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As a nonlinear dynamic system, the dynamic equation of triboelectric nanogenerator system is not only difficult to accurately establish, but also difficult to solve. The output signals of triboelectric nanogenerator system depends on the initial conditions of the system, which reflects the dynamic behavior of the system. Currently, there is little research on using the output signals to study the dynamic behavior of triboelectric nanogenerator system. Therefore, a dynamic simulation model of a freestanding triboelectric nanogenerator (F-TENG) system was established using COMSOL software. The mathematical model of the F-TENG system was fitted. Using chaos theory and its phase-space reconstruction technology, the phase-space trajectories of the F-TENG system were reconstructed through voltage signals under external factors, and the dynamic behavior of the F-TENG system was discussed and studied. Finally, validation experiments were carried out. The research shows that with the variation of internal factors and operating frequency, the mathematical model of the F-TENG system remains basically unchanged, and the variation law of output characteristics also remains basically unchanged. The F-TENG system is in periodic motion; with the variation of external factors such as resistance and friction layer spacing, the mathematical model of the F-TENG system will evolve in a complex direction, and the output characteristics will gradually distort. The F-TENG system will evolve from periodic motion to chaotic motion. The research results not only reveal the dynamic behavior and enrich the nonlinear dynamic theory of triboelectric nanogenerator system, but also provide theoretical guidance for increasing the working bandwidth and stability of triboelectric nanogenerator system. [ABSTRACT FROM AUTHOR]

Nonlinear dynamical problems, characterized by unpredictable and chaotic changes among variables over time, pose unique challenges in understanding. This paper explores the coupled nonlinear (2+1)-dimensional complex modified Korteweg-de-Vries (cmKdV) equation-a fundamental equation in applied magnetism and nanophysics. The study focuses on dynamic behaviors, specifically examining bifurcations and equilibrium points leading to chaotic phenomena by introducing an external term to the system. Employing chaos theory, we showcase the chaotic tendencies of the perturbed dynamical system. Additionally, a sensitivity analysis using the Runge-Kutta method reveals the solution's stability under slight variations in initial conditions. Innovatively, the paper utilizes the planar dynamical system technique to construct various solitons within the governing model. This research provides novel insights into the behavior of the (2+1)-dimensional cmKdV equation and its applications in applied magnetism and nanophysics. [ABSTRACT FROM AUTHOR]

Dynamic systems theory and complexity theory (DST/CT) is a framework explaining how complex systems change and adapt over time. In psychotherapy, DST/CT can be used to understand how a person's mental and emotional state changes during therapy incorporating higher levels of complexity. This study aimed to systematically review the variability of DST/CT methods applied in psychotherapy research. A primary studies search was conducted in the EBSCO and Web of Knowledge databases, extracting information about the analyzed DST/CT phenomena, employed mathematical methods to investigate these phenomena, descriptions of specified dynamic models, psychotherapy phenomena, and other information regarding studies with empirical data (e.g., measurement granularity). After screening 38,216 abstracts and 4,194 full texts, N = 41 studies published from 1990 to 2021 were identified. The employed methods typically included measures of dynamic complexity or chaoticity. Computational and simulation studies most often employed first-order ordinary differential equations and typically focused on describing the time evolution of client-therapist dyadic influences. Eligible studies with empirical data were usually based on case studies and focused on data with high time intensity of within-session dynamics. This review provides a descriptive synthesis of the current state of the proliferation of DST/CT methods in the psychotherapy research field. [ABSTRACT FROM AUTHOR]

In the work, we prove the presence of chaotic dynamics, for suitable values of the model parameters, for the discrete-time system, composed of two coupled logistic maps, as formulated in Yousefi et al. [Discrete Dyn. Nat. Soc. 5, 161–177 (2000)], which describes two interdependent economies, characterized by two competitive markets each, with a demand link between them. In particular, we rely on the SAP (Stretching Along the Paths) method, based on a stretching relation for maps defined on sets homeomorphic to the unit square and expanding the paths along one direction. Such technique is topological in nature and allows to establish the existence of a semiconjugacy between the considered dynamical system and the Bernoulli shift, so that the main features related to the chaos of the latter (e.g., the positivity of the topological entropy) are transmitted to the former. In more detail, we apply the SAP method both to the first and to the second iterate of the map associated with our system, and we show how dealing with the second iterate, although being more demanding in terms of computations, allows for a larger freedom in the sign and in the variation range of the linking parameters for which chaos emerges. Moreover, the latter choice guarantees a good agreement with the numerical simulations, which highlight the presence of a chaotic attractor under the conditions derived for the applicability of the SAP technique to the second iterate of the map. [ABSTRACT FROM AUTHOR]

In nonlinear dynamics, there is a continuous exploration of introducing systems with evidence of chaotic behavior. The presence of nonlinearity within system equations is crucial, as it allows for the emergence of chaotic dynamics. Given that quadratic terms represent the simplest form of nonlinearity, our study focuses on introducing a novel chaotic system characterized by only quadratic nonlinearities. We conducted an extensive analysis of this system's dynamical properties, encompassing the examination of equilibrium stability, bifurcation phenomena, Lyapunov analysis, and the system's basin of attraction. Our investigations revealed the presence of eight unstable equilibria, the coexistence of symmetrical strange repeller(s), and the potential for multistability in the system. [ABSTRACT FROM AUTHOR]

In a vast area of probabilistic limit theorems for dynamical systems with chaotic behaviors always only functional form (exponential, power, etc.) of the asymptotic laws and of convergence rates were studied. However, for basically all applications, e.g., for computer simulations, development of algorithms to study chaotic dynamical systems numerically, as well as for design and analysis of real (e.g., in physics) experiments, the exact values (or at least estimates) of constants (parameters) of the functions, which appear in the asymptotic laws and rates of convergence, are of primary interest. In this paper, we provide such estimates of constants (parameters) in the central limit theorem, large deviations principle, law of large numbers and the rate of correlations decay for strongly chaotic dynamical systems. [ABSTRACT FROM AUTHOR]

Using the sawtooth map as the basis of a coupled map lattice enables simple analytic results to be obtained for the global Lyapunov spectra of a number of standard lattice networks. The results presented can be used to enrich a course on chaos or dynamical systems by providing tractable examples of higher dimensional maps and links to a number of standard results in matrices. A number of suggestions for classroom use are given. [ABSTRACT FROM AUTHOR]

Chaos is an important dynamic feature, which generally occurs in deterministic and stochastic nonlinear systems and is an inherent characteristic that is ubiquitous. Many difficulties have been solved and new research perspectives have been provided in many fields. The control of chaos is another problem that has been studied. In recent years, a recurrent neural network has emerged, which is widely used to solve many problems in nonlinear dynamics and has fast and accurate computational speed. In this paper, we employ reservoir computing to control chaos in dynamic systems. The results show that the reservoir calculation algorithm with a control term can control the chaotic phenomenon in a dynamic system. Meanwhile, the method is applicable to dynamic systems with random noise. In addition, we investigate the problem of different values for neurons and leakage rates in the algorithm. The findings indicate that the performance of machine learning techniques can be improved by appropriately constructing neural networks. [ABSTRACT FROM AUTHOR]

The influence of orientation on the control of chaotic behaviors of the system in a porous medium is studied in this article based on the theory of dynamical systems. A three-dimensional system model is obtained using the spectral method. The results showed that chaos can be suppressed in the case of a low or moderate Prandtl number by choosing appropriate tilt angles. [ABSTRACT FROM AUTHOR]

The generalized complex Ginzburg–Landau equation is considered. An analytical condition for the existence of horseshoe chaos is obtained for the traveling wave reduction of the investigated equation by using the Melnikov method. A way to control chaos in the dynamical system is proposed. An analytical prediction is tested numerically by plotting attraction basins of attraction of the Poincare section of the studied system. [ABSTRACT FROM AUTHOR]

The fact that very small changes cause unpredictable big changes in nature has led to the chaos theory. Many researchers focusing on chaos have improved this theory in several directions. One of the prominent areas in chaos theory is controlled chaos. Due to the usefulness of controlled chaos, it has been used in many working areas as the iteration methods of fixed point theory. In this study, we have developed a chaos controlling mechanism using the Ishikawa iteration method. Firstly, Ishikawa fixed point type controlling mechanism is defined and the theorems determining the control parameters are proved. Afterward, the results of Ishikawa fixed point type controlling mechanism were obtained by taking the logistic map as an example. In addition, by using the MATLAB software program, the convergence, periodical behavior and chaotic regions occurring outside the range of the parameters forming the controlling mechanism have been investigated and numerical simulations studied. Subsequently, the effectiveness of the stability intervals obtained for the fixed point and periodic fixed points of the Ishikawa fixed point controlling mechanism by applying the Lyapunov exponent is demonstrated. Finally, we have given an application as control chaos of Ishikawa traffic model. [ABSTRACT FROM AUTHOR]

Complex systems, as interwoven miscellaneous interacting entities that emerge and evolve through self-organization in a myriad of spiraling contexts, exhibit subtleties on global scale besides steering the way to understand complexity which has been under evolutionary processes with unfolding cumulative nature wherein order is viewed as the unifying framework. Indicating the striking feature of non-separability in components, a complex system cannot be understood in terms of the individual isolated constituents' properties per se, it can rather be comprehended as a way to multilevel approach systems behavior with systems whose emergent behavior and pattern transcend the characteristics of ubiquitous units composing the system itself. This observation specifies a change of scientific paradigm, presenting that a reductionist perspective does not by any means imply a constructionist view; and in that vein, complex systems science, associated with multiscale problems, is regarded as ascendancy of emergence over reductionism and level of mechanistic insight evolving into complex system. While evolvability being related to the species and humans owing their existence to their ancestors' capability with regards to adapting, emerging and evolving besides the relation between complexity of models, designs, visualization and optimality, a horizon that can take into account the subtleties making their own means of solutions applicable is to be entailed by complexity. Such views attach their germane importance to the future science of complexity which may probably be best regarded as a minimal history congruent with observable variations, namely the most parallelizable or symmetric process which can turn random inputs into regular outputs. Interestingly enough, chaos and nonlinear systems come into this picture as cousins of complexity which with tons of its components are involved in a hectic interaction with one another in a nonlinear fashion amongst the other related systems and fields. Relation, in mathematics, is a way of connecting two or more things, which is to say numbers, sets or other mathematical objects, and it is a relation that describes the way the things are interrelated to facilitate making sense of complex mathematical systems. Accordingly, mathematical modeling and scientific computing are proven principal tools toward the solution of problems arising in complex systems' exploration with sound, stimulating and innovative aspects attributed to data science as a tailored-made discipline to enable making sense out of voluminous (-big) data. Regarding the computation of the complexity of any mathematical model, conducting the analyses over the run time is related to the sort of data determined and employed along with the methods. This enables the possibility of examining the data applied in the study, which is dependent on the capacity of the computer at work. Besides these, varying capacities of the computers have impact on the results; nevertheless, the application of the method on the code step by step must be taken into consideration. In this sense, the definition of complexity evaluated over different data lends a broader applicability range with more realism and convenience since the process is dependent on concrete mathematical foundations. All of these indicate that the methods need to be investigated based on their mathematical foundation together with the methods. In that way, it can become foreseeable what level of complexity will emerge for any data desired to be employed. With relation to fractals, fractal theory and analysis are geared toward assessing the fractal characteristics of data, several methods being at stake to assign fractal dimensions to the datasets, and within that perspective, fractal analysis provides expansion of knowledge regarding the functions and structures of complex systems while acting as a potential means to evaluate the novel areas of research and to capture the roughness of objects, their nonlinearity, randomness, and so on. The idea of fractional-order integration and differentiation as well as the inverse relationship between them lends fractional calculus applications in various fields spanning across science, medicine and engineering, amongst the others. The approach of fractional calculus, within mathematics-informed frameworks employed to enable reliable comprehension into complex processes which encompass an array of temporal and spatial scales notably provides the novel applicable models through fractional-order calculus to optimization methods. Computational science and modeling, notwithstanding, are oriented toward the simulation and investigation of complex systems through the use of computers by making use of domains ranging from mathematics to physics as well as computer science. A computational model consisting of numerous variables that characterize the system under consideration allows the performing of many simulated experiments via computerized means. Furthermore, Artificial Intelligence (AI) techniques whether combined or not with fractal, fractional analysis as well as mathematical models have enabled various applications including the prediction of mechanisms ranging extensively from living organisms to other interactions across incredible spectra besides providing solutions to real-world complex problems both on local and global scale. While enabling model accuracy maximization, AI can also ensure the minimization of functions such as computational burden. Relatedly, level of complexity, often employed in computer science for decision-making and problem-solving processes, aims to evaluate the difficulty of algorithms, and by so doing, it helps to determine the number of required resources and time for task completion. Computational (-algorithmic) complexity, referring to the measure of the amount of computing resources (memory and storage) which a specific algorithm consumes when it is run, essentially signifies the complexity of an algorithm, yielding an approximate sense of the volume of computing resources and seeking to prove the input data with different values and sizes. Computational complexity, with search algorithms and solution landscapes, eventually points toward reductions vis à vis universality to explore varying degrees of problems with different ranges of predictability. Taken together, this line of sophisticated and computer-assisted proof approach can fulfill the requirements of accuracy, interpretability, predictability and reliance on mathematical sciences with the assistance of AI and machine learning being at the plinth of and at the intersection with different domains among many other related points in line with the concurrent technical analyses, computing processes, computational foundations and mathematical modeling. Consequently, as distinctive from the other ones, our special issue series provides a novel direction for stimulating, refreshing and innovative interdisciplinary, multidisciplinary and transdisciplinary understanding and research in model-based, data-driven modes to be able to obtain feasible accurate solutions, designed simulations, optimization processes, among many more. Hence, we address the theoretical reflections on how all these processes are modeled, merging all together the advanced methods, mathematical analyses, computational technologies, quantum means elaborating and exhibiting the implications of applicable approaches in real-world systems and other related domains. [ABSTRACT FROM AUTHOR]

Pseudo‐random number generators (PRNGs) are the cornerstone of various fields including computer science, cryptography, and scientific simulation. To generate high‐quality and unpredictable pseudo‐random sequences, many works draw attention to the use of chaos theory and dynamical systems. However, most of them focus on continuous or low‐dimensional chaotic systems with strong temporal correlations or insufficient nonlinear dynamics. Thus, in this study, a new PRNG based on a four‐dimensional discrete system is proposed to solve the aforementioned flaws. The PRNG has been implemented on Xilinx Artix‐7 xc7a100tfgg484‐2, which can generate pseudo‐random sequences at a high speed of 12811.53 Mb/s without additional post‐processing. All these sequences have successfully passed the NIST SP800.22 standard test. Both throughput and randomness quality of the proposed PRNG outperform the state‐of‐the‐art. [ABSTRACT FROM AUTHOR]

Further exploration into the influence of a memristor on the behavior of chaotic systems deserves attention. When constructing memristor chaotic systems, it is commonly believed that increasing the number of memristors will lead to better system performance. This paper proposes a class of chaotic maps with different discrete memristors, achieved through internal perturbation based on the Sine map. The I-V curve of the discrete memristor has a symmetrical structure. The dynamic characteristics of the designed system are analyzed using the chaotic attractor phase diagram, Lyapunov exponent (LE) spectrum, and bifurcation diagram. Numerical simulations demonstrate that internal perturbations of discrete memristors enhance the Sine map's chaotic characteristics, expand the chaos range, and improve the ergodicity and LE value. Moreover, the type of discrete memristors has a significant impact on the dynamic characteristics of the system, while the number of discrete memristors has little influence. Therefore, in this paper, a direction for the design of a discrete memristor chaotic system is provided. Finally, a discrete memristor chaotic map with a simple structure and better performance is selected. Based on this, a pseudo-random sequence generator is designed, and the generated sequence passes the National Institute of Standards and Technology (NIST) test. [ABSTRACT FROM AUTHOR]

Theoretically, an eight-term chaos system is presented. The effect of changing the initial conditions values on behavior Chen system was studied. The basic dynamical properties of system are analyzed like time series, attractor, FFT spectrum, and bifurcation. Where the system appears steady state behavior at initial condition xi, yi, zi equal (0, 0, 0) respectively and it convert to quasi-chaotic at xi,yi,zi equal (-0.1, 0.5,-0.6). Finally, the system become hyper chaotic at xi,yi, zi equal(- 0.5, 0.5,-0.6 ) that can used it in many applications like secure communication. [ABSTRACT FROM AUTHOR]

Studies on fractional-order chaotic systems have increased significantly in the last decade. This paper presents Rucklidge chaotic system's dynamical analyses and its fractional-order circuit implementations. Component values required for realizing the circuit of the fractional-order system are calculated for different fractional-orders. The feasibility of the attractor is examined by implementing its electronic circuit with a fractional-order module. The module is constructed based on the Diyi-Chen model since it is easier to implement and cost-effective. In electronic circuit implementations, it is observed that the system's chaotic state disappears as the fractional degree decreases. Numerical and circuit simulation results are consistent well with the hardware experimental results. [ABSTRACT FROM AUTHOR]

Considering that chaotic systems are immersed in multiple areas of science and nature and that their dynamics are governed by a great sensitivity to the initial conditions and variations in their parameters, it is of great interest for the scientific community to have tools to characterize and reproduce these trajectories. Two dynamic chaotic systems whose equations are based on the jerky system are used as benchmarks, i.e., the Memristive Shaking Chaotic System (MSCS) and the Unstable Dissipative System of type I (UDSI). One characteristic common to them is their simple mathematical structure and the complexity of their solutions. Therefore, this paper presents a strategy for identifying chaotic trajectories using a recurrent wavelet first-order neural network (RWFONN) that is trained online with an error filtering algorithm and considering the Morlet-wavelet as an activation function. The parameters of the network are adjusted considering the Euclidean distance between the solutions. Finally, the results depict proper identification of the chaotic systems studied through analysis and numerical simulation to validate the behavior and functionality of the proposed network. [ABSTRACT FROM AUTHOR]

The coexistence of different attractors with fixed parameters affords versatility in the performance of the dynamical system. As a result, recent research has focused on defining nonlinear oscillators with infinite coexisting attractors, the vast majority of which are non-autonomous systems. In this study, we present an infinite number of equilibriums achieved with the simplest autonomous megastable oscillator. To begin, we explore the symmetry property of distinct coexisting attractors, such as periodic, quasi-periodic, and chaotic attractors. The dynamical properties of the proposed system are further investigated using phase portraits, Lyapunov exponents spectrum, and bifurcation diagram using the local maxima of the state variables. The detailed investigation reveals that the proposed system has a wide range of dynamical properties ranging from periodic oscillations to hyperchaos. Linear quadratic regulator (LQR) control-based controllers are designed to control the chaos in the proposed system and also achieve synchronization between proposed systems that have different initial conditions. The effectiveness of the controller designed based on LQR is presented by simulation results. It is shown that the proposed system converges asymptotically to the desired equilibrium point, and the synchronization between the slave and the master systems is achieved by converging the error to zero after controllers are activated. [ABSTRACT FROM AUTHOR]

A hydraulic generator regulating system with electrical, mechanical, and hydraulic constitution is a complex nonlinear system, which is analyzed in this research. In the present study, the dynamical behavior of this system is investigated. Afterward, the input/output feedback linearization theory is exerted to derive the controllable model of the system. Then, the chaotic behavior of the system is controlled using a robust controller that uses a Chebyshev neural network as a disturbance observer in combination with a non-singular robust terminal sliding mode control method. Moreover, the convergence of the system response to the desired output in the presence of uncertainty and unexpected disturbances is demonstrated through the Lyapunov stability theorem. Finally, the effectiveness and appropriate performance of the proposed control scheme in terms of robustness against uncertainty and unexpected disturbances are demonstrated through numerical simulations. [ABSTRACT FROM AUTHOR]

In this research, the ship power system is studied with a fractional-order approach. A 2-D model of a two-generator parallel-connected is considered. A chaotic attractor is observed for particular parameter values. The fractional-order form is calculated with the Adam–Bashforth–Moulton method. The chaotic response is identified even for the order 0.99. Phase portrait is generated using the Caputo derivative approach. Wolf's algorithm is used to calculate Lyapunov exponents. For the considered values of parameters, one positive Lyapunov exponent confirms the existence of chaos. Bifurcation diagrams are presented to analyze the various dynamical behaviors and bifurcation points. Interestingly, the considered system is multistable. Also, antimonotonicity, period-doubling, and period halving are observed in the bifurcation diagram. As the last step, a fractional-order controller is designed to remove chaotic dynamics. Time plots are simulated to show the effectiveness of the controller. [ABSTRACT FROM AUTHOR]

The chaotic behavior of the Solar System prevents the prediction of the precise evolution of the planetary secular frequencies. Here we use four cyclostratigraphic sequences, ∼259 Ma Wujiaping Formation, ∼455 Ma Pingliang Formation, ∼655 Ma Datangpo Formation and ∼2,465 Ma Dales Gorge Member, and Bayesian inversion to provide powerful geological constraints to this problem. The inversion provides constraining estimates for the Solar System fundamental secular frequencies g1, g2, g3, g4, and g5, and the Earth's precession rate k from 105.26 ± 2.68 (2σ) arcseconds/year at ∼2,465 Ma to 55.86 ± 2.60 (2σ) arcseconds/year at ∼259 Ma. Stochastic monotone interpolation enables the reconstruction of Earth‐Moon distance from 2.5 billion years to the present. Calculation of tidal dissipation supports the ocean model before 1.4 Ga, but is outside the error range after ∼1.8 Ga, suggesting lower tidal dissipation in the early Proterozoic Eon, and a possible high value in the early Paleozoic. Plain Language Summary: The chaotic history of the Solar System imposes unknown uncertainties on the fundamental secular frequencies of the planets over most of geologic time. This is complicated by a lack of knowledge about the evolution of Earth‐Moon distance, Earth's length of day and tidal dissipation. Geological evidence provides powerful constraints on all of these parameters. To shed light on the parameters, here we carry out Bayesian inversion of fundamental secular frequencies and Earth precession rate on four cyclostratigraphic sequences, and stochastic monotone interpolation to reconstruct Earth‐Moon distance from 2.5 billion years ago to present. From these results we estimate the Earth's tidal dissipation variation, which we compare with theoretical models. Our results provide critical geological constraints for paleotidal simulation, and shine a light on the unique potential of cyclostratigraphy to constrain Earth‐Moon system history across geologic time. Key Points: We obtain new estimates of Earth‐Moon and Solar System dynamical parameters from cyclostratigraphy at four ages from ∼2.5 Ga to ∼259 MaWe interpolate Earth‐Moon distances inferred from the cyclostratigraphy to reconstruct a continuous history from ∼2.5 Ga to the presentWe estimate Earth's tidal dissipation variations from ∼2.5 Ga to the present [ABSTRACT FROM AUTHOR]

Image encryption based on a chaos system can effectively protect the privacy of digital images. It is said that a 3D chaotic system has a larger parameter range, better unpredictability and more complex behavior compared to low-dimension chaotic systems. Motivated by this fact, we propose a new image cryptosystem that makes use of a 3D chaotic system. There are three main steps in our scheme. In the first step, the chaotic system uses the hash value of the plaintext image to generate three sequences. In step two, one of the sequences is used to dynamically select confusion and diffusion methods, where confusion and diffusion have three algorithms, respectively, and will produce 32n (n > 100) combinations for encryption. In step three, the image is divided into hundreds of overlapping subblocks, along with the other two sequences, and each block is encrypted in the confusion and diffusion process. Information entropy, NPCR, UACI results and various security analysis results show that the algorithm has a better security performance than existing, similar algorithms, and can better resist clipping, noise, statistical analysis and other attacks. [ABSTRACT FROM AUTHOR]

Purpose: The purpose of the study is to obtain explicit formulas to determine the stability of periodic solutions to the new system and study the extent of the stability of those periodic solutions and the direction of bifurcated periodic solutions. More than that, the authors did a numerical simulation to confirm the results that the authors obtained and presented through numerical analysis are the periodic and stable solutions and when the system returns again to the state of out of control. Design/methodology/approach: The authors studied local bifurcation and verified its occurrence after choosing the delay as a parameter of control in Zhou 2019's dynamical system with delayed feedback control. The authors investigated the normal form theory and the center manifold theorem. Findings: The occurrence of local Hopf bifurcations at the Zhou's system is verified. By using the normal form theory and the center manifold theorem, the authors obtain the explicit formulas for determining the stability and direction of bifurcated periodic solutions. The theoretical results obtained and the corresponding numerical simulations showed that the chaos phenomenon in the Zhou's system can be controlled using a method of time-delay auto-synchronization. Originality/value: As the delay increases further, the numerical simulations show that the periodic solution disappears, and the chaos attractor appears again. The obtained results can also be applied to the control and anti-control of chaos phenomena of system (1). There are still abundant and complex dynamical behaviors, and the topological structure of the new system should be completely and thoroughly investigated and exploited. [ABSTRACT FROM AUTHOR]

Bifurcation analysis and dynamical system studies are carried out to find the stability regime and chaotic scenario in electron acoustic waves in relativistic degenerate plasma. We have obtained the quantum hydrodynamic model and obtain the Korteweg–de Vries equation describing the nature and characteristics of solitary structures. The amplitude modulated envelop soliton formation due to external perturbations has been studied by analyzing the nonlinear Schrodinger equation. Further to study the stability factors and the parametric range for such stability, the dynamical system is studied and bifurcation analysis has been carried out. The chaotic behavior of the system is studied through largest Lyapunov exponent (LLE). This work will find application in theoretically predicting the stable modes in many solar plasma and stellar plasma applications and in laser plasma in future. [ABSTRACT FROM AUTHOR]

The Lyapunov exponent is the most-well-known measure for quantifying chaos in a dynamical system. However, its computation for any time series without information regarding a dynamical system is challenging because the Jacobian matrix of the map generating the dynamical system is required. The entropic chaos degree measures the chaos of a dynamical system as an information quantity in the framework of Information Dynamics and can be directly computed for any time series even if the dynamical system is unknown. A recent study introduced the extended entropic chaos degree, which attained the same value as the total sum of the Lyapunov exponents under typical chaotic conditions. Moreover, an improved calculation formula for the extended entropic chaos degree was recently proposed to obtain appropriate numerical computation results for multidimensional chaotic maps. This study shows that all Lyapunov exponents of a chaotic map can be estimated to calculate the extended entropic chaos degree and proposes a computational algorithm for the extended entropic chaos degree; furthermore, this computational algorithm was applied to one and two-dimensional chaotic maps. The results indicate that the extended entropic chaos degree may be a viable alternative to the Lyapunov exponent for both one and two-dimensional chaotic dynamics. [ABSTRACT FROM AUTHOR]

With the urgent demand of high-end equipment for high quality surfaces, the technique of ultrasonic vibration-assisted burnishing is introduced to strengthen the surface properties. To explore the influence of the ultrasonic vibration on the dynamic response of a burnishing system, the burnishing friction force generated from a multi-ball surface burnishing system was characterized by chaos theory. The system had four assisted forms: no ultrasonic vibration, one-dimensional (1D) ultrasonic in x-axis, 1D ultrasonic in z-axis, and 2D ultrasonic in xz-axis. The results showed that the burnishing system had chaotic nature. Under the 2D ultrasonic vibration-assisted burnishing, the burnishing friction force was reconstructed to be a chaotic attractor with high convergence degree. Moreover, the burnishing system has notable complexity and stability. The burnished Al7075 alloy sample has an excellent surface with a higher smoothness and hardness. The burnishing with 2D ultrasonic vibration in xz-axis is a technique to enhance surface properties. [ABSTRACT FROM AUTHOR]

Biological systems exhibit strong dynamic variations; for example, a system may have a tendency towards either a steady state, periodic behaviour or chaotic oscillations. These findings, explained by dynamic systems theory, have allowed researchers to formally relate a wide variety of systems, such as the theory of rumour propagation, glucose metabolism or the activity of a neuronal circuit. Interestingly, among all of the dynamic characteristics mentioned, qualitative aspects frequently observed in highly complex systems can be obtained through the recursive application of a very simple parabolic function known as the logistic equation. Currently, the importance of these dynamic phenomena for biomedicine is recognized, but simple models such as the logistic equation have not been sufficiently exploited in the training of biologists and health professionals. We briefly review some properties of the dynamics governed by the logistic equation, with a direct application to the theory of cancer based on the dynamics of aneuploidy. We believe that this already known mathematical application is an opportunity for integrated teaching of mathematics and biology. For this purpose we finally provide an appendix with some activities, all of them related to this article, which can be practiced in a simple spreadsheet that we have shared. [ABSTRACT FROM AUTHOR]

The existence and identification of chaos regarding in-plane pitch motions of tethered satellite systems are scientific problems of concern, and are directly related to normal system operations in the station-keeping phase. This paper studies the effect of microamplitude longitudinal oscillation on the occurrence of chaos in a tethered system subjected to atmospheric perturbations. Based on a simplified rod model that considers tether elasticity and satellite masses, the Melnikov method is used to identify a criterion that can predict chaos, and a chaotic zone is further proposed to recognize the relationship between the chaos and system parameters. Then the cell mapping method is used to describe the system's global dynamic behaviors, including chaos. A discrete model of the flexible system that consists of particles connected by massless springs and satellite rigid bodies is structured so that more accurate dynamic simulations can be used to verify the theoretical analysis. Finally, numerical examples demonstrate that the chaotic zone and criterion expression are useful tools for revealing chaos. The relationship between the dynamics and system parameters is also assessed. The results of the simplified model agree with those of the sophisticated model. • A chaotic criterion and a chaotic zone on a tethered satellite system are proposed. • The relationship between the dynamics and system parameters is estimated. • The global dynamic characteristics of the system are revealed via cell mapping. [ABSTRACT FROM AUTHOR]

We propose in this paper an investigation of the dynamical behaviors of a duopoly model with two Internet services providers (ISPs). The competition between them is assumed to take place in terms of their pricing decisions and their number of cached items. According to the ISPs' rationality level, we consider a scenario where both ISPs are bounded rational. Each ISP in any period uses the marginal profit observed from the previous period to choose its strategies. We compute explicitly the steady states of the dynamic system induced by bounded rationality and establish a necessary and sufficient condition for the stability of its Nash equilibrium (NE). Numerical simulations show that if some parameters of the model are varied, the stability of the NE point is lost and the complex (periodic or chaotic) behavior occurs. The chaotic behavior of the system is stabilized on the NE point by applying control. [ABSTRACT FROM AUTHOR]

This work is devoted to the study of global connections between typical generic singularities, named T -singularities, in piecewise smooth dynamical systems. Such a singularity presents the so-called nonsmooth diabolo, which consists on a pair of invariant cones emanating from it. We analyze global features arising from the communication between the branches of a nonsmooth diabolo of a T -singularity and we prove that, under generic conditions, such communication leads to a chaotic behavior of the system. More specifically, we relate crossing orbits of a Filippov system presenting certain crossing self-connections to a T -singularity, with a Smale horseshoe of a first return map associated to the system. The techniques used in this work rely on the detection of transverse intersections between invariant manifolds of a hyperbolic fixed point of saddle type of such a first return map and the analysis of the Smale horseshoe associated to it. From the specific case discussed in our approach, we present a robust chaotic phenomenon for which its counterpart in the smooth case seems to happen only for highly degenerate systems. [ABSTRACT FROM AUTHOR]

The close relationship between chaos and dynamical systems leads to naturally consider the iteration processes that are related to dynamical systems of fixed point theory. From this natural relationship, the control of chaos that occurs in fixed point iteration dynamics will be the main focus of the article. To achieve this goal, analytical solutions are obtained and used to control chaos that occurs at unstable fixed points of multistep iteration process. Later, we show an effective regime for the parameters of multistep iteration. To illustrate this claim, well-known special cases of multistep iteration process by Noor, Ishikawa, Mann, Krasnoselskij, Picard iteration processes are introduced. In particular, among these iterations, the Noor iteration process is studied in detail in terms of controlled chaos. The Lyapunov exponent is used to estimate the stability and unstability of fixed points and periods that generate chaos in iteration processes. Finally, with the help of MATLAB program, all these results are shown on logistic and cubic equations with chaotic properties. [ABSTRACT FROM AUTHOR]

In the applied research of nonlinear system, the low degree of chaos in the dynamical system leads to the limitation of using the chaos method to solve some practical problems. In this paper, we use the product trigonometric function and ternary polynomial to build a dynamical system, which has strong chaotic characteristics. The dynamical system is constructed by two product trigonometric functions and a ternary linear equation, and its chaotic properties are verified by bifurcation diagrams, Lyapunov exponents, fractal dimensions, etc. The system has many parameters and large parameter intervals and is not prone to cycles. The conditions for the non-divergence of this system are given by mathematical derivation, and it is found that the linear part of the system can be replaced by an arbitrary ternary polynomial system and still not diverge, and the bifurcation diagram is drawn to verify it. Finally, the chaotic sequence is distributed more uniformly in the value domain space by adding the modulo operation. Then, the bit matrix of multiple images is directly permuted by the above system, and the experiment confirms that the histogram, information entropy, and pixel correlation of its encrypted images are satisfactory, as well as a very large key space. [ABSTRACT FROM AUTHOR]

Understanding the asymptotic behavior of a dynamical system when system parameters are varied remains a key challenge in nonlinear dynamics. We explore the dynamics of a multistable dynamical system (the response) coupled unidirectionally to a chaotic drive. In the absence of coupling, the dynamics of the response system consists of simple attractors, namely, fixed points and periodic orbits, and there could be chaotic motion depending on system parameters. Importantly, the boundaries of the basins of attraction for these attractors are all smooth. When the drive is coupled to the response, the entire dynamics becomes chaotic: distinct multistable chaos and bistable chaos are observed. In both cases, we observe a mixture of synchronous and desynchronous states and a mixture of synchronous states only. The response system displays a much richer, complex dynamics. We describe and analyze the corresponding basins of attraction using the required criteria. Riddled and intermingled structures are revealed. [ABSTRACT FROM AUTHOR]

Chua's groundbreaking discovery in 1971 introduced the memristor as the fourth essential electrical component, joining resistance, capacitance, and inductance. Over the past few years, numerous discrete memristor models have been developed following its formal proposal, expanding our understanding of this subject. However, there is limited discussion about multistable discrete memristor models. This research combines a comprehensive methodology for generating memristive maps using the modulo function and an exponential memristance, resulting in the development of a multistable discrete memristor model. Nonlinear techniques, such as bifurcation diagrams, Lyapunov spectrum diagrams, and phase portraits were utilized to analyze the dynamic characteristics of the system. The findings uncovered a range of captivating phenomena associated with chaos theory, including regular orbits, chaotic orbits, the mechanism of period doubling leading to chaos, crisis phenomena, and the presence of multistability. • Diagrams resembling bifurcation plots with two parameters were used to illustrate the dynamic behavior of the system. • Variations in initial conditions brought to light the presence of coexisting attractors. • Modified general discrete memristor model presented multistability under different initial conditions. • The behavior of the memristor as an active element becomes apparent when its internal state assumes smaller values. [ABSTRACT FROM AUTHOR]

The performance of clock recovery phase-locked loop (PLL) can be severely degraded by unwanted spurious phase modulation, due to noise and nonlinearities associated with the transmission medium. Consequently, choosing appropriate circuit parameters to avoid or at least reduce these undesirable effects plays a crucial role in achieving a successful PLL circuit. This paper demonstrates that Melnikov's method, originally proposed to identify chaos in dynamic systems, can be successfully used to establish the appropriate circuit parameters by determining the regions that ensure the correct operation under conditions of noise and nonlinearities, represented by periodic perturbations. This study considers the use of one of the most commonly adopted configurations for clock recovery, the second-order PLL with lag-lead filter, and provides a valuable tool for circuit dimensioning and design of clock distribution networks. [ABSTRACT FROM AUTHOR]

An approach to the time-accurate prediction of chaotic solutions is by learning temporal patterns from data. Echo State Networks (ESNs), which are a class of Reservoir Computing, can accurately predict the chaotic dynamics well beyond the predictability time. Existing studies, however, also showed that small changes in the hyperparameters may markedly affect the network's performance. The overarching aim of this paper is to improve the robustness in the selection of hyperparameters in Echo State Networks for the time-accurate prediction of chaotic solutions. We define the robustness of a validation strategy as its ability to select hyperparameters that perform consistently between validation and test sets. The goal is three-fold. First, we investigate routinely used validation strategies. Second, we propose the Recycle Validation , and the chaotic versions of existing validation strategies, to specifically tackle the forecasting of chaotic systems. Third, we compare Bayesian optimization with the traditional grid search for optimal hyperparameter selection. Numerical tests are performed on prototypical nonlinear systems that have chaotic and quasiperiodic solutions, such as the Lorenz and Lorenz-96 systems, and the Kuznetsov oscillator. Both model-free and model-informed Echo State Networks are analysed. By comparing the networks' performance in learning chaotic (unpredictable) versus quasiperiodic (predictable) solutions, we highlight fundamental challenges in learning chaotic solutions. The proposed validation strategies, which are based on the dynamical systems properties of chaotic time series, are shown to outperform the state-of-the-art validation strategies. Because the strategies are principled – they are based on chaos theory such as the Lyapunov time – they can be applied to other Recurrent Neural Networks architectures with little modification. This work opens up new possibilities for the robust design and application of Echo State Networks, and Recurrent Neural Networks, to the time-accurate prediction of chaotic systems. [ABSTRACT FROM AUTHOR]

In this work, we look at the extension of classical discrete dynamical system to multidimensional discrete-time dynamical system by characterizing chaos notions on Z d -action. The Z d -action on a space X has been defined in a very general manner, and therefore we introduce a Z d -action on X which is induced by a continuous map, f : Z × X → X and denotes it as T f : Z d × X → X . Basically, we wish to relate the behavior of origin discrete dynamical systems (X, f) and its induced multidimensional discrete-time (X , T f) . The chaotic behaviors that we emphasized are the transitivity and dense periodicity property. Analogues to these chaos notions, we consider k-type transitivity and k-type dense periodicity property in the multidimensional discrete-time dynamical system. In the process, we obtain some conditions on (X , T f) under which the chaotic behavior of (X , T f) is inherited from the original dynamical system (X, f). The conditions varies whenever f is open, totally transitive or mixing. Some examples are given to illustrate these conditions. [ABSTRACT FROM AUTHOR]

Memristor, as a typical nonlinear element, is able to produce chaotic signals in chaotic systems easily. Chaotic systems have potential applications in secure communications, information encryption, and other fields. Therefore, it is of importance to generate abundant dynamic behaviors in a single chaotic system. In this paper, a novel memristor-based chaotic system without equilibrium points is proposed. One of the essential features is the absence of symmetry in this system, which increases the complexity of the new system. Then, the nonlinear dynamic behaviors of the system are analyzed in terms of chaos diagrams, bifurcation diagrams, Poincaré maps, Lyapunov exponent spectra, the sum of Lyapunov exponents, phase portraits, 0–1 test, recurrence analysis and instantaneous phase. The results of the sum of Lyapunov exponents show that the given system is a quasi-Hamiltonian system with certain initial conditions (IC) and parameters. Next, other critical phenomena, such as hidden multi-scroll attractors, abundant coexistence characteristics, are found characterized through basins of attraction and others. Especially, it reveals some rare phenomena in other systems that multiple hidden hyperchaotic attractors coexist. Finally, the circuit implementation based on Micro Control Unit (MCU) confirms theoretical analysis and the numerical simulation. [ABSTRACT FROM AUTHOR]

Dynamic systems exhibiting chaos with periodic windows and multistability are not recommended for cryptography due to the lack of security in these regimes. In this work, we use a linear augmentation control scheme to control a periodic attractor in the windows of multistability to a final chaotic attractor which survives to the variation of initial seed useful for image encryption. The empirical Chua's system with piecewise linear nonlinearity is used as a sample dynamical system but the idea can be applied using any other dynamical system. First, this system is analyzed to reveal multistability dynamics. Second, the technique of linear augmentation combined with the nonlinear system invariant sets like equilibrium points is used to choose a desired survive attractor among the coexisting ones. It is found that annihilation of multistability in the Chua's system when varying the coupling strength is obtained through several crises among which interior crisis and border collision. Finally, a survived chaotic attractor is jointly used with SHA-512 for image encryption algorithm using a simple diffusion-confusion structure. Security analysis shows that the encryption process based on control theory can resist various forms of attack. [ABSTRACT FROM AUTHOR]

Writing a history of a scientific theory is always difficult because it requires to focus on some key contributors and to "reconstruct" some supposed influences. In the 1970s, a new way of performing science under the name "chaos" emerged, combining the mathematics from the nonlinear dynamical systems theory and numerical simulations. To provide a direct testimony of how contributors can be influenced by other scientists or works, we here collected some writings about the early times of a few contributors to chaos theory. The purpose is to exhibit the diversity in the paths and to bring some elements—which were never published—illustrating the atmosphere of this period. Some peculiarities of chaos theory are also discussed. [ABSTRACT FROM AUTHOR]

In this paper, based on chaos theory and dynamic system simulation, a dynamic model of a relational leadership system with pressure factors is established and simulated. The study found that the excessive self-leadership and the relational leadership cannot effectively improve the leadership effectiveness of leaders. In addition, with the adjustment of the self-leadership adjustment coefficient, leadership effectiveness will change from a stable state to a chaotic state. Only by adopting limited rational decision-making strategies in multicycle decision-making can leaders maintain a high level of leadership effectiveness. Previous research mainly focuses on the theoretical analysis of the complexity of the leadership system or the issuance of questionnaires, but seldom on the dynamic complex process of the leadership system. This study reveals the complexity and chaos characteristics of the dynamic evolution of the multifactor relational leadership system, puts forward the relational leadership theory and the complex leadership theory under the influence of pressure, and makes some suggestions on how to improve the leadership effectiveness under the complex environment. The research results are of great significance to theoretical innovation and practical application. [ABSTRACT FROM AUTHOR]

Does complexity make multilingualism special? Since there is no unequivocal notion of complexity on which researchers agree, several characteristics that have been considered crucial for complexity are brought to bear on multilingualism. While multilingualism is fairly complex in some senses, for instance, because it requires that many variables be studied, it is less clear whether multilingualism becomes special in this way. The most salient possible way in which multilingualism might be special due to its complexity is that qualitatively new features emerge as we move from mono‐ or bilingualism to multilingualism. While research in mathematics and physics has shown examples in which novel features emerge at a certain level, related results are not easily transferred to multilingualism. This is shown by analyzing the application of dynamic systems theory to multilingualism by Herdina and Jessner. It is ultimately a matter of empirical research whether there are suitable novel features that make multilingualism special. [ABSTRACT FROM AUTHOR]

The purpose of this article is to build an introductory approach to chaos theory from the Chua' oscillator for students of mathematics, physics and engineering. The Chua' circuit is one of the simplest dynamic systems that produce irregular behavior. Despite its simplicity, this oscillating circuit is a very useful device for studying basic principles of chaos theory. This study begins with the definition of equilibrium points, this is done in a graphic and analytical way. The analysis of the stationary solution of differential equations system allows to show the occurrence of pitchfork bifurcation. The eigenvalues are calculated and the trajectories of the Chua' oscillator are analyzed, then the occurrence of Hopf bifurcation is demonstrated. Period-doubling cascade are observed using phase portrait and orbit diagram. Finally, sensitivity to initial conditions is studied by calculating Lyapunov exponents. [ABSTRACT FROM AUTHOR]