. Furthermore, the investigation constructs general solutions for solitary wave solutions, such as exponential and hyperbolic function, singular, dark, and bright soliton solutions, by using the new Kudryashov methodology to determine the investigated equation analytically. These solutions are shown graphically as 2D, 3D, and contour plots with specifically selected values. They include as well with the related constraint circumstances. Additionally, a discussion and a visual illustration of the considered equation's sensitivity analysis are presented. The observations demonstrate that the aforementioned approach is an effective procedure for treating a variety of nonlinear PDE systems that arise in nonlinear physics analytically. The plot of the Lyapunov exponents is employed to validate the chaotic dynamics of the studied model. Additionally, the multiplier method is employed to determine the conserved vectors for the analyzed problem. • Coupled Konno–Oono equation within the domain of magnetic fields is considered. • The Lie invariance condition is applied to identify symmetry generators. • Soliton solutions are derived and phase portrait analysis has been done. • Chaotic behaviors are examined with different characteristic settings. • Conserved vectors for the studied equation are calculated. [ABSTRACT FROM AUTHOR]

This paper is subjected to study the nonlinear integrable model which is the (3+1)-dimensional Boussinesq equation which has a lot of applications in engineering and modern sciences. To find and examine the analytical exact solitary wave solutions of (3+1)-dimensional Boussinesq equation, a modified generalized exponential rational functional method is exerted. As a result, waves, singular periodic, hyperbolic, and trigonometric type solutions are obtained. These acquired solutions are more innovative and encouraging to researchers in their endeavor to study physical marvels. To illustrate how some selected exact solutions propagate, the graphical representation in 2D, Contour, and 3D of those solutions is provided with various parametric values. The considered equation is additionally transformed into the planar dynamical structure by applying the Galilean transformation. All potential phase portraits of the dynamical system are investigated using the theory of bifurcation. The Hamiltonian function of the dynamical system of differential equations is established to see that, the system is conservative over time. The presentation of energy levels through graphics provides valuable insights, and it demonstrates that the model has solutions that can be expressed in closed form. The periodic, quasi-periodic, and chaotic behaviors of the 2D, 3D, and time series are also observable once the dynamical system is subjected to an external force. Meanwhile, the sensitivity of the derived solutions is carefully examined for a range of initial conditions. [ABSTRACT FROM AUTHOR]

Simple-form chaotic systems with complex behaviors have always attracted the attention of researchers. This article introduces a novel 4D chaotic system that has only a quadratic nonlinear term as compared with the existing no-equilibrium continuous chaotic ones with two quadratic or higher nonlinearity. Despite the simplicity of its model, this new system exhibits complicated chaotic behaviors. Many interesting phenomena including the coexistence of different hidden attractors, offset boosting and antimonotonicity have been discovered by various numerical analysis such as bifurcation diagrams, Lyapunov exponent spectra, and phase portraits. The further circuit simulation of this new system verifies its feasibility and practicality. Moreover, the new system is applied to construct Pseudo-Random Number Generators (PRNG), following a thorough assessment of its complexity and randomness by sample entropy. The generated numbers are subsequently assessed using 15 statistical tests. The results confirm that the new system can produce unpredictable numbers with high randomness, making it ideal for information encryption. [ABSTRACT FROM AUTHOR]

The driving mechanism of spatiotemporal chaos is one of the most important questions in the dynamics of predator-prey systems. In this paper, we investigate the spatiotemporal chaotic behaviors driven by refuge effect and their complexity in a Lotka-Volterra discrete predator-prey system. By choosing refuge effect as the bifurcation parameter, the existence conditions of Neimark-Sacker and flip bifurcations near the stable equilibrium point are analyzed. The bifurcation diagrams, maximum Lyapunov exponents, phase diagrams, diagrams of space-amplitude and space-time, spatial return maps and Kolmogorov-Sinai entropy are used to analyze the complex dynamical behaviors of the system. The results show that the refuge effect has a two-way effect on the stability of the system, and both Neimark-Sacker and flip bifurcations caused by it open the route to chaos. The increase in unpredictability of chaotic attractors makes the system increasingly chaotic characteristics. In addition, the characteristics of spatiotemporal dynamics on the route to chaos are captured. The chaotic behaviors of the system on the two types of bifurcations show transitions among several generic features, including frozen random pattern, defect chaotic diffusion pattern, pattern competition intermittency and fully developed turbulence. Our study promotes the understanding of the complex spatiotemporal dynamics of refuge-driven spatiotemporal chaos in predator-prey systems. • The refuge effect drives spatiotemporal chaos in the predator-prey system. • The system opens the route to chaos through two types of bifurcations. • The characteristics of chaos are captured and four chaotic patterns are found. • It provides a new perspective for future experiments on the refuge effect. [ABSTRACT FROM AUTHOR]

For finite time series, it is difficult to provide a convincing determination scheme for specifying relevant parameters of entropy. This article proposes a new entropy, σ k E n , based on the amplitude fluctuations of time series segments. It uses the optimal variance estimation under the principle of minimizing A M S E ( σ ˆ 2) to adaptively select the window width, and then realizes the no-argument calculation process of volatility entropy through kernel density estimation. By testing with artificial data and applying to real-world data, it has been verified anti-noise and stability properties of σ k E n. In a background noise environment with an intensity of 20%, σ k E n distinguish various chaotic systems and random signals efficiently. In addition, this manuscript also proposes the plane of (σ k E n , N k E n) and (σ ̄ , σ k E n) , which also successfully identifies 6 types of chaotic systems, and 3 random noise signals. Furthermore, (< σ ̄ > , < σ k E n >) curve exhibits different dynamical behavior patterns for these chaotic systems. Finally, we employ (σ ̄ , σ k E n) as a feature and take 8 usual machine learning models, such as Linear Discriminant, Gaussian Naive Bayes and Subspace Discriminant, etc, to perform anomaly detection and fault classification tests on 6 open datasets. Given the appropriate classifiers, our method can achieve a classification accuracy of 100% for four tasks, and an average accuracy of over 96% for the remaining two. This is significantly superior than the accuracy obtained by using Permutation Entropy as a feature. • Based on volatility of fragments, we introduce an parameter-free entropy, σ k E n. • Optimal estimator and KDE yield a stable, anti-noise complexity measurement. • Distinguish various deterministic and stochastic singles in noisy environment. • Test on 6 open dataset, get 100% accuracy in 4 diagnosis tasks, 96% in other 2. [ABSTRACT FROM AUTHOR]

Recently, the color image encryption algorithm based on chaos theory has become the focus of current research. When encrypting color images, the common practice is to treat color images as different gray components and process them severally, which results in more redundancy and low efficiency. The security of chaotic cryptosystems also depends on the performance of the chaotic systems adopted. The structures of many current chaotic systems are fixed, making their behaviors highly predictable. Additionally, the range of chaotic region parameters is limited and discontinuous. To solve the above-mentioned problems, a new color image encryption algorithm (CIEA) using fractal and chaos theory is presented, which fully considers the inherent connection among the RGB components of color images. First, we propose a variable-structure discrete hyperchaotic system (VSDHS) to solve the dilemma encountered by existing chaotic systems. The excellent dynamic properties of VSDHS are verified by rigorous mathematical proof and simulation performance analyses. Then, VSDHS-CIEA is designed using fractal theory and VSDHS. The algorithm makes full use of the inherent connection among the different components of color images and performs cross-plane confusion. Simulations and performance analyses prove that VSDHS-CIEA has higher security and better performance than some representative image encryption algorithms. [ABSTRACT FROM AUTHOR]

The paper introduces a novel memristive chaotic system characterized by an infinite number of index-2 saddle foci, enabling it to generate multiscroll chaos and exhibit extreme multistability. Bifurcation diagrams, phase portraits and other methods are employed to examine the stabilities of equilibria and complex dynamics. It shows that by modifying the function of memristor, the system can produce multiscroll attractors with varying scroll counts. Furthermore, it can be decomposed into coexisting chaotic attractors at different locations, and this decomposition is influenced by adjustments in parameters and initial values, illustrating the impact of initial-relied and parameter-relied offset boosting. With variations in the parameter, the coexisting chaotic attractors will undergo a bifurcation, ultimately transforming into coexisting periodic attractors. The image encryption application of the system is explored, introducing an efficient chaos-based algorithm applied to encrypt Internet of Medical Things (IoMT) images, followed by a comprehensive performance evaluation. • New memristive system with multiscroll chaos and offset boosting is presented. • Complex dynamics of the system are theoretically and numerically studied. • New chaos-based image encryption algorithm with high security is designed. [ABSTRACT FROM AUTHOR]

The objective of this manuscript is to examine the non-linear characteristics of the modified equal width-Burgers equation, known as the generalized Kadomtsive–Petviashvili equation, and its ability to generate a long-wave with dispersion and dissipation in a nonlinear medium. We employ the Lie symmetry approach to reduce the dimension of the equation, resulting in an ordinary differential equation. Utilizing the newly developed generalized logistic equation method, we are able to derive solitary wave solutions for the aforementioned ordinary differential equation. In order to gain a deeper understanding of the physical implications of these solutions, we present them using various visual representations, such as 3D, 2D, density, and polar plots. Following this, we conduct a qualitative analysis of the dynamical systems and explore their chaotic behavior using bifurcation and chaos theory. To identify chaos within the systems, we utilize various chaos detection tools available in the existing literature. The results obtained from this study are novel and valuable for further investigation of the equation, providing guidance for future researchers. • Application of the generalized logistic equation method to address the dynamics of solitary wave solutions for the generalized KP–MEW-Burgers equation. • Computation of the invariant transformations and symmetry reductions using Lie symmetry analysis. • Bifurcation and phase portraits of the unperturbed dynamical system using the idea of bifurcation theory of dynamical systems. • Chaos behavior of the perturbed dynamical system is identified through different chaos detecting tools using chaos theory of dynamical systems. [ABSTRACT FROM AUTHOR]

Communication technologies play a key role in various fields. Hybrid Soft Computing approaches have significant potential for design and investigation of complex systems. In this paper, we use combination of fuzzy logic and chaos theory to model uncertainty and complexity of a secure communication system. Fuzzy rule base is used to describe dependence of behavior of a chaotic system on its parameters and initial conditions. The rule base is constructed by applying fuzzy clustering to a large data set. The approach is characterized by its relatively low computational complexity due to the use of fuzzy rule base instead of intensive simulations of a fuzzy chaotic system. Complexity of this hybrid fuzzy-chaotic approach assures an increased level of security. Trade-off between complexity and security may be achieved by generation of transmitted information using fuzzy modeling of chaotic behavior. Computer simulations are used to verify feasibility and effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

No isolated system may ever support complexity. Emergent phenomena may however appear in an open system, if, as established by the Edge of Chaos theory, some of its constitutive elements feature the capability to amplify infinitesimal fluctuations in energy, provided an external source supplies them with a sufficient amount of DC power, which is known to be a signature for locally-active behaviour. In particular, complex behaviours, including static and dynamic pattern formation, may emerge in arrays of identical diffusively-coupled cells, if and only if the basic unit is poised on a particular sub-domain of the Local Activity regime, referred to as Edge of Chaos, within which a quiet state hides in fact a high degree of excitability. Here we show, for the first time, that these counterintuitive phenomena may emerge in a basic memristor cellular neural network, consisting of two identical diffusively-coupled second-order cells. The proposed bio-inspired array represents the simplest ever-reported open system, which reproduces the shocking phenomenon, reported by Smale in 1974, when, while studying a model from cellular biology, he observed two identical reaction cells, “mathematically dead” on their own, pulsating together upon diffusive coupling. Impressively, the bio-inspired two-cell reaction-diffusion network contains only nine circuit elements, specifically two DC voltage sources, three linear resistors, two linear capacitors, and two functional niobium oxide (NbO) memristors from NaMLab. Applying the theory of Local Activity to an accurate model of the memristor oscillator, a comprehensive picture for its local and global dynamics may be drawn, providing a systematic method to tune the design parameters of the two-cell array to enable diffusion-driven instabilities therein. [ABSTRACT FROM AUTHOR]

The author reflects on his struggle to find a topic for his column due to the decision by the major newspapers in Melbourne, Australia to distribute their papers through a single company instead of through a range of news agencies. Topics discussed include his description on their suburb's newsagent, where much of the coverage of public life is generated, and limited value of newspaper commentary on public life.

This paper introduces a real-time experimental set-up design to realize the robust synchronization of chaotic systems with Markov-Jump behavior, where the control law is designed based on a finite-time sliding-mode controller. First, master and slave chaotic systems are designed with the dynamics of biological Hindmarsh–Rose (HR) neurons using electronic circuit elements. In order to create the Markov-Jump behavior, the printed circuit board is adjusted to add or remove the resistors so that it is possible to obtain the time-varying dynamic of a chaotic system in the synchronization process. In the realization of chaotic neurons, Matlab and Multisim environments are used for the simulations, and Proteus software is utilized for the design of PCB layout. The Arduino microcontroller is used for signal processing and closed-loop control, where the states of chaotic neurons are recorded and control signals are produced and applied to the slave chaotic system to be synchronized. The real-time data corresponding to the behavior of the circuits was recorded, and Lyapunov exponents were calculated to check whether the neuron circuits are chaotic or not. Second, a sliding-mode controller (SMC) was designed to guarantee a finite-time convergence of synchronization error where its parameters are optimized using a recently developed efficient optimization method, namely the adolescent-identity search algorithm. Robust finite-time synchronization results with sliding-mode control were recorded in simulation and real-time experiments. Finally, the difficulties encountered in general and unachievable results related to experimental system design and finite-time synchronization are discussed for future studies. • Design of real-time HR neuron circuits with Markov jump dynamics. • Positive Lyapunov exponents of real-time HR neuron circuits. • Finite-time sliding-mode synchronization using analog circuit simulation. • Finite-time sliding-mode synchronization using the experimental set-up. [ABSTRACT FROM AUTHOR]

Most of the explored chaotic systems are either of dissipative or conservative behavior. The dissipative or conservative behavior of a chaotic system is governed by its divergence of volume. The system shows a dissipative nature for negative divergence whereas for zero divergence, a conservative nature is shown by the system. In most of the literature, these behaviors are shown in a chaotic system either with the change in system parameters or with the initial conditions. However, a chaotic system that can show both dissipative and conservative behavior based on the initial conditions and also on the system parameters is rare in the literature. Therefore, a novel chaotic system is proposed that can meet such peculiarities. In addition, the proposed system also possesses attractors of self-excited and hidden nature. The system is developed by using a Hamiltonian approach of modifying the skew-symmetric matrix and adjusting the rest of the system parameters to control the system divergence. The proposed system boasts the presence of heterogeneous multistability, extreme multistability and the coexistence of conservative flow and dissipative attractor. Many qualitative inspections are done using phase portraits, Lyapunov spectrums, bifurcation diagrams and Poincare maps to support the claims. Further, to demonstrate the controllability of the chaotic system a nonlinear active controller with single control input is designed. The real-time application of the proposed system is validated by the circuit simulation in NI Multisim and hardware implementation on a Arduino board. [ABSTRACT FROM AUTHOR]

In this paper, we establish the theory of propagation of chaos and propose an Euler–Maruyama method for McKean–Vlasov SDEs driven by fractional Brownian motion with Hurst parameter H ∈ (0 , 1 / 2) ∪ (1 / 2 , 1). Meanwhile, upper bounds for errors in the Euler–Maruyama method are obtained. Two numerical examples are demonstrated to verify the theoretical results. • The convergence rates of the numerical method for solving McKean–Vlasov stochastic differential equations driven by fractional Brownian motion with Hurst parameter H ∈ (0 , 1 / 2) ∪ (1 / 2 , 1) are shown. • The corresponding propagation of chaos is obtained. [ABSTRACT FROM AUTHOR]

The chaotic jerk system represents a class of nonlinear systems with unique, varied, and interesting characteristics and behavior. In this study, we proposed a modified fractional-order chaotic Jerk system by the introduction of two novel nonlinear terms and two constant terms. The nonlinear terms introduced are the exponential and tangential functions while the two parameters are a and b. Stability analysis of the fractional-order Jerk system was used to establish the parameter ranges for chaotic behavior in the system. Chaos in the system was confirmed via the Lyapunov exponents. The bifurcation diagram with respect to the parameters, a and b , as well as the fractional order, α were examined. The synchronization of the system was implemented using the active control method. Finally, the studied novel circuit was realized using MultiSIM electronic software for experimental validation. • New novel Fractional order Jerk system with exponential and Tangential nonlinearity. • Active control for detecting complete synchronization and anti-synchronization. • Electronic realization of the new novel system using off-shelf component. [ABSTRACT FROM AUTHOR]

The article urges readers to embrace life's challenges, acknowledge their own humanity, and find beauty and purpose amidst the chaos, ultimately calling for a collective prayer for guidance and resilience.

The dynamical and chaotic behaviors of natural convection flow in the semi-annular and full-annular cylindrical domains are studied using the method of isolation by spectral expansion. For each of the flow cases, spectral models with different orders are obtained and dynamical characteristics of the flow are investigated using numerical simulations. For the semi-annular case of study, results reveal that in all the generated spectral models, chaotic solution appears, but the onset of chaos between each system requires different set of control parameters. Bifurcation diagrams are provided for all the systems which show the ranges of periodic and chaotic behavior of the flow. Strange attractors are captured for all the systems with different orders. For the lowest-order model, a Lorenz-like attractor is produced, which has two distinguishable scrolls similar to Lorenz's famous attractor and for the other high-order models, the shape of strange attractor gets different. For the full-annular case of study, the physical characteristics of the flow are obtained. The results show that the produced thermal plume on the top of annulus gets higher fluctuations as the flow Rayleigh number gets higher. At a specific Rayleigh number, the flow behavior gets completely unstable, which shows a transitional regime change. At this state of flow, the domain is dominated by several vortices. [Display omitted] • The dynamical behavior of natural convection in a cylindrical annulus is studied. • Spectral models with different orders are obtained for the fluid flow system. • Numerical simulations are made in order to capture the flow field of study. • Bifurcation diagrams are provided for the high-order spectral models of flow system. • The flow transition from a stable steady to an unstable point is discussed. [ABSTRACT FROM AUTHOR]

The author focuses on the chaos theory referring to version of director Ashton Kutcher in his film "The Butterfly Effect." Topics discussed include study of chaos theory by meteorologist Edward Norton Lorenz in 1963, book "The Essence Of Chaos" by Lorenz, and television (TV) program "Heartbreak High."

Recent studies demonstrate that the density of prey population is not only affected by direct killing by the predator, but the fear in prey caused by predator also reduces it by cutting down the reproduction rate of prey community, and prey shows anti-predator behavior in response to this fear. In this study, we propose a prey–predator model with fear in prey due to predator and anti-predator behavior by prey against the predator with fear response delay and gestation delay. It is assumed that the predator consumes prey via simplified Holling Type-IV functional response. We evaluate the equilibrium points and study the local and global stability behavior of the system around them. It is observed that our system undergoes Hopf-bifurcation with respect to the fear parameter. Moreover, the system shows the attribute of bi-stability involving two stable equilibriums. Further, we study the dynamics of the delayed system by incorporating fear response delay and gestation delay. We observe that the delayed system suffers Hopf-bifurcation with respect to both delays. Using the normal form method and center manifold theory, the direction and stability of Hopf-bifurcation are studied. Chaotic behavior for delayed system is observed for large values of fear response delay. All these findings are supported by numerical simulation. [ABSTRACT FROM AUTHOR]

The fast growth in digital image transmission technologies requires more secure and effective image encryption schemes to provide essential security. In this paper, we present a novel one-dimensional chaotic map amplifier (1-DCMA). The evaluation of the proposed chaotic system shows that the 1-DCMA improve the chaotic behavior, control parameters' structure, and sensitivity of the 1-D chaotic maps used as input. We further implement a chaotic map generated by the 1-DCMA in a new asymmetric image encryption scheme (Amp-Lg-IE). Using the secret key, the proposed encryption algorithm adds rows and implement a new index representation (IR) concept with shifting sequences to manipulate the pixels' positions and values synchronously. Finally, we execute bit-level operations to obtain the ciphered image. The simulation and security analysis prove that the Amp-Lg-IE, in a satisfying time, can encrypt a plain image into an unidentified random-like one with high resistance to different types of threats and attacks. [ABSTRACT FROM AUTHOR]

The dynamic behavior of a fractional governor system is studied in this paper. The Stability and bifurcation of the equilibrium points of the system are investigated. We derive specific conditions for which the Hopf bifurcation of the fractional governor system may occur. It can be seen that different results are obtained compared to the classical mode. In the non-autonomous system, the tendency towards chaos is investigated using diagrams of bifurcation and Poincare maps analysis. Finally, the numerical results are given to illustrate the theoretical results. Analytical and numerical simulations result could be extended to other systems and ultimately, these results could be applied as a technical tool for the control and rotary machine designers. • Bifurcation analysis of a fractional governor system is determined analytically and numerically. • The conditions under which a Hopf bifurcation may occur are investigated. • Numerical simulations are performed to substantiate our analytical findings. • The tendency towards chaos in the non-autonomous system is investigated using diagrams of bifurcation and Poincare maps analysis. [ABSTRACT FROM AUTHOR]

Adaptive behavior on networks leads to bifurcation and other special phenomenon, which has been proved for differential systems. In this paper, one will discuss the stability, bifurcation and chaos of the discrete pair epidemic model on adaptive networks, which will bring some new challenges. The stability of the disease free equilibrium and endemic equilibrium with respect to basic reproduction number R 0 is studied. Under certain conditions, as the time step-size increases, flip bifurcation that occurs at the endemic equilibrium is discussed, the period-doubling bifurcation and the chaos phenomenon of the system are exhibited, some figures are provided for illustration. According to the results in this paper, the dynamical behaviors of the discrete model are multitudinous and quite different from the continuous model. [ABSTRACT FROM AUTHOR]

The article looks at how chance shapes the lives of humans and their attempts to influence and understand it in their everyday lives, through the science of basic quantum processes. Topics include scientist Isaac Newton's view of the universe, which the author argues left little room for randomness, a historical overview of the connection between a universe that admits randomness and one that admits free will, with comments from physicist Valerio Scarani, the beginnings of chaos theory, which are systems that appear to behave at random even though they are controlled by strict mathematical processes, and physicist John Bell's test for determining hidden variables in quantum systems.

Chaos theory has been widely used in the design of image encryption schemes. Some low-dimensional chaotic maps have been proved to be easily predictable because of their small chaotic space. On the other hand, high-dimensional chaotic maps have a larger chaotic space. However, their structures are too complicated, and consequently, they are not suitable for real-time image encryption. Motivated by this, we propose a new fractional one-dimensional chaotic map with a large chaotic space. The proposed map has a simple structure and a high chaotic behavior in an extensive range of its control parameters values. Several chaos theoretical tools and tests have been carried out to analyze and prove the proposed map's high chaotic behavior. Moreover, we use the proposed map in the design of a novel real-time image encryption scheme. In this new scheme, we combine the substitution and permutation stages to simultaneously modify both of the pixels' positions and values. The merge of these two stages and the use of the new simple one-dimensional chaotic map significantly increase the proposed scheme's security and speed. Besides, the simulation and experimental analysis prove that the proposed scheme has high performances. [ABSTRACT FROM AUTHOR]

The dynamics of COVID-19 is investigated with regard to complex contributions of the omitted factors. For this purpose, we use a fractional order SEIR model which allows us to calculate the number of infections considering the chaotic contributions into susceptible, exposed, infectious and removed number of individuals. We check our model on Wuhan, China-2019 and South Korea underlying the importance of the chaotic contribution, and then we extend it to Italy and the USA. Results are of great guiding significance to promote evidence-based decisions and policy. [ABSTRACT FROM AUTHOR]

This paper presents a controller-based synchronization scheme in a leader-follower architecture for chaotic maps of fractional variable-order. The proposed scheme demonstrates successful synchronization of three different chaotic maps through simulation results. The implementation on Arduino UNO boards is used to showcase the relatively low algorithm complexity of the proposed scheme. Additionally, the paper explores a potential application of the proposed scheme in an RGB image encryption system, leveraging the chaotic behavior of the system for pseudo-random sequence generation. The encoding scheme utilizes the complementary property of Deoxyribonucleic acid (DNA) molecule bases. The presented encryption scheme employs the variable-order of the fractional derivative as a key for improved safety. This enhancement improves the methodology resistance against brute-force attacks while maintaining resilience against other cryptoanalysis techniques. • We present a controller-based synchronization scheme. • The proposed scheme demonstrates successful synchronization of chaotic maps. • We implemented the algorithm on Arduino UNO boards. • We explore a potential application in an RGB image encryption system. • The encoding scheme utilizes the Deoxyribonucleic acid (DNA) molecule bases. [ABSTRACT FROM AUTHOR]

In the world of economics, financial risk is a significant issue that concerns countries and all economic agents. It refers to the potential for experiencing losses due to unpredictable fluctuations in financial or investment activities caused by endogenous factors. Chaos theory provides an explanation for the complexity observed in economic time series, as it supports an endogenous view. This article proposed an enhanced chaos-based financial risk system with one stable and two unstable equilibrium points. The existence of stable and unstable equilibrium points shows that the proposed system has both hidden and self-excited chaotic attractors. By analyzing its dynamic behavior, the study unveils the transition from stability to chaos in the improved financial risk system, while also describing how the complex dynamics of the system affect changes in financial market risk. In addition, this study explores the fundamental dynamic behavior of the financial risk system, including stability analysis, Lyapunov exponent, bifurcation analysis, coexisting attractors, 0–1 test, algorithm and spectral entropy (SE). Finally, with the help of integral sliding mode control (ISMC), we discover the outcomes for the asymptotic synchronization of two chaotic financial systems that are constructed as master and slave systems. Simulations are provided to accompany and support the findings from this research. [ABSTRACT FROM AUTHOR]

The problem of heteroclinic chaos detecting is considered with the help of the Melnikov method. This paper presents the Melnikov method adaptation based on investigation of a nanosatellite attitude dynamics in the presence of internal dissipation properties. The need for this adaptation is determined by dynamical aspects of perturbing oscillations acting with damping amplitudes. In this case formal analytical exponential growth of perturbations values in reverse time takes place while the classical Melnikov integral does not converge. The adaptation of the Melnikov method allows investigating the heteroclinic chaos in mechanical systems with internal dissipation properties. As a prime example of this adaptation, the research presents the complete study of chaotic attitude dynamics of a nanosatellite with a slightly movable unit under action of its damped oscillations considered as perturbations. Moreover, the considered research of the nanosatellite attitude dynamics discovers a range of important tasks in the area of a rigid body dynamics under different perturbations with damping. • The problem of detecting heteroclinic chaos using the modified Melnikov method is considered. • Some modifications of the classical Melnikov method are fulfilled. • The chaotic dynamics of a nanosatellite with a slightly movable unit is investigated. • Boundary values of the damping are found, which define the parametric border of the heteroclinic chaos arising. [ABSTRACT FROM AUTHOR]

This article analyzes a discrete-time three-species model of phytoplankton and zooplankton with infochemical producing, non-infochemical producing, and zooplankton. It is assumed that the grazing of infochemical producing phytoplankton indirectly increases the mortality rate of zooplankton. The fixed points of the system and their respective local stability has been examined. The conditions for Neimark–Sacker bifurcation along with period doubling bifurcation at the positive fixed point are outlined. The sufficient conditions for global asymptotic stability of the positive fixed point are obtained. The application of a hybrid control strategy effectively stabilizes the chaotic behavior of the system. Lastly, our theoretical discussion is solidified through a numerical simulation which enables us to witness the complex behavior of the system. [ABSTRACT FROM AUTHOR]

The concept of self-similarity, widely employed in specialized fields of applied sciences, has been investigated through numerous works utilizing both analytical elements and experimental observations. As a significant component of both fractal and chaos theories, self-similarity can naturally manifest around us or be artificially simulated using mathematical arguments and algorithms. In this paper, we utilize analytical, numerical, and experimental circuit elements to study, analyze, and simulate a generalized chaotic system of attractors with multiple scrolls. This system presents various types of processes, including chaotic and hyperchaotic dynamics, as well as hidden attractors. We discuss the well-posedness of the system before examining its numerical solvability, error tolerance, and simulations for the model involving fractal and fractional operations. Simulations are conducted for various values of the model's parameters, demonstrating that the system displays fractal features combined with chaotic behavior. The system is implemented using a Field Programmable Gate Array (FPGA) board, generating self-similar results akin to those obtained numerically. • A chaotic model of attractors with multiple scrolls is analyzed. • The model presents processes that are chaotic, hyperchaotic & with hidden attractors. • Self-similarity is generated using analytical and numerical methods. • Circuit implementation is performed to recover and verify the self-similarity. [ABSTRACT FROM AUTHOR]

This paper investigates the global dynamics of a new 3-dimensional fractional-order (FO) system that presents just cross-product nonlinearities. Firstly, the FO forced Lorenz-84 system is introduced and the stability of its equilibrium points, as well as the chaos control for their stabilization, are addressed. Secondly, dynamical behavior is further analyzed and bifurcation diagrams, phase portraits, and largest Lyapunov exponent (LE) are discussed. Then, the global Mittag-Leffler attractive sets (MLASs) and Mittag-Leffler positive invariant sets (MLPISs) of the FO forced Lorenz-84 system are presented. Finally, the Hamilton energy function (HEF) of the Lorenz-84 system is calculated by using the Helmholtz theorem. The calculation of the Hamilton energy has an essential role on the estimation of chaos in dynamical systems, the guidance of orbits, and stability. In fact, any control action on the dynamical system completely changes the HEF. Numerical simulations are presented for illustrating the theoretical findings. • Presents global dynamics of a new 3-dimensional fractional-order system. • Discusses the stability of equilibrium points as well as chaos control. • Studies global Mittag-Leffler attractive and positive invariant sets. • Calculates Hamilton energy function by using the Helmholtz theorem. • Presents numerical simulations for illustrating the theoretical findings. [ABSTRACT FROM AUTHOR]

This paper investigates the global dynamics of a new 3-dimensional fractional-order (FO) system that presents just cross-product nonlinearities. Firstly, the FO forced Lorenz-84 system is introduced and the stability of its equilibrium points, as well as the chaos control for their stabilization, are addressed. Secondly, dynamical behavior is further analyzed and bifurcation diagrams, phase portraits, and largest Lyapunov exponent (LE) are discussed. Then, the global Mittag-Leffler attractive sets (MLASs) and Mittag-Leffler positive invariant sets (MLPISs) of the FO forced Lorenz-84 system are presented. Finally, the Hamilton energy function (HEF) of the Lorenz-84 system is calculated by using the Helmholtz theorem. The calculation of the Hamilton energy has an essential role on the estimation of chaos in dynamical systems, the guidance of orbits, and stability. In fact, any control action on the dynamical system completely changes the HEF. Numerical simulations are presented for illustrating the theoretical findings. • Presents global dynamics of a new 3-dimensional fractional-order system. • Discusses the stability of equilibrium points as well as chaos control. • Studies global Mittag-Leffler attractive and positive invariant sets. • Calculates Hamilton energy function by using the Helmholtz theorem. • Presents numerical simulations for illustrating the theoretical findings. [ABSTRACT FROM AUTHOR]

Recently, the majority of research in chaos field has focused on dissipative chaos systems, while conservative chaos systems, particularly conservative hyperchaos systems, have been largely overlooked. However, these systems possess more intricate dynamic properties and greater potential for practical applications than general chaos systems. This paper aims to address this gap by introducing two linear terms to the system equation of an existing five-dimensional Hamiltonian conservative hyperchaos system, based on the construction techniques of the existing conservative chaos system. As a result, a five-dimensional non-Hamiltonian conservative hyperchaos system without time-reversal symmetry is obtained, which exhibits complex dynamic properties. Specifically, the system displays multi-stable characteristics with consistent parameters but varying initial conditions. Furthermore, during the study of the system's transient characteristics, a unique phenomenon was observed, namely intermittent chaos and intermittent quasi-periodic characteristics. Finally, the system's sequences were evaluated using the NIST test, and the results indicated their good level of pseudo-randomness. [ABSTRACT FROM AUTHOR]

Chaos theory is a branch of mathematics that studies systems highly sensitive to initial conditions. Various methods for analyzing and controlling these systems, such as bifurcation theory and amplitude control, have been developed and applied in diverse fields, including cryptography, physics, and engineering. In this study, we introduce a new class of Bernoulli-Like Shift (BLS) maps. These maps are generated by using a specific class of Piecewise Linear (PWL) maps combined with a modulo operation. We conduct a fixed point analysis by establishing propositions and corollaries, which outline the conditions that guarantee the presence or absence of fixed points. Additionally, we provide numerical examples to validate these theoretical results. Utilizing bifurcation diagrams and Lyapunov exponent calculations, we demonstrate that the range of chaotic behavior in the BLS maps is broader than those of the logistic and Bernoulli Shift maps. Moreover, we show that the BLS maps possess higher values of Lyapunov exponents and a better distribution than both the Bernoulli shift and logistic maps. Lastly, we introduce a transformation that generates a modified BLS map, allowing for amplitude control. This modification enables the regulation of amplitude while maintaining constant Lyapunov exponents. • A new class of Bernoulli-like shift maps is introduced overcoming numerical simulation limitations of other PWL maps. • Theoretical and numerical results validate the existence or non-existence of fixed points in these maps. • A transformation is proposed to allow amplitude control in the Bernoulli-like shift maps. • A broad range of chaotic behavior than the logistic and Bernoulli-shift maps. [ABSTRACT FROM AUTHOR]

In this paper, the modified Kadomtsev Petviashvili (mKP) equation is derived by considering the dynamics of the dust-acoustic waves (DAWs) with kappa-distributed hot and cold ions and Boltzmannian electrons. The reductive perturbation technique is applied for deriving mKP. The bifurcation theory of the planar dynamical system is used to obtain the phase portrait of the DAWs in the framework of mKP equation. The formation of dust-acoustic solitary waves (DASWs) is analyzed for different plasma parameters in the Saturn's magnetosphere. In the current plasma model, the external periodic force is introduced to study the quasiperiodic and chaotic behavior of the DAWs. It is noted that periodic initial conditions lead to the emergence of many outlandish features in the system by comparison with the solitary initial conditions. The frequency of the external periodic force ω plays an important role in the transition from quasiperiodic to chaotic behavior. Moreover, the strength of the external periodic force, the value of the nonlinear coefficient of the mKP equation, and the nonlinear acoustic speed are also found to have a significant effect on the chaotic behavior of the system. • Dynamical system and phase portrait analysis of the mKP Equation. • The bifurcation theory is applied to obtain the phase portrait of the dust-acoustic waves (DAWs). • Dust-acoustic solitons are investigated for Saturn's magnetosphere plasma parameters. • External periodic force (EPF) is introduced for studying the chaotic behavior of DAWs. • The EPF frequency has a great role in the transition from quasiperiodic to chaotic behavior. [ABSTRACT FROM AUTHOR]

A Duffing-Holmes system containing nonlinear damping is used to investigate some dynamical properties of the system under the combined action of constant excitation and simple harmonic excitation. The harmonic balance method is used to find the main vibration equation of the system and obtain the amplitude-frequency response relationship. In the analysis of the global characteristics of the system, the Melnikov method is applied to analyze the necessary, insufficient conditions for the analysis of the system chaos, and the correctness of the analytical solution is verified by numerical calculations; the effect of the constant excitation amplitude on the global characteristics of the system is analyzed by the cell mapping method. The paper shows: gradually increasing the constant excitation amplitude causes the system amplitude-frequency response to appear rigidly asymptotic phenomenon; the boundary curve of the system generating chaos in the sense of Smale's horseshoe, with the increase of excitation frequency ω, the curve first decreases, then rises, finally tends to infinity, and the possibility of chaos occurs most near the resonance frequency of the system; with the change of the constant excitation amplitude, The number of attractor domains and attractors in the global attraction domain of the system changes substantially with the change of the constant excitation amplitude, and the attractor domains are entangled with each other. • The harmonic balance method is used to solve the main resonance equation of the system, and the amplitude-frequencyresponse relationship is obtained. • In the analysis of the global characteristics of the system, the necessary conditions for the chaos of the system are studied by Melnikov method, and the stability of the system is studied by cell mapping method. [ABSTRACT FROM AUTHOR]