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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]

This is a call for papers that should contribute to the unification of behavioral sciences and team management, focusing on the biological origin of cooperation and swarm intelligence, moving from biology to psychology and from sociology to political science, with the help of the theoretical tools of complex networks. This issue should shed light into the origin of ergodicity breaking and contribute to establishing a connection, still lacking theoretical support, between complexity properties that are expected to be correlated. Examples are: non-Poisson renewal events and multi-fractality; complexity matching and chaos synchronization; criticality and extended criticality of small size systems. Although the emphasis is on systems of small size, and especially on the search of the size maximizing both information transport and cooperation emergence, special attention will be devoted to the interaction between small groups and their larger societies. [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]

Self-sustained chaotic system based on active materials, where energy is absorbed directly from the environment to maintain one's own motion, furnishes an extensive scope of applications in energy harvesters, encrypted communication, bionic heart devices and other fields. This paper seeks to put forward a self-sustained chaotic pendulum system consisting of a liquid crystal elastomer fiber and a mass sphere under steady illumination. To investigate the self-sustained chaotic behavior of the pendulum system, we combine the dynamic liquid crystal elastomer model with principles of dynamics to establish the corresponding theoretical model of the system. Numerical results suggest that three typical motion modes, namely, static mode, self-sustained oscillation mode and self-sustained chaotic motion mode, are involved in the liquid crystal elastomer pendulum. The self-sustained motion is maintained by the work done by the contraction of the liquid crystal elastomer fiber with a light-blocking coating, which compensates for the energy dissipated by the damping. Furthermore, this study also explores the influences of five system parameters on the motion behavior of the LCE pendulum, and determines the key parameter values for the three distinct motion modes through detailed calculations and bifurcation diagrams. The present research findings demonstrate that introducing a new degree of freedom into the self-sustained periodic vibration system, it is possible to achieve self-sustained chaotic motion, providing significant insights into the development of self-sustained chaotic systems derived from active materials. [ABSTRACT FROM AUTHOR]

• A novel piezoelectric inertial stick-slip driven pipeline robot is proposed to meet the detection requirements. • The main body is hollowed out to reduce mass and install actuators. • The nonlinear dynamics model of the pipeline robot is established, and analysis results for the chaos are obtained. • The occurrence conditions of chaotic behavior are determined, which provides theoretical reference for the design of the robot. Pipeline robots using the conventional driving mode have encountered a bottleneck in miniaturization. To address this problem, a micro piezoelectric pipeline robot based on the inertia stick-slip driving principle is proposed in this paper. The robot is well suited to the inspection needs of micro pipes. However, undesirable design parameters found during the structural optimization phase can lead to unstable operation and reduced load capacity of the robot, and can even cause the drive system to stop with chaotic behavior. Therefore, the nonlinear dynamics model of the four degree of freedom discrete system of the pipeline robot is established. The Runge-Kutta method is used to solve the dynamic response of each nonlinear subsystem. The nonlinear dynamical behavior of the system is analyzed through the bifurcation diagram, time domain diagram, phase diagram, Poincaré map and power spectrum diagram of each subsystem response. The stable operation intervals of the machine and electrical parameters in the drive system are given, and the conditions for the occurrence of chaotic behavior are determined. This research can be applied to most of the drive systems consisting of piezoelectric stacks and flexure hinges. It helps to determine and optimize the structural parameters of the piezoelectric actuator, thus avoiding chaotic behavior and ensuring that the actuator maintains good operational performance. [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]

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]

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]

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]

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]

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]

The economic system is an irreversible entropy increase process which is constructed by many elements and is far away from the equilibrium point; and affected by various parameters change, it is quite common that its motion state appears chaotic phenomenon due to instability. The extremely complex and not completely random aperiodic motion form of chaotic phenomenon is strongly sensitive to initial conditions. The development of nonlinear science, especially the emergence and development of chaos and fractal theory, has gradually become a powerful tool for economists to study the complexity, uncertainty and nonlinearity of social economic systems; and some visionary economists began to apply the research results of nonlinear science to economics, which has produced nonlinear economics. On the basis of summarizing and analyzing previous research works, this paper first obtains the non-degenerate equilibrium point of some typical fractional-order chaotic economic systems and transforms the equilibrium points of those systems to the origin through coordinate transformation, and then analyzes the Jacobi matrixes of new systems obtained through coordinate translation, and the parameter conditions of bifurcation in the economic systems are finally given and the numerical simulation of the fractional-order chaotic economic system evolution is carried out through bifurcation diagram, phase diagram and time series diagram. The study results of this paper provide a reference for the further study of the evolution of fractional-order chaotic economic systems with non-degenerate equilibrium points. [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]

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]

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]

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]

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 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]

Abstract This paper investigates the problem of stability and stabilization of a class of chaotic system through the use of sampled-data control. By employing a Takagi–Sugeno (T-S) fuzzy model to describe the chaotic system and using a time-dependent Lyapunov function, an exponential stability condition is derived for the resulting closed-loop systems with input saturation constraint. Based on this condition, a fuzzy sampled-data controller is designed to stabilize the systems under consideration. The results obtained in this paper are based on the actual characteristic of sampling model. They depend explicitly on both the upper and lower bounds of sampling intervals. The chaotic Lorenz system is considered and solved by using the proposed approach so as to demonstrate the benefits and the superiority of the proposed approach over existing methods. [ABSTRACT FROM AUTHOR]

Highlights • This paper is a response. • Some ideas are to address and discuss the comments. • Parameter identification of fractional-order chaotic systems is analyzed when there has noise and noiseless. • We published the relevant codes of the key program (Matlab) on a website. Abstract Parameter identification of chaotic systems without or with noise are two different cases of ideal and real situations. These two cases have been widely studied. This paper is a reply to the comments by Jafari et al. (2013) regarding the objective function in the process of parameter identification. The following scripts are to address and discuss the comments. Positive and satisfactory responses are provided to all the ambiguous questions. Furthermore, we have also analyzed parameter identification in fractional-order chaotic systems with noise and published the related key codes. Experimental results show that the proposed technique and adopted objective function is reliable, even with the presence of noise. [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]

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]

For non-invasively investigating the interaction between insulin and glucose, mathematical modeling is very helpful. In this paper, we propose a new model for insulin-glucose regulatory system based on the well-known prey and predator models. The results of previous researches demonstrate that chaos is a common feature in complex biological systems. Our results are in accordance with previous studies and indicate that glucose-insulin regulatory system has various dynamics in different conditions. One interesting feature of this new model is having hidden attractor for some set of parameters. The result of this paper might be helpful for better understanding of regulatory system that contains glucose, insulin, and diseases such as diabetes, hypoglycemia, and hyperinsulinemia. [ABSTRACT FROM AUTHOR]

In this paper, the fault-tolerant control (FTC) problem of nonlinearly chaotic systems via adaptive slide-mode control is investigated. Both the Denial-of-Service (DoS) attack and network faults are considered in the system. Perturbations caused by general nonlinear couplings, the signal attenuation, and network deterioration are regarded as network faults in this paper. A slide-mode control strategy is studied to guarantee the compensation of the faulted and perturbed couplings by means of adaptive estimations of unknown controller parameters. Then, based on the Lyapunov stability theory and mathematical analysis method, the adaptive slide-mode strategy is successful in achieving the asymptotic synchronization of nonlinearly chaotic systems. Finally, the simulations are performed to illustrate the effectiveness and feasibility of the design method. [ABSTRACT FROM AUTHOR]

In this paper we compute topological entropy with prescribed accuracy for different economic models, showing the existence of a topologically chaotic regime for them. In order to make the paper self-contained, a general overview on the topological entropy of continuous interval maps is given. More precisely, we focus on piecewise monotone maps which often appear as dynamical models in economy, but also in population growth and physics. Our main aim is to show that when topological entropy can be approximated up to a given error, it is a useful tool which helps to analyze the chaotic dynamics in one dimensional models. [ABSTRACT FROM AUTHOR]

This paper presents a new chaotic system that exhibits a two wing (2W) chaotic attractor in its integer order dynamics, three-wing (3W) and four-wing (4W) chaotic attractors in its fractional-order (FO) dynamics, and an eight-wing (8W) attractor in its nonautonomous fractional dynamics. An interesting feature of the proposed system is that two distinct periodic orbits coexist with a strange attractor that gradually evolves into a 4W attractor. The asymmetry, dissimilarity and topological structure of this proposed system with respect to those available in literature, manifest increased irregularity, which in turn indicate more chaos. Besides, the authors have drawn its comparison with various well-known fractional-order chaotic systems (FOCS)s to prove its enhanced features in terms of higher Lyapunov Exponent, fractional order orbital velocities, bandwidth, density, range of dynamical behaviour, etc. A control scheme is proposed to enable switching hybrid synchronisation between the 8W nonautonomous FOCS and the 4W autonomous FOCS, using the former as master and the latter as slave. This work throws light on the potential practical applicability of the proposed system by designing a circuit using minimum circuit components possible, thus signifying the objectives of the paper are finally achieved. [ABSTRACT FROM AUTHOR]

This paper deals with the stability and bifurcation of equilibria in a new chaotic fractional-order system in the sense of the Caputo fractional derivative with the chaos entanglement function. We derive conditions under which the system undergoes a Hopf bifurcation and obtain critical parameter value in the Hopf bifurcation. Moreover, the linear feedback control technique is used to control and stabilize the system to equilibrium point in order to eliminate the chaotic vibration. We then design control laws to synchronize two identical chaotic fractional-order systems. Furthermore, by means of numerical simulation, we support the validity of analytical results and reveal more dynamical behaviors consisting chaos, local bifurcation, limit cycles, quasiperiodic and asymptotic stability behaviors. We further emphasize that the order of fractional derivative plays significant roles as the chaos controlling parameter and the Hopf bifurcation parameter. [ABSTRACT FROM AUTHOR]

With the rapid growth of data transmission technology, image encryption is becoming a widely discussed topic in the field of information security. In this paper, we introduce a novel one-dimensional sine powered chaotic system (1DSP). Evaluation of the proposed chaotic system indicates the existence of chaotic behavior with high sensitivity and randomness. We further present a new image encryption scheme using 1DSP (1DSP-IE). The proposed scheme uses the 1DSP for sequence addition to improve its security. Furthermore, row-by-row and column-by-column concepts are used for confusion and diffusion operations. This approach breaks the correlation of adjacent pixels by scrambling pixels out of their original rows and columns and applies bit-level pixel value manipulation for each row and then each column to spread any change in the plain image throughout the ciphered image. The simulation results of the proposed 1DSP-IE indicate an effective encryption and decryption process with good speed. In addition, the security analysis illustrates the ability of 1DSP-IE to provide a satisfying level of security in comparison with other image encryption schemes. [ABSTRACT FROM AUTHOR]

Individuals diagnosed with mild cognitive impairment (MCI) are at a high risk of transition to Alzheimer's disease (AD), but a diagnosis of MCI is challenging. Resting-state functional magnetic resonance imaging (rs-fMRI) is a promising tool for identifying patients with MCI, but an accurate and robust analysis method is needed to extract discriminative rs-fMRI features for classification between MCI patients and healthy people. In this paper, a new rs-fMRI data analysis approach based on Chaotic Binary Grey Wolf Optimization (CBGWO) and Adaptive Neuro-Fuzzy Inference System (ANFIS), namely (CBGWO-ANFIS), is presented to distinguish MCI patients based on rs-fMRI. CBGWO is a new feature selection model that attempts to reduce the number of features without loss of significant information for classification, and it uses the naïve Bayes criterion as a part of the objective function. Based on the chaos theory, the important parameters of GWO are estimated and tuned by using ten different chaos sequence maps. Subsequently, ANFIS is used to classify MCI patients and healthy people based on the subset of features retained by CBGWO. Experiments were carried out on 62 MCI patients and 65 normal controls (NC). Fractional amplitude of low frequency fluctuation (f-ALFF) was extracted from rs-fMRI as features. The results indicate that the proposed CBGWO-ANFIS approach with the Chebyshev chaos map shows a higher accuracy (around 86%), higher convergence speed, and shorter execution time than other chaos maps. Further, the proposed approach outperforms the conventional machine learning techniques and the recent meta-heuristic optimization algorithms. This study indicates that the proposed CBGWO-ANFIS approach on rs-fMRI could be a potential tool for early diagnosis of MCI. [ABSTRACT FROM AUTHOR]

• We investigate the complex behavior of a plant under air flow through electrical analog circuit. • We use the isospike diagram to instantly inform how the chaotic behavior of the system occurs as its parameters change. • It appears a coexistence of multiple attractors in the system that makes its chaotic character less robust. • The obtained experimental results corroborate perfectly with the numerical ones established via Matlab and PSpice environments. This paper considers a specific plant (pinus family) and examines its complex behavior under the air flow while checking the real-time dynamics of the proposed analog electronic simulator with an active RC realization. It appears that the system can be chaotic and its dynamics depend on the chosen initial conditions. We show the coexistence of multiple attractors in the system and observe that their occurrence makes its chaotic character less robust. We also establish through the basin of attraction that the region of mixed-mode oscillations can be extended by increasing values of the wind amplitude ratio. Furthermore, the isospike diagram is introduced to instantly inform how the dynamics of the plant moves from periodic to chaotic motion as the main parameters of the wind augment all together. Our experimental searches yield results that are in perfect agreement with the numerical outcomes established via Matlab and Pspice environments. Those experimental surveys also display the coexistence of asymmetric and symmetric attractors that confirms the complex behavior of the plants subjected to the wind loads. [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]

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]

This paper investigates the vibrational resonance and control of the response of a nonlinear tilted cantilever beam under combined multi-harmonic low and non-resonant, high-frequency excitations. The basic system considers the first mode of vibration of a nonlinear tilted beam under parametric and direct base excitation with the frequency components locked in 2:1 ratio. Such a system is susceptible to parametric amplification of resonance, which may be detrimental for structural health in many mechanical systems. The primary objective of the research is to investigate the non-trivial effects of non-resonant high-frequency excitation on parametric amplification, vibrational resonance and chaos in the system, which, to the best of the authors' knowledge, is not reported previously in the literature. The slow dynamic equations of the system are obtained using the method of direct partition of motion (MDPM), which is then analysed using the harmonic balance method (HBM). It has been shown that the high-frequency base excitation considerably influences the effective damping and stiffness of the system under consideration. The results of the research demonstrate that high-frequency excitation can effectively control the parametric amplification by shifting the resonance curve in the frequency axis and eliminating undesirable bifurcations by increasing the effective damping. The phenomenon of vibrational resonance has been found for specific parameter values in the presence of high-frequency base excitation. Detail parametric studies are performed to reveal the system's rich nonlinear dynamics and bifurcations, like multiple jumps in response, dual peaks, isolated resonance response (isola), loop formation in frequency response plot, etc. The numerical results reveal that high-frequency excitation transforms chaotic responses into regular periodic responses. Direct numerical simulations are performed in MATLAB SIMULINK to validate the analytical results. • Non-trivial effects of high-frequency excitation (HFE) on a nonlinear cantilever beam under multi-frequency excitation are studied. • Effective nonlinear damping of the system can be significantly modified by HFE. • Parametric amplification and chaotic response is controlled. • Vibrational jump resonance can be induced and controlled by HFE. [ABSTRACT FROM AUTHOR]

This review paper aims at answering a basic question on the sign of Lyapunov exponents. A few recent papers reported hyperchaotic system having the sign of Lyapunov exponents as (+, +, −, −). Such (+, +, −, −) sign of Lyapunov exponents is in contradiction with the well known (+, +, 0, −) sign of Lyapunov exponents for a 4-D hyperchaotic system. This paper thus discusses various issues related to Lyapunov exponents and proves that the reported sign of (+, +, −, −) Lyapunov exponents is actually (+, +, 0, −) or (+, 0, −, −). This clarification is very important and essential since Lyapunov exponents are the only quantitative measure for the existence of hyperchaos. Three different algorithms are used for calculating the Laypunov exponents to prove the actual sign of Lyapunov exponents for a hyperchaotic system. [ABSTRACT FROM AUTHOR]

Tristable system with negative stiffness has attracted extensive attention in the low frequency vibration isolation and vibration energy harvester. As a low frequency vibration isolator, it can achieve high static stiffness and low dynamic stiffness. As a vibration energy harvester, it had a wider bandwidth for resonance than the bistable one. The introduction of negative stiffness may induce subharmonic resonance and chaos in the tristable system. Chaos usually brings disorder to mechanical vibration system. Subharmonic resonance plays the negative effect on low frequency vibration isolation because they will transfer the high frequency energy of the system to the low frequency, but it is beneficial to broaden the working frequency band of vibration energy harvester. In this paper, the subharmonic bifurcation and chaos of a class of tristable system with negative stiffness are studied. The piecewise linearized systems are established to approximate the system with tristable potential. In order to conduct Melnikov analysis, the homoclinic-heteroclinic orbits and periodic orbits for the unperturbed piecewise linearized system are obtained respectively. The subharmonic Melnikov method for nonsmooth systems with four switched manifolds is developed. The thresholds for homoclinic-heteroclinic chaos and subharmonic resonance are derived by using non-smooth Melnikov method. It provides a theoretical support not only for design of the vibration energy harvester to obtain wider working frequency band, but also for design of the vibration isolation system to avoid high frequency energy transfer to low frequency. Moreover, the phenomena for infinite subharmonic bifurcations to chaos from odd order subharmonic orbit and the coexistence for chaotic and subharmonic attractors are revealed. The subharmonic Melnikov method with four switching manifolds developed in this paper lays a foundation for the subharmonic resonance analysis of other nonsmooth tristable systems. [ABSTRACT FROM AUTHOR]

Accurately detecting a homoclinic cycle and chaos in a concrete system takes a huge challenge. This paper gives a sufficient condition of coexisting homoclinic cycles (degenerate) with respect to a saddle-focus in a class of three-dimensional piecewise systems with two switching manifolds. Furthermore, by constructing the Poincaré map, it is rigorously shown that chaotic behavior is induced by the coexisting cycles without necessary symmetry in the considered system, which is evidently different from the usual dynamics in smooth system theory that a pair of symmetrical homoclinic cycles can generate chaos but the asymmetric ones cannot. It finally provides an example to illustrate the effectiveness of the results established. • Obtain a criterion for accurately detecting coexisting homoclinic cycles (degenerate and non-degenerate) in a class of piecewise systems. • Propose existence conditions of chaos, which generalizes the Shilnikov theorem of smooth systems to non-smooth systems. • Rigorously prove the coexisting cycles possibly without symmetry can give rise to topological horseshoes, which is in sharp contrast with the smooth systems that need symmetrical homoclinic cycles to produce chaos. • The new findings and theoretical results contribute to better understanding the geometry and generation mechanism of chaotic attractors. • The analysis method in this paper is also applicable to a investigation on similar objects caused by other types of coexisting singular cycles in non-smooth dynamical systems. [ABSTRACT FROM AUTHOR]

• Several chaotic systems with interesting behaviors are presented. • Atangana–Baleanu fractional derivative operator is considered. • Existence and uniqueness of general system as well as local stability analysis are examined. • A range of chaotic patterns examined through time series are obtained for different instances of fractional orders. In this paper, a range of chaotic systems with some interesting behaviors such as multi-scroll attractors, self-excited and hidden attractors, period-doubling to chaos, periodic and chaotic bursting oscillations, and different multiple coexisting attractors have been considered and modelled with the new Atangana–Baleanu fractional derivative operator in time. Existence and uniqueness of general system as well as local stability analysis are examined. In the simulation framework, a range of chaotic patterns examined through time series were obtained for different instances of fractional orders. Comparison between the integer (with p = 1) and noninteger (0 < p < 1) order results are given. [ABSTRACT FROM AUTHOR]

The main concern of this paper is to address the synchronization problem of chaotic fractional-order fuzzy cellular neural networks (FOFCNNs) through designing the novel adaptive control scheme. The objective of the study is to explore the importance of considering fractional order derivatives (FODs) and time-varying delays. Even though numerous works have been reported in the literature regarding the derivation of sufficient conditions, there has been a lack of research on involving the dynamical analysis of FOFCNNs. Hence, this study focuses on the dynamical analysis of FOFCNNs. Particularly, both asymptotical and exponential synchronization of drive-response FOFCNN model is guaranteed via sufficient conditions that are derived by constructing the fractional Lyapunov functional candidate and solvable linear matrix inequalities (LMIs). Besides that, numerical simulations are performed to reveal the significance of the FODs. Also, an image encryption algorithm is designed based on the chaotic FOFCNNs solutions that result in better security measures. In summary, the overall contribution of the study is categorized into two: (1) sufficient conditions which ensure the global asymptotic and exponential stability are derived in a novel manner; (2) an image encryption algorithm is proposed by considering the FOFCNN as pseudo-random number generator (PRNG), which outperforms the existing encryption algorithms. [ABSTRACT FROM AUTHOR]

Highlights • Delayed FOMASs with external disturbances and unknown parameters are considered. • An artificial bee colony algorithm-based parameters estimation method is proposed. • A distributed discontinuous nonlinear consensus tracking protocol is designed. • A distributed continuous nonlinear consensus tracking algorithm is further designed. Abstract In this paper, the distributed consensus tracking of unknown nonlinear chaotic delayed fractional-order multi-agent systems (FOMASs) with external disturbances is investigated, where the fractional orders and system parameters are unknown. Firstly, in order to identify the unknown parameters of the delayed nonlinear FOMASs, an efficient artificial bee colony algorithm (ABC)-based parameters estimation approach is put forward. Secondly, based on the estimated parameters, by using fractional derivative inequality and comparison principle of linear fractional equation with delay, a discontinuous distributed control protocol is proposed to make the tracking errors converge to zero asymptotically. Thirdly, to overcome the undesirable chattering phenomenon caused by the discontinuous controller, a continuous distributed control algorithm is further designed and uniformly ultimately bounded (UUB) tracking errors can be obtained and reduced as small as desired. Finally, numerical simulations are given to test the effectiveness of the proposed parameters estimation scheme and the deigned control algorithms. [ABSTRACT FROM AUTHOR]

Abstract This paper proposes a set of novel fuzzy controllers with adjustable control forces for the robust synchronization of different chaotic systems that include uncertain terms or unexpected disturbances. The proposed fuzzy rule-based control system is developed through Lyapunov stability theory and the fuzzy logic concept to form a multi-input multi-output (MIMO) system. This system provides flexible control inputs based on the error dynamics of the system, and the inputs satisfy the synchronization condition of being asymptotically stable. In this case, the error derivatives always have signs opposite those of the error states, which yields negative definite first-order Lyapunov function derivatives. Furthermore, the proposed system appropriately adjusts the control inputs through the constructed fuzzy if-then rules. This method is robust to both unmodeled terms and unexpected interference signals during the synchronization process. The simulation results demonstrate the feasibility and effectiveness of the proposed fuzzy rule-based control system. [ABSTRACT FROM AUTHOR]

Highlights • DC 1, 2, and 5/2 are invariants under iterations in non-autonomous discrete systems. • DC 1, 2, and 5/2 are invariants of topological equi-conjugacy in non-autonomous discrete systems. • Instead of uniformly convergent sequence of maps, a sequence of equi-continuous maps are considered. • Those related existing results are generalized to non-autonomous systems. Abstract This paper is concerned with some weak versions of distributional chaos in a non-autonomous discrete system generated by a given sequence of maps { f n } n = 0 ∞ in a metric space (X, d). It is shown that three versions named DC1, DC2, and DC 2 1 2 are invariants under iterations when { f n } n = 0 ∞ is equi-continuous in X , which weakens the condition in the literature that { f n } n = 0 ∞ uniformly converges in a compact space X. It is also proved that DC1, DC2, and DC 2 1 2 are invariants of topological equi-conjugacy. One example is provided with computer simulations for illustration. [ABSTRACT FROM AUTHOR]

Highlights • We study multistability in a master-slave piecewise linear system. • We investigate the collective behavior of systems with unidirectional links and ring topology. • Symbolic dynamics of the nodes is useful to detect synchronization. • Itinerary synchronization is used to analyze and classify multistable behavior. • ε Itinerary synchronization is displayed in a ring topology network. Abstract In this paper the collective dynamics of N -coupled piecewise linear (PWL) systems with different number of scrolls is studied. The coupling is in a master-slave sequence configuration, with this type of coupling we investigate the synchrony behavior of a ring-connected network and a chain-connected network both with unidirectional links. Itinerary synchronization is used to detect synchrony behavior. Itinerary synchronization is defined in terms of the symbolic dynamics arising by assigning different numbers to the regions where the scrolls are generated. A weaker variant of this notion, ϵ-itinerary synchronization is also introduced and numerically investigated. It is shown that in certain parameter regimes if the inner connection between nodes takes account of all the state variables of the system (by which we mean that the inner coupling matrix is the identity matrix), then itinerary synchronization occurs and the coordinate motion is determined by the node with the smallest number of scrolls. Thus the collective behavior in all the nodes of the network is determined by the node with least scrolls in its attractor. Results about the dynamics in a directed chain topology are also presented. Depending on the inner connection properties, the nodes present multistability or preservation of the number of scrolls of the attractors. [ABSTRACT FROM AUTHOR]

Abstract Supercomputer power has been doubling approximately every 14 months for several decades, increasing the capabilities of scientific modelling at a similar rate. However, to utilize these machines effectively for applications such as computational fluid dynamics, improvements to strong scalability are required. Here, the particular focus is on semi-implicit, viscous-flow CFD, where the largest bottleneck to strong scalability is the parallel solution of the linear pressure-correction equation — an elliptic Poisson equation. State-of-the-art linear solvers, such as Krylov subspace or multigrid methods, provide excellent numerical performance for elliptic equations, but do not scale efficiently due to frequent synchronization between processes. Complete desynchronization is possible for basic, Jacobi-like solvers using the theory of 'chaotic relaxations'. These non-deterministic, chaotic solvers scale superbly, as demonstrated herein, but lack the numerical performance to converge elliptic equations — even with the relatively lax convergence requirements of the example CFD application. However, these chaotic principles can also be applied to multigrid solvers. In this paper, a 'chaotic-cycle' algebraic multigrid method is described and implemented as an open-source library. It is tested on a model Poisson equation, and also within the context of CFD. Two CFD test cases are used: the canonical lid-driven cavity flow and the flow simulation of a ship (KVLCC2). The chaotic-cycle multigrid shows good scalability and numerical performance compared to classical V-, W- and F-cycles. On 2048 cores the chaotic-cycle multigrid solver performs up to 7. 7 × faster than Flexible-GMRES and 13. 3 × faster than classical V-cycle multigrid. Further improvements to chaotic-cycle multigrid can be made, relating to coarse-grid communications and desynchronized residual computations. It is expected that the chaotic-cycle multigrid could be applied to other scientific fields, wherever a scalable elliptic-equation solver is required. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we construct a duopoly model of technological innovation based on constant conjectural variation under the assumption of bounded rationality. The equilibrium points of constant conjectural variation and their local stability are investigated. By the theoretical proof, we prove that the model displays Li-Yorke chaos and distributional chaos in the ranges of the output adjusted coefficients by employing the snap-back repeller theory. The technological innovation is the main source of the core competitiveness of a firm, and technological content is an important factor of technological innovation. We find that the firms have the optimal technological contents while they could have the maximal profits in the two different situations, non-cooperative technological innovation and cooperative technological innovation. It's proved that the cooperation of technological innovation can increase the marginal technological content, and benefit to technological progress. Then Numerical simulations are used to illustrate the dynamic analysis of the model under the situations with different technological innovation and conjectural variation. When the output adjusted coefficient changes, a series of complex phenomena including bifurcation, chaos and strange attractor can be observed in our model. And when the technological content is larger, the chaos of the system finally appears. [ABSTRACT FROM AUTHOR]