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We examine the relationship between mysticism and mathematical creativity through case studies from the history of mathematics., (Copyright © 2017 Elsevier Ltd. All rights reserved.)

Offers several LOGO programs to study the behavior of simple nonlinear systems. Suggests that LOGO is an excellent tool for studying chaotic systems. Offers suggestions for different forms of LOGO. Builds upon programs presented in a previous article. (MVL)

This research explores the complex dynamics of a Novel Four-Dimensional Fractional Supply Chain System (NFDFSCS) that integrates a quadratic interaction term involving the actual demand of customers and the inventory level of distributors. The introduction of the quadratic term results in significantly larger maximal Lyapunov exponents (MLE) compared to the original model, indicating increased system complexity. The existence, uniqueness, and Ulam–Hyers stability of the proposed system are verified. Additionally, we establish the global Mittag-Leffler attractive set (MLAS) and Mittag-Leffler positive invariant set (MLPIS) for the system. Numerical simulations and MATLAB phase portraits demonstrate the chaotic nature of the proposed system. Furthermore, a dynamical analysis achieves verification via the Lyapunov exponents, a bifurcation diagram, a 0–1 test, and a complexity analysis. A new numerical approximation method is proposed to solve non-linear fractional differential equations, utilizing fractional differentiation with a non-singular and non-local kernel. These numerical simulations illustrate the primary findings, showing that both external and internal factors can accelerate the process. Furthermore, a robust control scheme is designed to stabilize the system in finite time, effectively suppressing chaotic behaviors. The theoretical findings are supported by the numerical results, highlighting the effectiveness of the control strategy and its potential application in real-world supply chain management (SCM).

In this paper, we present a new class of extended oscillators in light of chaos theory. It is based on dynamical complex systems built on the concept of self-describing with a stopping criterion process. We offer an effective studying approach with a specific focus on learning, provoking students’ thinking through the triad of enigmatics–creativity–acmeology. Dynamic processes are the basis of mathematical modeling; thus, we can reach the goal of the above-mentioned triad by the proposed differential systems. The results we derive strongly confirm the presence of symmetry in the outcomes of the proposed models. We suggest a stochastic approach to structuring the proposed dynamical systems by modeling the coefficients that drive them by some discrete probability distribution that exhibits symmetry or asymmetry. We propose specific tools for researching the behavior of these systems.

Smartphones, digital cameras, and other imaging devices generate vast amounts of high-resolution colored images daily, stored on devices equipped with multi-core central processing units or on the cloud. Safeguarding these images from potential attackers has become a pressing concern. This paper introduces a set of six innovative image ciphers designed to be stronger, faster, and more efficient. Three of these algorithms incorporate the State-Based Tent Map (SBTM) Chaotic Pseudo Random Number Generator (CPRNG), while the remaining three employ a proposed modified variant, SBTMPi. The Grayscale Image Cipher (GIC), Colored Image Cipher Single-Thread RGB (CIC1), and Colored Image Cipher Three-Thread RGB (CIC3) showcase the application of the proposed algorithms. By incorporating novel techniques in the confusion and diffusion phases, these ciphers demonstrate remarkable performance, particularly with large colored images. The study underscores the potential of SBTM-based image ciphers, contributing to the advancement of secure image encryption techniques with robust random number generation capabilities.

György Ligeti was very interested in many artistic and scientific fields and drew inspiration for his compositional work from them (his engagement with mathematics - particularly fractal geometry and chaos theory - is perhaps the best known). In this chapter I compare the concept of artistic research with Ligeti's practice and oeuvre. While the notion of artistic research was only appearing in embryonic form during the latter stages of Ligeti's career, many - though not all - of his statements seem to be suitable for describing his artistic processes. The benefit of this investigation is expected to be twofold: applying the concept of artistic research to Ligeti's approaches and practices should yield new insights on the relationship between his work and his interest in other humanities and sciences. Yet this look at Ligeti may also help to refine the concept of artistic research as discussed and applied to the artistic output of today. [ABSTRACT FROM AUTHOR]

This paper introduces a new 3D chaotic attractor in a gene regulatory network. The proposed model has eighteen parameters. Formulas for characteristic numbers of critical points for three-dimensional systems were considered. We show that the three equilibrium points of the new chaotic 3D system are unstable and deduce that the three-dimensional system exhibits chaotic behavior. The possible outcomes of this 3D model were compared with the results of the Chua circuit. The bifurcation structures of the proposed 3D system are investigated numerically, showing periodic solutions and chaotic solutions. Lyapunov exponents and Kaplan-Yorke dimension are calculated. For calculations, the Wolfram Mathematica is used.

Starting from the contribution of such thinkers as the famous Giordano Bruno (1583) and the great mathematician and physicist Henri Poincaré (1889) and the surprising discovery of the meteorologist Edward Lorenz (1963), we consider the expansion of the mathematics of chaos in this article, paying attention to topology, qualitative geometry, and Catastrophe Theory, on the one hand, and addressing the possibilities derived from the new Computer Science as Quantum Algorithms and the advances in Artificial Intelligence, on the other. We especially highlight the section on computing chaos, which we consider to be new calculation and analysis instruments, such as machine learning and its algorithm called reservoir computing, through which we can know the dynamics of a chaotic system. With past data, with equations like Karamoto–Sivashinsky, one can improve predictions of the system eight times further ahead than in previous methods. Integrating the machine learning approach and traditional model-based prediction, one could obtain accurate predictions twelve Lyapunov times. As we know, in the framework of chaos theory, it is habitually accepted that the idea of long-term prediction seems impossible because we live under a veil of uncertainty. But with technological advances, the landscape begins to change, both in chaos theory and in its applications, especially in the field of economics, to which we devote particular attention, carrying out as an example the analysis of the evolution of the Madrid Stock Exchange in the 2006–2013 crisis. Above all this, a reflection of a general nature is necessary to enlighten us on the possibility of opening a new horizon.

Generally, in most applications of engineering, the parameters of the mathematical models are considered deterministic. Although, in practice, there are always some uncertainties in the model parameters; these uncertainties may be made wrong representation of the mathematical model of the system. These uncertainties can be generated from different reasons like measurement error, inhomogeneity of the process, chaotic behavior of systems, etc. This problem leads researchers to study these uncertainties and propose solutions for this problem. The iterative analysis is a method that can be utilized to solve these kinds of problems. In this paper, a new combined method based on interval chaotic and iterative decomposition method is proposed. The validation of the proposed method is performed on a chaotic Rossler system in stable Intervals. The simulation results are applied on 2 practical case studies and the results are compared with the interval Chebyshev method and Runge–Kutta method of order four (RK4) method. The final results showed that the proposed method has a good performance in finding the confidence interval for the Rossler models with interval uncertainties; the results also showed that the proposed method can handle the wrapping effect in a better manner to sharpen the range of non-monotonic interval. [ABSTRACT FROM AUTHOR]

Image segmentation is one of the pivotal steps in image processing due to its enormous application potential in medical image analysis, data mining, and pattern recognition. In fact, image segmentation is the process of splitting an image into multiple parts in order to provide detailed information on different aspects of the image. Traditional image segmentation techniques suffer from local minima and premature convergence issues when exploring complex search spaces. Additionally, these techniques also take considerable runtime to find the optimal pixels as the threshold levels are increased. Therefore, in order to overcome the computational overhead and convergence problems of the multilevel thresholding process, a robust optimizer, namely the Levy flight and Chaos theory-based Gravitational Search Algorithm (LCGSA), is employed to perform the segmentation of the COVID-19 chest CT scan images. In LCGSA, exploration is carried out by Levy flight, while chaotic maps guarantee the exploitation of the search space. Meanwhile, Kapur’s entropy method is utilized for segmenting the image into various regions based on the pixel intensity values. To investigate the segmentation performance of ten chaotic versions of LCGSA, firstly, several benchmark images from the USC-SIPI database are considered for the numerical analysis. Secondly, the applicability of LCGSA for solving real-world image processing problems is examined by using various COVID-19 chest CT scan imaging datasets from the Kaggle database. Further, an ablation study is carried out on different chest CT scan images by considering ground truth images. Moreover, various qualitative and quantitative metrics are used for the performance evaluation. The overall analysis of the experimental results indicated the efficient performance of LCGSA over other peer algorithms in terms of taking less computational time and providing optimal values for image quality metrics.

To each colored graph one can associate its closure in the universal space of isomorphism classes of pointed colored graphs, and this subspace can be regarded as a generalized subshift. Based on this correspondence, we introduce two definitions for chaotic (colored) graphs, one of them analogous to Devaney’s. We show the equivalence of our two novel definitions of chaos, proving their topological genericity in various subsets of the universal space.

Substitution boxes are the key factor in symmetric-key cryptosystems that determines their ability to resist various cryptanalytic attacks. Creating strong substitution boxes that have multiple strong cryptographic properties at the same time is a challenging task for cryptographers. A significant amount of research has been conducted on S-boxes in the past few decades, but the resulting S-boxes have been found to be vulnerable to various cyberattacks. This paper proposes a new method for creating robust S-boxes that exhibit superior performance and possess high scores in multiple cryptographic properties. The hybrid S-box method presented in this paper is based on Chua’s circuit chaotic map, two-dimensional cellular automata, and an algebraic permutation group structure. The proposed 16×16 S-box has an excellent performance in terms of security parameters, including a minimum nonlinearity of 102, the absence of fixed points, the satisfaction of bit independence and strict avalanche criteria, a low differential uniformity of 5, a low linear approximation probability of 0.0603, and an auto-correlation function of 28. The analysis of the performance comparison indicates that the proposed S-box outperforms other state-of-the-art S-box techniques in several aspects. It possesses better attributes, such as a higher degree of inherent security and resilience, which make it more secure and less vulnerable to potential attacks.

In this work, a color image encryption and decryption algorithm for digital images is presented. It is based on the modular discrete derivative (MDD), a novel technique to encrypt images and efficiently hide visual information. In addition, Langton’s ant, which is a two-dimensional universal Turing machine with a high key space, is used. Moreover, a deterministic noise technique that adds security to the MDD is utilized. The proposed hybrid scheme exploits the advantages of MDD and Langton’s ant, generating a very secure and reliable encryption algorithm. In this proposal, if the key is known, the original image is recovered without loss. The method has demonstrated high performance through various tests, including statistical analysis (histograms and correlation distributions), entropy, texture analysis, encryption quality, key space assessment, key sensitivity analysis, and robustness to differential attack. The proposed method highlights obtaining chi-square values between 233.951 and 281.687, entropy values between 7.9999225223 and 7.9999355791, PSNR values (in the original and encrypted images) between 8.134 and 9.957, the number of pixel change rate (NPCR) values between 99.60851796% and 99.61054611%, unified average changing intensity (UACI) values between 33.44672377% and 33.47430379%, and a vast range of possible keys >5.8459×1072. On the other hand, an analysis of the sensitivity of the key shows that slight changes to the key do not generate any additional information to decrypt the image. In addition, the proposed method shows a competitive performance against recent works found in the literature.

After an investigation of the well-known basic properties of muons conducted by the standard model (SM), this paper presents the results obtained for the phase space reconstruction, for the correlation dimension and for the largest Lyapunov exponent of a muon time series detected for a period of about three years (2019–2021) in an Italian laboratory at the sea level. These results confirm that the dynamics of such a time series is chaotic in nature, and therefore open new perspectives in the study of cosmic rays. In the following studies, we will explore if such muon time series have a mono- or a multifractal regime with a complete analysis of all the parameters that usually involve such studies.

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.

In this study, Multiple-Chaotic maps were merged by using multiplicative-additive form to generate the chaotic sequences which are used to track the addresses of shuffled bits in steganography. Three techniques are introduced for image steganography in the spatial domain. The first system is based on the well-known LSB technique, the second system is based on looking for the identical bits between the secret message and the cover image and the third system is based on the concept of LSB substitution, it is employed the mapping of secret data bits with cover pixel bits. It was tested and evaluated security levels for the proposed techniques by using the Peak Signal-to-Noise Ratio (PSNR), Mean Square Error (MSE), histogram analysis and correlative analysis and tested the Chaotic sequences generated by using correlation, Lypaunov exponents, Poincaré section and 0-1Test. The results show that the proposed methods perform better than existed systems.

The surrounding reality can be analyzed due to the interaction of complex nonlinear dynamic systems. The article’s main objective is to develop and analyze the models that best describe the efficient behavior of the Romanian capital market that generated the analyzed time series. The empirical analysis carried out in this paper does not aim to classify the Romanian market capital as efficient or ineffective but rather to identify the degree of deviation from efficiency relative to other markets, namely, an analysis of the dynamics of the degree of deviation over time. To describe the distribution of returns, we focused on the family of generalized hyperbolic distributions, which have statistical properties similar to financial returns. The presence of wide tails in the distributions (of extreme values) suggests using statistical tests and measures to detect dependencies, which take this behavior into account. Statistical methods and efficiency indicators are used, such as the Hurst exponent, Taken’s theorem, and the fractal dimension, which facilitate the detection of the main types of dependencies that could be present in the return series; measures that are robust to the heteroscedastic behavior of the returns. These statistical measures are applied to the entire period and sliding windows.

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]

This paper tackles to some linear feedback control strategies, where we take a 3D chaotic system with a five critical point of unstability, which is discovered by scientist [Zhu Congxu, 2010]. So we applied some linear feedback strategies: first strategy Ordinary Feedback Control and the second strategy Dislocate Feedback Control on this system at origin point and we noticed that a necessary condition for suppression is getting positive feedback coefficient; but this condition fails at some strategies. For this reason, we focused on these cases in our search, and design more than a strategy for studying these different situations. Theoretical analysis and numerical simulation check the validity of the results obtained.

In this work, we attempted to find a non-linear dependency in the time series of strawberry production in Huelva (Spain) using a procedure based on metric tests measuring chaos. This study aims to develop a novel method for yield prediction. To do this, we study the system’s sensitivity to initial conditions (exponential growth of the errors) using the maximal Lyapunov exponent. To check the soundness of its computation on non-stationary and not excessively long time series, we employed the method of over-embedding, apart from repeating the computation with parts of the transformed time series. We determine the existence of deterministic chaos, and we conclude that non-linear techniques from chaos theory are better suited to describe the data than linear techniques such as the ARIMA (autoregressive integrated moving average) or SARIMA (seasonal autoregressive moving average) models. We proceed to predict short-term strawberry production using Lorenz’s Analog Method.

The Raychaudhuri equation is derived by assuming geometric flow in space–time M of n+1 dimensions. The equation turns into a harmonic oscillator form under suitable transformations. Thereby, a relation between geometrical entropy and mean geodesic deviation is established. This has a connection to chaos theory where the trajectories diverge exponentially. We discuss its application to cosmology and black holes. Thus, we establish a connection between chaos theory and general relativity.

Research investigating the dynamics of coupled physical systems has demonstrated that small feedback delays can allow a dynamic response system to anticipate chaotic behavior. This counterintuitive phenomenon, termed anticipatory synchronization, has been observed in coupled electrical circuits, laser semi-conductors, and artificial neurons. Recent research indicates that the same process might also support the ability of humans to anticipate the occurrence of chaotic behavior in other individuals. Motivated by this latter work, the current study examined whether the process of feedback delay induced anticipatory synchronization could be employed to develop an interactive artificial agent capable of anticipating chaotic human movement. Results revealed that incorporating such delays within the movement-control dynamics of an artificial agent not only enhances an artificial agent’s ability to anticipate chaotic human behavior, but to synchronize with such behavior in a manner similar to natural human-human anticipatory synchronization. The implication of these findings for the development of human-machine interaction systems is discussed. [ABSTRACT FROM AUTHOR]

Code modulated Visual Evoked Potentials (c-VEP) based BCI studies usually employ m-sequences as a modulating codes for their broadband spectrum and correlation property. However, subjective fatigue of the presented codes has been a problem. In this study, we introduce chaotic codes containing broadband spectrum and similar correlation property. We examined whether the introduced chaotic codes could be decoded from EEG signals and also compared the subjective fatigue level with m-sequence codes in normal subjects. We generated chaotic code from one-dimensional logistic map and used it with conventional 31-bit m-sequence code. In a c-VEP based study in normal subjects (n = 44, 21 females) we presented these codes visually and recorded EEG signals from the corresponding codes for their four lagged versions. Canonical correlation analysis (CCA) and spatiotemporal beamforming (STB) methods were used for target identification and comparison of responses. Additionally, we compared the subjective self-declared fatigue using VAS caused by presented m-sequence and chaotic codes. The introduced chaotic code was decoded from EEG responses with CCA and STB methods. The maximum total accuracy values of 93.6 ± 11.9% and 94 ± 14.4% were achieved with STB method for chaotic and m-sequence codes for all subjects respectively. The achieved accuracies in all subjects were not significantly different in m-sequence and chaotic codes. There was significant reduction in subjective fatigue caused by chaotic codes compared to the m-sequence codes. Both m-sequence and chaotic codes were similar in their accuracies as evaluated by CCA and STB methods. The chaotic codes significantly reduced subjective fatigue compared to the m-sequence codes. [ABSTRACT FROM AUTHOR]

Highlights • Permutation instead of raw vector is adopted to simplify time irreversibility analysis. • Considering the forbidden permutation, a subtraction-based parameter, Y S , is proposed. • Chaotic and reversible processes and their surrogate data validate Y S. • Healthy and diseased brain activities manifest time irreversibility. • Epilepsy leads loss of time irreversibility of EEG during seizure-free intervals. Abstract To simplify the quantification of time irreversibility, we employ order patterns instead of the raw multi-dimension vectors in time series, and considering the existence of forbidden permutation, we propose a subtraction-based parameter, Y S , to measure the probabilistic differences between symmetric permutations for time irreversibility. Two chaotic models, the logistic and Henon systems, and reversible Gaussian process and their surrogate data are used to validate the time-irreversible measure, and time irreversibility of epileptic EEGs from Nanjing General Hospital is detected by the parameter. Test results prove that it is promising to quantify time irreversibility by measuring the subtraction-based probabilistic differences between symmetric order patterns, and our findings highlight the manifestation of nonlinearity of whether healthy or diseased EEGs and suggest that the epilepsy leads to a decline in the nonlinearity of brain electrical activities during seize-free intervals. [ABSTRACT FROM AUTHOR]

We consider billiards with a single cusp where the walls meeting at the vertex of the cusp have zero one-sided curvature, thus forming a flat point at the vertex. For Hölder continuous observables, we show that properly normalized Birkhoff sums, with respect to the billiard map, converge in law to a totally skewed α-stable law. [ABSTRACT FROM AUTHOR]

Recently, Lu et al. claimed that Xie et al.’s three-party password-authenticated key agreement protocol (3PAKA) using chaotic maps has three security vulnerabilities; in particular, it cannot resist offline password guessing attack, Bergamo et al.’s attack and impersonation attack, and then they proposed an improved protocol. However, we demonstrate that Lu et al.’s attacks on Xie et al.’s scheme are unworkable, and their improved protocol is insecure against stolen-verifier attack and off-line password guessing attack. Furthermore, we propose a novel scheme with enhanced security and efficiency. We use formal verification tool ProVerif, which is based on pi calculus, to prove security and authentication of our scheme. The efficiency of the proposed scheme is higher than other related schemes. [ABSTRACT FROM AUTHOR]

This paper investigates the chaotic behavior of a modified jerk circuit with Chua’s diode. The Chua’s diode considered here is a nonlinear resistor having a symmetric piecewise linear voltage-current characteristic. To describe the system, we apply fundamental laws in electrical circuit theory to formulate a mathematical model in terms of a third-order (jerk) nonlinear differential equation, or equivalently, a system of three first-order differential equations. The analysis shows that this system has three collinear equilibrium points. The time waveform and the trajectories about each equilibrium point depend on its associated eigenvalues. We prove that all three equilibrium points are of type saddle focus, meaning that the trajectory of (x(t),y(t)) diverges in a spiral form but z(t) converges to the equilibrium point for any initial point (x(0),y(0),z(0)). Numerical simulation illustrates that the oscillations are dense, have no period, are highly sensitive to initial conditions, and have a chaotic hidden attractor.

Bi-level programming problem (BLPP) is an optimization problem consists of two interconnected hierarchical optimization problems. Solving BLPP is one of the hardest tasks facing the optimization community. This paper proposes a modified genetic algorithm and a chaotic search to solve BLPP. Firstly, the proposed algorithm solves the upper-level problem using a modified genetic algorithm. The genetic algorithm has modified with a new selection technique. The new selection technique helps the upper-level decision-maker to take an appropriate decision in anticipation of a lower level’s reaction. It distinguishes the proposed algorithm with a very small number of solving the lower-level problem, enhances the algorithm performance and fasts convergence to the solution. Secondly, a local search based on chaos theory has applied around the modified genetic algorithm solution. Chaotic local search enables the algorithm to escape from local solutions and increase convergence to the global solution. The proposed algorithm has evaluated on forty different test problems to show the proposed algorithm effectiveness. The results have analyzed to illustrate the new selection technique effect and the chaotic search effect on the algorithm performance. A comparison between the proposed algorithm results and other state-of-the-art algorithms results has introduced to show the proposed algorithm superiority.

In the field of complex networks, how to identify influential nodes is a significant issue in analyzing the structure of a network. In the existing method proposed to identify influential nodes based on the local dimension, the global structure information in complex networks is not taken into consideration. In this paper, a node information dimension is proposed by synthesizing the local dimensions at different topological distance scales. A case study of the Netscience network is used to illustrate the efficiency and practicability of the proposed method. [ABSTRACT FROM AUTHOR]

The aim was to assess the effect of fatigue on linear and nonlinear parameters in swimming. Twenty-four fitness-oriented swimmers performed a maximal bout of 100 m at front-crawl to elicit fatigue. Before (pre-) and immediately after (post-test) the bout, participants swam an all-out 25 m to derive the speed fluctuation (dv), approximate entropy (ApEn) and fractal dimension (FD) from the speed-time series collected by a speedo-meter. Swim speed was 10.85% slower in the post-test than in the pre-test (p < .001, η2 = 0.72). There was an effect of the fatigue on the dv with a moderate effect size. The dv increased shifting the 95CI band from 0.116-0.134 to 0.140-0.161. The ApEn showed non-significant variations between the pre- and post-test having the 95CI of pre- and post-test overlapped (pre: 0.659-0.700; post: 0.641-0.682). The FD showed as well a significant variation (the 95CI moved from 1.954-1.965 to 1.933-1.951). It can be concluded that in swimming there are changes in classical and nonlinear parameters under fatigue. [ABSTRACT FROM AUTHOR]

In this paper, a new four-variable dynamical system is proposed to set chaotic circuit composed of memristor and Josephson junction, and the dependence of chaotic behaviors on nonlinearity is investigated. A magnetic flux-controlled memristor is used to couple with the RCL-shunted junction circuit, and the dynamical behaviors can be modulated by changing the coupling intensity between the memristor and the RCL-shunted junction. Bifurcation diagram and Lyapunov exponent are calculated to confirm the emergence of chaos in the improved dynamical system. The outputs and dynamical behaviors can be controlled by the initial setting and external stimulus as well. As a result, chaos can be suppressed and spiking occurs in the sampled outputs under negative feedback, while applying positive feedback type via memristor can be effective to trigger chaos. Furthermore, it is found that the number of multi-attractors in the Jerk circuit can be modulated when memristor coupling is applied on the circuit. These results indicate that memristor coupling can be effective to control chaotic circuits and it is also useful to reproduce dynamical behaviors for neuronal activities. [ABSTRACT FROM AUTHOR]

Stroke survivors are more prone to developing cognitive impairment and dementia. Dementia detection is a challenge for supporting personalized healthcare. This study analyzes the electroencephalogram (EEG) background activity of 5 vascular dementia (VaD) patients, 15 stroke-related patients with mild cognitive impairment (MCI), and 15 control healthy subjects during a working memory (WM) task. The objective of this study is twofold. First, it aims to enhance the discrimination of VaD, stroke-related MCI patients, and control subjects using fuzzy neighborhood preserving analysis with QR-decomposition (FNPAQR); second, it aims to extract and investigate the spectral features that characterize the post-stroke dementia patients compared to the control subjects. Nineteen channels were recorded and analyzed using the independent component analysis and wavelet analysis (ICA-WT) denoising technique. Using ANOVA, linear spectral power including relative powers (RP) and power ratio were calculated to test whether the EEG dominant frequencies were slowed down in VaD and stroke-related MCI patients. Non-linear features including permutation entropy (PerEn) and fractal dimension (FD) were used to test the degree of irregularity and complexity, which was significantly lower in patients with VaD and stroke-related MCI than that in control subjects (ANOVA; p ˂ 0.05). This study is the first to use fuzzy neighborhood preserving analysis with QR-decomposition (FNPAQR) dimensionality reduction technique with EEG background activity of dementia patients. The impairment of post-stroke patients was detected using support vector machine (SVM) and k-nearest neighbors (kNN) classifiers. A comparative study has been performed to check the effectiveness of using FNPAQR dimensionality reduction technique with the SVM and kNN classifiers. FNPAQR with SVM and kNN obtained 91.48 and 89.63% accuracy, respectively, whereas without using the FNPAQR exhibited 70 and 67.78% accuracy for SVM and kNN, respectively, in classifying VaD, stroke-related MCI, and control patients, respectively. Therefore, EEG could be a reliable index for inspecting concise markers that are sensitive to VaD and stroke-related MCI patients compared to control healthy subjects. [ABSTRACT FROM AUTHOR]

In this paper, we study a network of Izhikevich neurons to explore what it means for a brain to be at the edge of chaos. To do so, we first constructed the phase diagram of a single Izhikevich excitatory neuron, and identified a small region of the parameter space where we find a large number of phase boundaries to serve as our edge of chaos. We then couple the outputs of these neurons directly to the parameters of other neurons, so that the neuron dynamics can drive transitions from one phase to another on an artificial energy landscape. Finally, we measure the statistical complexity of the parameter time series, while the network is tuned from a regular network to a random network using the Watts-Strogatz rewiring algorithm. We find that the statistical complexity of the parameter dynamics is maximized when the neuron network is most small-world-like. Our results suggest that the small-world architecture of neuron connections in brains is not accidental, but may be related to the information processing that they do. [ABSTRACT FROM AUTHOR]

Small-angle scattering (SAS) technique is applied to study the nano and microstructural properties of spatial patterns generated from chaos game representation (CGR). Using a simplified version of Debye formula, we calculate and analyze in momentum space, the monodisperse scattering structure factor from a system of randomly oriented and non-interacting 2D Sierpinski gaskets (SG). We show that within CGR approach, the main geometrical and fractal properties, such as the overall size, scaling factor, minimal distance between scattering units, fractal dimension and the number of units composing the SG, can be recovered. We confirm the numerical results, by developing a theoretical model which describes analytically the structure factor of SG. We apply our findings to scattering from single scale mass fractals, and respectively to a multiscale fractal representing DNA sequences, and for which an analytic description of the structure factor is not known a priori. [ABSTRACT FROM AUTHOR]

Scientific analysis often relies on the ability to make accurate predictions of a system’s dynamics. Mechanistic models, parameterized by a number of unknown parameters, are often used for this purpose. Accurate estimation of the model state and parameters prior to prediction is necessary, but may be complicated by issues such as noisy data and uncertainty in parameters and initial conditions. At the other end of the spectrum exist nonparametric methods, which rely solely on data to build their predictions. While these nonparametric methods do not require a model of the system, their performance is strongly influenced by the amount and noisiness of the data. In this article, we consider a hybrid approach to modeling and prediction which merges recent advancements in nonparametric analysis with standard parametric methods. The general idea is to replace a subset of a mechanistic model’s equations with their corresponding nonparametric representations, resulting in a hybrid modeling and prediction scheme. Overall, we find that this hybrid approach allows for more robust parameter estimation and improved short-term prediction in situations where there is a large uncertainty in model parameters. We demonstrate these advantages in the classical Lorenz-63 chaotic system and in networks of Hindmarsh-Rose neurons before application to experimentally collected structured population data. [ABSTRACT FROM AUTHOR]

Predicting extrema of chaotic systems in high-dimensional phase space remains a challenge. Methods, which give extrema that are valid in the long term, have thus far been restricted to models of only a few variables. Here, a method is presented which treats extrema of chaotic systems as belonging to discretised manifolds of low dimension (low-D) embedded in high-dimensional (high-D) phase space. As a central feature, the method exploits that strange attractor dimension is generally much smaller than parent system phase space dimension. This is important, since the computational cost associated with discretised manifolds depends exponentially on their dimension. Thus, systems that would otherwise be associated with tremendous computational challenges, can be tackled on a laptop. As a test, bounding manifolds are calculated for high-D modifications of the canonical Duffing system. Parameters can be set such that the bounding manifold displays harmonic behaviour even if the underlying system is chaotic. Thus, solving for one post-transient forcing cycle of the bounding manifold predicts the extrema of the underlying chaotic problem indefinitely. [ABSTRACT FROM AUTHOR]

It has been proposed that neural noise in the cortex arises from chaotic dynamics in the balanced state: in this model of cortical dynamics, the excitatory and inhibitory inputs to each neuron approximately cancel, and activity is driven by fluctuations of the synaptic inputs around their mean. It remains unclear whether neural networks in the balanced state can perform tasks that are highly sensitive to noise, such as storage of continuous parameters in working memory, while also accounting for the irregular behavior of single neurons. Here we show that continuous parameter working memory can be maintained in the balanced state, in a neural circuit with a simple network architecture. We show analytically that in the limit of an infinite network, the dynamics generated by this architecture are characterized by a continuous set of steady balanced states, allowing for the indefinite storage of a continuous parameter. In finite networks, we show that the chaotic noise drives diffusive motion along the approximate attractor, which gradually degrades the stored memory. We analyze the dynamics and show that the slow diffusive motion induces slowly decaying temporal cross correlations in the activity, which differ substantially from those previously described in the balanced state. We calculate the diffusivity, and show that it is inversely proportional to the system size. For large enough (but realistic) neural population sizes, and with suitable tuning of the network connections, the proposed balanced network can sustain continuous parameter values in memory over time scales larger by several orders of magnitude than the single neuron time scale. [ABSTRACT FROM AUTHOR]

The classical spaces ℓp+, 1 ≤ p < ∞, and Lp−, 1

Perpetual points represent a new interesting topic in the literature of nonlinear dynamics. This paper introduces some chaotic flows with four different structural features from the viewpoint of fixed points and perpetual points. [ABSTRACT FROM AUTHOR]

Data compression and encryption are key components of commonly deployed platforms such as Hadoop. Numerous data compression and encryption tools are presently available on such platforms and the tools are characteristically applied in sequence, i.e., compression followed by encryption or encryption followed by compression. This paper focuses on the open-source Hadoop framework and proposes a data storage method that efficiently couples data compression with encryption. A simultaneous compression and encryption scheme is introduced that addresses an important implementation issue of source coding based on Tent Map and Piece-wise Linear Chaotic Map (PWLM), which is the infinite precision of real numbers that result from their long products. The approach proposed here solves the implementation issue by removing fractional components that are generated by the long products of real numbers. Moreover, it incorporates a stealth key that performs a cyclic shift in PWLM without compromising compression capabilities. In addition, the proposed approach implements a masking pseudorandom keystream that enhances encryption quality. The proposed algorithm demonstrated a congruent fit within the Hadoop framework, providing robust encryption security and compression. [ABSTRACT FROM AUTHOR]

Highly connected recurrent neural networks often produce chaotic dynamics, meaning their precise activity is sensitive to small perturbations. What are the consequences of chaos for how such networks encode streams of temporal stimuli? On the one hand, chaos is a strong source of randomness, suggesting that small changes in stimuli will be obscured by intrinsically generated variability. On the other hand, recent work shows that the type of chaos that occurs in spiking networks can have a surprisingly low-dimensional structure, suggesting that there may be room for fine stimulus features to be precisely resolved. Here we show that strongly chaotic networks produce patterned spikes that reliably encode time-dependent stimuli: using a decoder sensitive to spike times on timescales of 10’s of ms, one can easily distinguish responses to very similar inputs. Moreover, recurrence serves to distribute signals throughout chaotic networks so that small groups of cells can encode substantial information about signals arriving elsewhere. A conclusion is that the presence of strong chaos in recurrent networks need not exclude precise encoding of temporal stimuli via spike patterns. [ABSTRACT FROM AUTHOR]

Complex nonlinear phenomena are investigated in a basic power system model of the single-machine-infinite-bus (SMIB) with a synchronous generator modeled by a classical third-order differential equation including both angle dynamics and voltage dynamics, the so-called flux decay equation. In contrast, for the second-order differential equation considering the angle dynamics only, it is the classical swing equation. Similarities and differences of the dynamics generated between the third-order model and the second-order one are studied. We mainly find that, for positive damping, these two models show quite similar behavior, namely, stable fixed point, stable limit cycle, and their coexistence for different parameters. However, for negative damping, the second-order system can only collapse, whereas for the third-order model, more complicated behavior may happen, such as stable fixed point, limit cycle, quasi-periodicity, and chaos. Interesting partial collapse phenomena for angle instability only and not for voltage instability are also found here, including collapse from quasi-periodicity and from chaos etc. These findings not only provide a basic physical picture for power system dynamics in the third-order model incorporating voltage dynamics, but also enable us a deeper understanding of the complex dynamical behavior and even leading to a design of oscillation damping in electric power systems. [ABSTRACT FROM AUTHOR]

Algorithms using 4-pixel Feistel structure and chaotic systems have been shown to resolve security problems caused by large data capacity and high correlation among pixels for color image encryption. In this paper, a fast color image encryption algorithm based on the modified 4-pixel Feistel structure and multiple chaotic maps is proposed to improve the efficiency of this type of algorithm. Two methods are used. First, a simple round function based on a piecewise linear function and tent map are used to reduce computational cost during each iteration. Second, the 4-pixel Feistel structure reduces round number by changing twist direction securely to help the algorithm proceed efficiently. While a large number of simulation experiments prove its security performance, additional special analysis and a corresponding speed simulation show that these two methods increase the speed of the proposed algorithm (0.15s for a 256*256 color image) to twice that of an algorithm with a similar structure (0.37s for the same size image). Additionally, the method is also faster than other recently proposed algorithms. [ABSTRACT FROM AUTHOR]

In this paper, control of uncertain fractional-order financial chaotic system with input saturation and external disturbance is investigated. The unknown part of the input saturation as well as the system’s unknown nonlinear function is approximated by a fuzzy logic system. To handle the fuzzy approximation error and the estimation error of the unknown upper bound of the external disturbance, fractional-order adaptation laws are constructed. Based on fractional Lyapunov stability theorem, an adaptive fuzzy controller is designed, and the asymptotical stability can be guaranteed. Finally, simulation studies are given to indicate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

The main advantage of detecting chaos is that the time series is short term predictable. The prediction accuracy decreases in time. A strong evidence of chaotic dynamics is the existence of a positive Lyapunov exponent (i.e. sensitivity to initial conditions). In chaotic time series prediction theory the methods used can be placed in two classes: global and local methods. Neural networks are global methods of prediction. The paper tries to find a relation between the two parameters used in reconstruction of the state space (embedding dimension m and delay time τ) and the number of input neurons of a multilayer perceptron (MLP). For two of three time series studied, the minimum absolute error value is minimum for a MLP with the number of inputs equal to m*τ.