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I analyse in detail the construction of asymptotic surfaces in Sections 16-19 of Poincaré (1890), also known as the prize paper. There are two prime reasons for doing so. Firstly, this part of the prize paper contains an interesting argumentative strategy, which I call Poincaré's gestalt switch. Secondly, it has been claimed that the prize paper contains one of the first descriptions of chaotic motion. I will argue that the latter claim is false, although both the gestalt switches and the graphical representation which Poincare (1890) chose for the asymptotic surfaces might well have provided the inspiration for later works in chaos theory. [ABSTRACT FROM AUTHOR]

I analyse in detail the construction of asymptotic surfaces in Sections 16–19 of Poincaré (1890) , also known as the prize paper. There are two prime reasons for doing so. Firstly, this part of the prize paper contains an interesting argumentative strategy, which I call Poincaré ׳ s gestalt switch . Secondly, it has been claimed that the prize paper contains one of the first descriptions of chaotic motion. I will argue that the latter claim is false, although both the gestalt switches and the graphical representation which Poincaré (1890) chose for the asymptotic surfaces might well have provided the inspiration for later works in chaos theory. [ABSTRACT FROM AUTHOR]

Artificial Intelligence and Machine learning have been widely used in various fields of mathematical computing, physical modeling, computational science, communication science, and stochastic analysis. Approaches based on Deep Artificial Neural Networks (DANN) are very popular in our days. Depending on the learning task, the exact form of DANNs is determined via their multi-layer architecture, activation functions and the so-called loss function. However, for a majority of deep learning approaches based on DANNs, the kernel structure of neural signal processing remains the same, where the node response is encoded as a linear superposition of neural activity, while the non-linearity is triggered by the activation functions. In the current paper, we suggest to analyze the neural signal processing in DANNs from the point of view of homogeneous chaos theory as known from polynomial chaos expansion (PCE). From the PCE perspective, the (linear) response on each node of a DANN could be seen as a 1st degree multi-variate polynomial of single neurons from the previous layer, i.e. linear weighted sum of monomials. From this point of view, the conventional DANN structure relies implicitly (but erroneously) on a Gaussian distribution of neural signals. Additionally, this view revels that by design DANNs do not necessarily fulfill any orthogonality or orthonormality condition for a majority of data-driven applications. Therefore, the prevailing handling of neural signals in DANNs could lead to redundant representation as any neural signal could contain some partial information from other neural signals. To tackle that challenge, we suggest to employ the data-driven generalization of PCE theory known as arbitrary polynomial chaos (aPC) to construct a corresponding multi-variate orthonormal representations on each node of a DANN. Doing so, we generalize the conventional structure of DANNs to Deep arbitrary polynomial chaos neural networks (DaPC NN). They decompose the neural signals that travel through the multi-layer structure by an adaptive construction of data-driven multi-variate orthonormal bases for each layer. Moreover, the introduced DaPC NN provides an opportunity to go beyond the linear weighted superposition of single neurons on each node. Inheriting fundamentals of PCE theory, the DaPC NN offers an additional possibility to account for high-order neural effects reflecting simultaneous interaction in multi-layer networks. Introducing the high-order weighted superposition on each node of the network mitigates the necessity to introduce non-linearity via activation functions and, hence, reduces the room for potential subjectivity in the modeling procedure. Although the current DaPC NN framework has no theoretical restrictions on the use of activation functions. The current paper also summarizes relevant properties of DaPC NNs inherited from aPC as analytical expressions for statistical quantities and sensitivity indexes on each node. We also offer an analytical form of partial derivatives that could be used in various training algorithms. Technically, DaPC NNs require similar training procedures as conventional DANNs, and all trained weights determine automatically the corresponding multi-variate data-driven orthonormal bases for all layers of DaPC NN. The paper makes use of three test cases to illustrate the performance of DaPC NN, comparing it with the performance of the conventional DANN and also with plain aPC expansion. Evidence of convergence over the training data size against validation data sets demonstrates that the DaPC NN outperforms the conventional DANN systematically. Overall, the suggested re-formulation of the kernel network structure in terms of homogeneous chaos theory is not limited to any particular architecture or any particular definition of the loss function. The DaPC NN Matlab Toolbox is available online and users are invited to adopt it for own needs. • Kernel structure reformulation of deep artificial neural networks in terms of homogeneous chaos theory. • Response on each node of the deep network represented through polynomial chaos expansion. • Orthonormal decomposition mitigates the non-optimal representation while processing the neural signal. • Accounting for high-order effects reflecting simultaneous neural impacts. • Analytical estimation of mean and variance of neural signal on each node is provided. [ABSTRACT FROM AUTHOR]

• Chaos theory is used to eliminate differences in the feature distribution of images with different modalities. • We propose a structure-aware feature extraction method to retain more features at the detail layer. • In this paper, proportional integral differential (PID) control is introduced into the fusion process. • Fusion experiments and object detection experiments demonstrate the superiority of our method. Differences in the imaging mechanisms of infrared and visible light images lead to differences in the way their visually meaningful gradients are formed. Existing fusion methods use the same feature extractor to extract features from the source video frames, which ignores the differences in the gradients of video frames from different modalities. In this paper, we propose an infrared and visible light video fusion method based on chaos theory and proportional integral differential (PID) control. Firstly, for the Lorentz chaotic system, we give it initial values and parameters, and iterate it to obtain three scrambling sequences, through which the source video frames are scrambled in the rows, columns, and diagonal directions respectively to eliminate their visually meaningful gradients, so that the features extracted are of comparable scales in the same layer, and the fusion process can be carried out in the scale-consistent space. Second, we propose a structure-aware relative total variation feature extraction method (saRTV) for the two-scale decomposition of source video frames, which can transfer more features of source video frames to the detail layer. Then, based on our previous work, this paper introduces PID control to construct a closed-loop control system through transfer function design, controller design and measurement function design. This control system is used to fuse the detail layer, so as to realize the real-time guidance of the source video frames to the fusion process. Experiments on public datasets demonstrate that our method has better performance compared to some state-of-the-art methods. [ABSTRACT FROM AUTHOR]

• Constructing pseudo-Hamiltonian system, the topology of dynamic motion is able to distinguish whether the system is in chaos. • The fractal process of the attraction basin verifies the chaotic distribution solved by Melnikov function. • Larger force fluctuation leads to the critical point hysteresis of damping and average force occurring bifurcation or chaos. • The quantitative optimal couple of structural with process parameters determine the priority of the parameter control. This paper investigates the stability of horizontal nonlinear vibration of four-high cold mill under the effects of gyro-precession and eccentricity between rolls. The threshold value of chaos about Smale horseshoe commutation is given from Melnikov method, and the correctness of the result is verified by the fractal process of the attraction basin. Through the analyzes of bifurcation, maximum Lyapunov exponents and bi-parameter bifurcation, it is revealed for the effect of different parameters variation on the nonlinear dynamic behavior of horizontal vibration in rolling system. The results show, the fluctuation amplitude of horizontal force increases, the stable domain decreases, and once its value ≥3.5, it is necessary to prefer to the adjustment of parameter-exciting stiffness; meanwhile, it is noticeable that the critical points of damping and average force of the system with bifurcation or chaos appear hysteresis. The evolutions of system dynamics with four pairs of parameter changes confirm the quantitative optimal couple of one structural plus one process parameter in the system. Based on this, we can also find a balance between the average horizontal force and eccentricity, so as to optimize the structural design so that there is a proper and reliable eccentricity between the rolls. The results provide a theoretical reference for service stability and dynamic reliability of the rolling mill system. [ABSTRACT FROM AUTHOR]

This paper studies the synchronization behavior of multistable chaotic systems with coexisting symmetric attractors. Specifically, the focus is on the attractive and repulsive couplings of the attractors in single-variable couplings. It is shown that in the self-couplings, both attractors have the same synchronization pattern either in the attractive or repulsive coupling. In the cross-couplings, the synchronization pattern of the attractors is dependent on the variables involved in the coupling and the symmetry transformation. If the coupling scheme is defined such that only one of the variables in the coupling participates in the symmetry transformation, the synchronization patterns of the symmetric attractors are symmetric in the attractive and repulsive couplings. The master stability function is applied to four chaotic systems with different symmetry transformations to represent the results. The corresponding chaotic circuit of two coupled symmetric systems is also implemented and their symmetric responses are shown. • Synchronization stability of coexisting symmetric chaotic attractors is investigated. • Coupling is considered to be both attractive and repulsive through single variable. • In self-couplings, symmetric attractors have the same synchronization patterns. • symmetry synchrony is observed if one transformation variable is in the coupling. [ABSTRACT FROM AUTHOR]

Due to the attractiveness of household gas-electric tools, in this paper, an optimization technique is suggested based on the integrated demand response (IDR) and degree of tolerance for household energy management. The proposed method is mostly used to express the dynamic change in the forms of energy and undetermined variables in the systems, resulting from household and photovoltaic (PV) load. Thermostatically controlled demands include gas-electricity and air conditioning, and cut-able loads include gas-electric stove and washing machine. The interval optimization is modeled for optimizing the operation and greenhouse gas emission costs in multi-purpose systems. The undetermined variables are formulated as interval statistics and the limitations are simplified by degree of tolerance. In order to solve it, the interval optimization technique is converted into certainty optimization with the interval order relationship and the delayed probability degree. Then, the developed grasshopper search algorithm is based on the chaos theory to solve the interval optimization model in order to respond to uncertainty and demands of the users, such that degree of tolerance of cost that is acceptable by users is optimized. Contrary to other optimization algorithms, the grasshopper search algorithm can be combined with other methods. In this paper, the chaos theory is adopted to find a better solution. Since the information is placed in the search space without order, using this technique considerably leads to good convergence speed, precise final solution finding, not being trapped in local minima, lower SD, and robustness. Both methods of IDR and degree of tolerance for the household gas-electric equipment manage to reduce energy consumption by about 25% compared to traditional methods. • Gas-electric equipment is integrated in household energy management. • A novel IDR method in HMES is proposed, while uncontrollable loads are also executed. • To deal with HMES uncertainty with electric equipment, interval optimization is used. • The users' degree of tolerance in HMES with gas-electric equipment is modeled. • New model of the GSA based on chaos theory is introduced. [ABSTRACT FROM AUTHOR]

[Display omitted] • Shape and size Optimization of truss Structures is considered. • Chaos Game Optimization (CGO) is utilized for optimization purposes. • Benchmark 10-bar, 37-bar, 52-bar, 72-bar and 120-bar truss structures are utilized. An engineering system consists of properly established activities and put together to achieve a predefined goal. These activities include analysis, design, construction, research, and development. Designing and constructing structural systems, including buildings, bridges, highways, and other complex systems, have been developed over the centuries. However, the evolution of these systems has been prolonged because the overall process is very costly and time-consuming, requiring primary human and material resources to be utilized. One of the options for overcoming these shortcomings is the utilization of metaheuristic algorithms as recently developed intelligent techniques. These algorithms can be utilized as upper-level search techniques for optimization procedures to achieve better results. Shape and size optimization of truss structures are considered in this paper utilizing the Chaos Game Optimization (CGO) as one of the recently developed metaheuristic algorithms. The principles of chaos theory and fractal configuration are considered inspirational concepts. For the numerical purpose, the 10-bar, 37-bar, 52-bar, 72-bar, and 120-bar truss structures as four of the benchmark problems in this field are considered as design examples in which the frequency constraints are considered as limits that have to be dealt with during the optimization procedure. Multiple optimization runs are also conducted for having a comprehensive statistical analysis, while a comparative investigation is also conducted with other algorithms in the literature. Based on the results of the CGO and other approaches from the literature, the CGO can provide better and competitive results in dealing with the considered truss design problems. In summary, the CGO can provide better solutions in dealing with the considered real-size structural design problems with higher levels of complexity. [ABSTRACT FROM AUTHOR]

Nonlinear kinetic energy harvesters are becoming more and more popular as well as advanced and efficient. This paper presents the study of the dynamics of such a system in a wide range of excitation parameters, assuming at the same time the possibility of a cyclical and smooth change of the potential function. We have designed a system that allows to obtain a wide spectrum of potential characteristics, from a single well to a three-well system, and we have analyzed its effectiveness. Next, we checked the influence of parameters characterizing the change of potential using bifurcation diagrams and their comparison with the effective voltage values. We also analyzed the behavior of the system in chaotic and periodic motion zones and presented selected sections of Poincare and Fourier amplitude-frequency spectra of chaotic solutions. The last element of the analysis was the impact of cyclic potential change on coexisting solutions. We have shown that the best effectiveness is achieved when the frequency of the external load is equal to the resonant frequency of the flexible cantilever beam and the change of potential is limited to extreme positions. • A new way of the potential function dynamic change in EH was proposed. • EH effectiveness analysis was made for chaotic and periodic motion zones. • Influence of the range and frequency of potential changes on the dynamics of the system was checked. [ABSTRACT FROM AUTHOR]

Memory resistor, or memristor, has been realized as a discrete electronic device and has a perspective application in the field of cryptography. The physical implementation of the memristor in chaotic circuits has been scarcely explored. In this paper, a memristor is fabricated by spin-coating a cobalt ferrite precursor on a processed silicon and is then electro-sputtered with silver to act as the anode with the base silicon as the cathode. This fabrication process has a scalability potential in conjunction with integrated circuit fabrication techniques and complementary metal oxide semiconductor (CMOS) technologies. The fabricated cobalt ferrite memristor has shown a ratio between the on and off resistance of >1000 and has been implemented in a chaotic Chua's circuit, making it one of few physical implementations of a physical memristor in a physical circuit. The analysis and characterization of this circuit using bifurcation diagrams and Lyapunov exponent prove the chaotic behavior of a real Chua's circuit. This chaotic behavior can be useful in chaotic cryptography as nonperiodic oscillations can be leveraged to make sensitive information more difficult to interpret by bad actors. [Display omitted] • A memristor is constructed on silicon to make fabrication process more scalable. • The memristor consists of spin-coated cobalt ferrite layer with active electrodes. • The memristor is implemented in Chua's circuit that exhibits chaotic behavior. • Chaotic behavior is proven with bifurcation and Lyapunov exponent analysis. [ABSTRACT FROM AUTHOR]

The accurate prediction of digging load serves as a fundamental cornerstone for advancing the development of intelligent and unmanned excavators. Given the complex nonlinear dynamics of digging load, this paper proposes a novel prediction model for excavator load power based on the chaos theory and support vector regression (SVR). The presence of chaos in the dynamic digging load system is detected through phase space reconstruction. SVR is utilized for nonparametric modeling and prediction, with the reconstructed phase space capturing the essential characteristics of excavator load and serving as inputs for SVR. To optimize the hyperparameters, an improved particle swarm optimization (IPSO) algorithm is presented. Excavation experiments conducted under two typical load conditions demonstrate the superiority of the proposed chaos-based IPSO-SVR model in terms of prediction accuracy. This research lays a solid foundation for practical load prediction in industrial excavator settings. [ABSTRACT FROM AUTHOR]

Synchronization of satellite systems offers numerous advantages, including cost-effectiveness, flexibility, and extended area coverage. However, the presence of chaotic behavior in such systems poses substantial challenges. This paper investigates chaos in satellite systems and proposes a novel approach - the Novel Integral Terminal Sliding Mode (NITSM) controller - specifically designed for synchronizing the angular velocities of chaotic Leader-Follower satellite systems. The NITSM controller leverages limited-time features to eliminate chattering and exhibits remarkable robustness against external disturbances such as cosmic rays and solar storms. MATLAB simulations are conducted to validate the effectiveness of the proposed method, comparing the synchronization error of the NITSM controller with a common approach. The results demonstrate the superior performance of the Novel Integral Terminal Sliding Mode controller, showcasing reduced response time, robust behavior, and a substantial six-fold reduction in angular velocity synchronization error. [ABSTRACT FROM AUTHOR]

• The dynamic model of axially moving piezoelectric rectangular thin plate in thermal-electric-mechanical field is established. • The coupling relationship between thermal-electric-mechanical field and elastic deformation of thin plate are considered. • The influence mechanism on the resonance amplitude under the effects of thermal, force, electric and damping coefficients are clarified. In this paper, the combination resonance of parametric-forced excitation characteristics for an axially moving rectangular piezoelectric plate under a thermal-electromechanical field is studied. Based on Kirchhoff-Love plate theory and Von Karman theory, the transverse vibration governing equations are derived from Hamilton's principle. The equations are discretized to ordinary differential equations by the Galerkin method. Then, the multiple scales method is applied to solve the system combination resonance equation, two different resonance states and corresponding amplitude-frequency response equations are obtained by eliminating the secular term, respectively. Additionally, the stability of the steady-state responses is analyzed by Lyapunov stability. Based on the numerical analysis, the influence of axial velocity, external voltage, central temperature difference, structural damping, and other parameters on nonlinear combination resonance response are investigated. The effect of parameter variation on period-doubling bifurcation and the chaotic motion of the system are also discussed by the system global bifurcation diagram. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the stochastic dynamics of a class of nonlinear ship rolling motion with multiplicative noise under both displacement and velocity delay feedback controls. The I t o ˆ -stochastic differential equation for the amplitude and phase of the roll motion is derived using the stochastic center manifold method and stochastic average method. Subsequently, the stochastic stability and bifurcation behaviors of the system are analyzed. Furthermore, using the stationary probability density method, we derive the parameter conditions for the occurrence of stochastic D -bifurcation and stochastic P -bifurcation. We also analyze the properties and shape changes of the system's probability density function under different parameters through numerical simulation. It has been determined that the system exhibits stochastic bifurcation behavior, specifically P -bifurcation and D -bifurcation. The validity of the method is verified by a numerical model. The theoretical chaos threshold of the system is determined using the random Melnikov method, and the impact of delayed feedback parameters on the chaotic motion of the system is analyzed by combining the bifurcation diagram, phase portrait, and time series. • We investigate the stochastic dynamics of a class of nonlinear ship rolling motion with multiplicative noise under both displacement and velocity delay feedback controls. • The Ito-stochastic differential equation for the amplitude and phase of the roll motion is derived using the stochastic center manifold method and stochastic average method. • Using the stationary probability density method, we derive the parameter conditions for the occurrence of stochastic D-bifurcation and stochastic P-bifurcation. • The theoretical chaos threshold is determined using the random Melnikov method. [ABSTRACT FROM AUTHOR]

This paper considers system identification (ID) of linear and nonlinear non-autonomous systems from noisy and sparse data. We analyze an optimization objective derived from Bayesian inference for the dynamics of hidden Markov models. We then relate this objective to that used in several state-of-the-art approaches for both linear and nonlinear system ID. In the former, we analyze least squares approaches for Markov parameter estimation, and in the latter, we analyze the multiple shooting approach. We demonstrate the limitations of the optimization problems posed by these existing methods by showing that they can be seen as special cases of the proposed optimization objective under certain simplifying assumptions: conditional independence of data and zero model error. Furthermore, we observe that the proposed approach has improved smoothness and inherent regularization that make it well-suited for system ID and provide mathematical explanations for these characteristics' origins. Finally, numerical simulations demonstrate a mean squared error over 8.7 times lower compared to multiple shooting when data are noisy and/or sparse. Moreover, the proposed approach identifies accurate and generalizable models even when there are more parameters than data or when the system exhibits chaotic behavior. • Modeling of model uncertainty with process noise leads to inherent regularization. • Process noise induces objective function smoothness similarly to multiple shooting. • Certain Markov parameter estimation methods treat data as conditionally independent. • Bayesian system ID outperforms state-of-the-art methods on small, noisy datasets. [ABSTRACT FROM AUTHOR]

This paper deals with the attitude motion of LEO satellites with deployable side panels designed for passive aerodynamic stabilization in a rarefied atmosphere. The influence of the aerodynamic and gravitational torques on the planar attitude motion near the unstable and stable equilibrium positions is studied. The presence of the unstable equilibrium position and small perturbations such as the oscillations of the flexible panels is the cause of chaos. A critical altitude is found above which the chaos is possible. The equations of planar attitude motion of the satellite with deployed flexible panels are obtained. The chaotic behavior of the system is demonstrated through numerical simulations of the attitude motion of a 3U CubeSat. The results of this paper can be used to analyze the applicability of passive aerodynamic stabilization for LEO satellites. • Planar attitude motion of a LEO satellite with deployable side panels is studied. • The satellite as a rigid body with the flexible appendages is considered. • For a wide range of altitudes, stable and unstable equilibrium positions are found. • The chaotic behavior of the satellite with flexible panel is proven. • It's been shown that flexible panels do not always lead to stabilization of motion. [ABSTRACT FROM AUTHOR]

In this paper, a three-dimensional conservative system is proposed. By analyzing the Hamiltonian energy, it is concluded that it is a non-Hamiltonian conservative system. By changing the values of the parameters b, c, the system presents different attractors. Then, by studying the multiple attractors coexisting of the system under different initial values, it can be found that the system has rich coexistence phenomena by changing the parameters. There are various types of coexistence of systems with chaos and chaos coexistence, periods and periods coexistence and chaos and periods coexistence. In addition, the offset-boosting under parameter control is studied by phase diagram. By studying the complexity of the system at two different initial values, the more complex initial value is chosen as the initial state when the system is synchronized. The analog simulation circuit is implemented using Multisim, the actual digital circuit is implemented with field programmable gate array (FPGA). The Matlab phase diagram, the numerical simulation results and the simulated circuit all agree well, proving the feasibility of the new system. Finally, the system is applied to backstepping synchronization, which lays a foundation for the realization of engineering application. • In this paper, a three-dimensional conservative system is proposed. It is proved to be a non-Hamiltonian three-dimensional conservative system by Hamiltonian energy analysis. • Successful implementation of Multisim simulation and FPGA hardware proved the feasibility of the system. • Through a series of studies, it has been found that conservative systems are more conducive to confidential communication. Therefore, based on this 3D conservative system, the system is applied to backstepping synchronization. [ABSTRACT FROM AUTHOR]

External disturbances are always inevitable in complex application scenarios, especially in synchronizing chaotic systems. This paper proposes a noise-restraint zeroing neural network (NRZNN) model to expedite the synchronisation of chaotic systems under external disturbances. Its associative controller is then evolved to suppress the interference of external noise. Theoretical analysis shows that the NRZNN model and its associated controller have inherent robustness. For comparison, the conventional zeroing neural network (CZNN) approach is utilized for the synchronisation of chaotic systems. Numerical comparison results validate the efficiency of the NRZNN model for synchronising chaotic systems under the constant noise disturbance. Moreover, through additional tests, it is found that the proposed NRZNN model can also suppress time-dependent noise during the synchronization of chaotic systems. Finally, the effect on the convergence performance is further investigated by adjusting the values of design parameters. [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]

In this paper, a multiscroll snap oscillator with hyperbolic tangent function is proposed. There is no limitation in the number of scrolls and it can be increased by proper choice of a specific function. The Lyapunov exponents of the proposed system are obtained to testify the chaotic behavior of the system. Fractional order multiscroll system is derived from its integer order model by using the Adams–Bashforth–Moulton algorithm. A new scheme is applied in order to investigate the synchronization of the multiscroll systems. The main objective of the paper is to propose a multiscroll attractor and show that the number of scrolls can be controlled by the only nonlinear function. Such systems are less investigated in the literatures and has many real time applications like image and voice encryption, random number generators, chaos based communication systems and so on. • We propose a multiscroll snap oscillator with hyperbolic tan function. • We have demonstrated generating up to five scrolls but not limited to five. The number of scrolls may be increased by proper choice of the multiscroll function. • Fractional order multiscroll system is derived from its integer order model using the Adams-Bashforth-Moulton algorithm. • A new synchronization scheme is derived and presented to synchronize the multiscroll systems. [ABSTRACT FROM AUTHOR]

This paper investigates n -scrambled sets in the sense of distribution and Li-Yorke, focusing on the equivalence of n -scrambled sets and the existence of uncountable n -scrambled sets. For the equivalence, n -scrambled sets can imply m -scrambled sets in both senses for any 2 ≤ m < n , but their inverses require certain conditions to hold. A method is provided for constructing an uncountable n -scrambled set in the distribution and Li-Yorke senses, which also verifies the equivalence between n -scrambled sets. For the existence, it is shown that distributionally n -scrambled tuples can be derived from the weak specification property (WSP), a 1-periodic and a non-1-periodic point. And the Li-Yorke (n + 1)-scrambled tuple can be characterized by an n -scrambled tuple utilizing the SP. Finally, the uncountable Li-Yorke n -scrambled set is established based on the weak mixing, and the uncountable mean Li-Yorke n -scrambled set is induced by the shadowing and Devaney chaos. • Obtain the equivalence between n -scrambled sets in the distribution and Li-York sense. • Provide a method for constructing an uncountable n -scrambled set in two chaotic senses. • Establish criteria for the existence of n -scrambled set and n -scrambled tuple. • Present a method to induce an n -scrambled tuple by weak specification property. • Build a connection between global and local chaotic behaviors of a dynamical system. [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]

Highlights • Multiple coupled map lattices (MCML) was proposed. • The method of nonlinear-diffusion was proposed. • The strategy of simultaneously shuffling and diffusion was adopted. • The proposed scheme only needs one round of encryption to get good effects. • The superior performance of the proposed algorithm was proved by experiments. Abstract This paper proposes a new spatiotemporal chaos system named Multiple coupled map lattices (MCML). The proposed spatiotemporal chaos system has outstanding cryptographic features, which is very suitable for encryption algorithms. Based on this system, this paper proposes a new image encryption algorithm. The proposed algorithm employs the strategy of nonlinear-diffusion for the first time, and simultaneously performs shuffling and diffusion. The ciphertext value of each pixel in the diffusion phase depends on a chaotic interference value, a pixel value of the plain image, and two values of the ciphered image non-adjacent to it. This strategy reduces the correlation between adjacent pixels of the plain image as well as the correlation between the R , G and B components of color image. Theoretical analysis and experimental results prove the high efficiency and security of the proposed algorithm. [ABSTRACT FROM AUTHOR]

Abstract As for the dynamical networks which consist of some high-dimensional nonlinear systems, the problems that researchers are concerned with are usually the asymptotic convergence on some components (rather than all components) of node's state variables under certain condition. This means that partial component synchronization is more meaningful than identical synchronization in some cases. In this paper, the definition of partial component synchronization is given, and then the problem of partial component synchronization on a class of chaotic dynamical networks is investigated. By using matrix theory, stability theory and the hypothesis that several components in the solution vector of a single uncoupled node are ultimately dissipative, some sufficient conditions on partial component synchronization in the chaotic dynamical networks are derived. Finally, numerical simulations are shown to demonstrate the correctness of the theoretical results. Highlights • Partial component synchronization is a kind of group dynamics behavior weaker than identical synchronization. • In this paper, the definition of partial component synchronization is given, and the stability theory of partial variables is applied to study it. • Several sufficient conditions for partial component synchronization to be realized on the network are derived. [ABSTRACT FROM AUTHOR]

In this paper, we give solution to the quickest drift change detection problem for a Lévy process consisting of both a continuous Gaussian part and a jump component. We consider here Bayesian framework with an exponential a priori distribution of the change point using an optimality criterion based on a probability of false alarm and an expected delay of the detection. Our approach is based on the optimal stopping theory and solving some boundary value problem. Paper is supplemented by an extensive numerical analysis related with the construction of the Generalized Shiryaev-Roberts statistics. In particular, we apply this method (after appropriate calibration) to analyse Polish life tables and to model the force of mortality in this population with a drift changing in time. [ABSTRACT FROM AUTHOR]

Abstract Stochastic resonance is a phenomenon that a weak signal can be amplified and optimized by the assistance of noise in bistable system. There is still not enough research on the mutual interplay among system, noise and signal. In this paper, we study the role of every parameter in nonlinear transfer and discover chaos phenomenon in stochastic resonance. To measure the influence of chaos, a trajectory decision function was proposed. Based on this function, we found two forms of stochastic resonance, clockwise resonance and counterclockwise resonance. Highlights • In this paper, we study the role of every parameter in stochastic resonance and discover chaos phenomenon in it. • A trajectory decision function was proposed to measure the influence of chaos. • Based on this function, we found two forms of stochastic resonance, clockwise resonance and counterclockwise resonance. [ABSTRACT FROM AUTHOR]

• A synchronous-scrambling-diffusion image encryption method is presented based on our new spatiotemporal chaotic system and sorting vector. Experimental results show that our method is more efficient and secure than other algorithms. • Based on the fractal sorting matrix and fractal curve, the Hilbert sorting vector (HSV) is proposed. • HSV is more flexible during applications in the chaotic system with arbitrary lattice numbers and realises the parameter control of the generated sorting number scaling. • The proposed HSV effectively realises the uncertainty of coupling node selection in the spatiotemporal chaotic system, thus improving the chaotic performance of the system and presenting a more stable chaotic state, making the image encryption algorithm more secure and reliable. • Hilbert-sorting-vector coupled map lattice (HSVCML) has the potential to be applied effectively in various fields, such as information security and secure communication. The design of the lattice coupling mode in current spatiotemporal chaotic systems lacks dynamic characteristics. When it is applied to the cryptosystem, its chaotic performance defects weaken the security of the cryptosystem. In this context, it is urgent to design new coupling rules and secure cryptosystems based on fractal and chaos theory. In this paper, a kind of sorting vector based on the fractal sorting matrix and fractal curve, the Hilbert sorting vector (HSV), is proposed creatively, and its iterative generation process is introduced. HSV is irregular and infinitely iterable. According to the requirements of the actual situation, HSV has a flexible adjustment vector length, which improves the multiplicity and efficiency of changing information positions. Then, HSV is used to reconstitute the the interaction of nodes during iteration in a new spatiotemporal chaotic system named Hilbert-sorting-vector coupled map lattice (HSVCML). Using this new sorting vector, the dynamic characteristics of spatiotemporal chaotic systems can be effectively improved. This is proved by comparing their Lyapunov exponent, Kolmogorov-Sinai entropy, bifurcation diagram, and information entropy with the coupled map lattice (CML). Moreover, the rich spatio-temporal behaviours of HSVCML are studied. Therefore, HSVCML is more appropriate for image encryption than CML. Finally, HSV is combined with a spatiotemporal chaotic system to frame an image encryption method. For the purpose of proving the effectiveness and security of this algorithm against different types of attacks, a large number of tests related to security analysis and time complexity analysis are carried out. Simulation results prove that our encryption algorithm is more secure and efficient than the previous algorithms and can resist various attacks. [ABSTRACT FROM AUTHOR]

First, this study proposes the use of the newly developed Stochastic Fractal Search (SFS) algorithm for training MLP NNs to design the evolutionary classifier. Evolutionary classifiers, often experience problems of slow convergence speed, trapping in local minima, and non-real-time classification. This paper also use four chaotic maps to improve the performance of the SFS. This modified version of SFS has been called Chaotic Fractal Walk Trainer (CFWT). To assess the performance of the proposed classifiers, these networks will be evaluated using the two benchmark datasets and a high-dimensional practical sonar dataset. For endorsement, the results are compared to four popular meta-heuristics trainers. The results show that new classifiers suggest better performance than the other benchmark algorithms, in terms of entrapment in local minima, classification accuracy, and convergence speed. This paper also implements the designed classifier on the Filed Programmable Field Array (FPGA) substrate for testing the real-time processing ability of the proposed method. The results of the real application prove that the designed classifiers are applicable to high-dimension challenging problems with unknown search spaces. [ABSTRACT FROM AUTHOR]

Wind speed oscillations are known to exhibit varying characteristics at different time scales. Our recent analysis has shown that a collection of autoregressive models fitted separately on the frequency components of wind speed data can significantly increase the prediction accuracy. In this paper, we report the results of the investigation of dynamical behaviour across a broad frequency spectrum of wind speed measurements. The results show the existence of diverse characteristics such as stochastic, deterministic and chaotic behaviour apart from the variation of the dimensionality of underlying dynamics as well as the degree of fluctuations. It is also demonstrated that a cluster of deterministic models built upon separate frequency components of a wind speed time series can enhance the prediction accuracy by as much as 80%, on the average, consistently for predictions up to 12 h. The comparison shows the definite advantage of deterministic prediction models over autoregressive models. The f-index introduced in this paper to measure the fluctuations of wind speed over a period indicates that the observed seasonal variations of prediction errors can be correlated with changes in the f-index of the component series contributed mostly by the lower scales of decomposition. [ABSTRACT FROM AUTHOR]

In the commented paper, the authors consider a three-dimensional system and analyze the presence of Shilnikov chaos as well as a Hopf bifurcation. On the one hand, they state that the existence of a chaotic attractor is verified via the homoclinic Shilnikov theorem. The homoclinic orbit of this system is determined by using the undetermined coefficient method, introduced by Zhou et al. in [Chen's attractor exists, Int. J. Bifurcation Chaos 14 (2004) 3167–3178], a paper that presents very serious shortcomings. However, it has been cited dozens of times and its erroneous method has been copied in lots of papers, including the commented paper where an even expression for the first component of the homoclinic connection is used. It is evident that this even expression cannot represent the first component of a Shilnikov homoclinic connection, an orbit which is necessarily non-symmetric. Consequently, the results stated in Section 3, the core of the paper, are worthless. On the other hand, the study of the Hopf bifurcation presented in Section 4 is also wrong because the first Lyapunov coefficient provided by the authors is incorrect. [ABSTRACT FROM AUTHOR]