Chaotic Behavior Of System Research Articles

Generalized Gaussian Distribution Improved Permutation Entropy: A New Measure for Complex Time Series Analysis.

To enhance the performance of entropy algorithms in analyzing complex time series, generalized Gaussian distribution improved permutation entropy (GGDIPE) and its multiscale variant (MGGDIPE) are proposed in this paper. First, the generalized Gaussian distribution cumulative distribution function is employed for data normalization to enhance the algorithm's applicability across time series with diverse distributions. The algorithm further processes the normalized data using improved permutation entropy, which maintains both the absolute magnitude and temporal correlations of the signals, overcoming the equal value issue found in traditional permutation entropy (PE). Simulation results indicate that GGDIPE is less sensitive to parameter variations, exhibits strong noise resistance, accurately reveals the dynamic behavior of chaotic systems, and operates significantly faster than PE. Real-world data analysis shows that MGGDIPE provides markedly better separability for RR interval signals, EEG signals, bearing fault signals, and underwater acoustic signals compared to multiscale PE (MPE) and multiscale dispersion entropy (MDE). Notably, in underwater target recognition tasks, MGGDIPE achieves a classification accuracy of 97.5% across four types of acoustic signals, substantially surpassing the performance of MDE (70.5%) and MPE (62.5%). Thus, the proposed method demonstrates exceptional capability in processing complex time series.

Out-of-time-ordered-correlators for the pure inverted quartic oscillator: classical chaos meets quantum stability

Out-of-time-ordered-correlators (OTOCs) have been suggested as a means to diagnose chaotic behavior in quantum mechanical systems. Recently, it was found that OTOCs display exponential growth for the inverted quantum harmonic oscillator, mirroring the fact that this system is classically and quantum mechanically unstable. In this work, I study OTOCs for the inverted anharmonic (pure quartic) oscillator in quantum mechanics, finding only oscillatory behavior despite the classically unstable nature of the system. For higher temperature, OTOCs seem to exhibit saturation consistent with a value of –2⟨x2⟩T ⟨p2⟩T at late times. I provide analytic evidence from the spectral zeta-function and the WKB method as well as direct numerical solutions of the Schrödinger equation that the inverted quartic oscillator possesses a real and positive energy eigenspectrum, and normalizable wave-functions.

Hamilton energy, competitive modes and ultimate bound estimation of a new 3D chaotic system, and its application in chaos synchronization

Abstract This paper discusses the dynamical behavior of a new 3D chaotic system of integer and fractional order. To get a comprehensive knowledge of the dynamics of the proposed system, we have studied competitive modes and Hamilton energy for different parameter values. In order to get the ultimate bound set for the proposed system, we employed the Lagrange coefficient approach to solve the optimization problem. We have also explored the use of the bound set in synchronization. Furthermore, we have examined the Hamilton energy, time series, bifurcation diagrams, and Lyapunov exponents for the fractional version of the proposed chaotic system. Finally, we looked at the Mittage-Leffler positive invariant sets and global attractive sets by merging the Lyapunov function approach with the Mittage-Leffler function. Numerical simulations have shown the obtained bound sets and other analytical outcomes.

Effect of autaptic synapse on signal transmission performance of dressed Hodgkin–Huxley neuron

Astrocytes, which have complex relationships with neurons, are the most important biological arguments that support neurons in the performance of cognitive functions. Another important biological argument in nervous systems is autapse, a special type of synapse through which the neuron provides feedback to itself. It is well known that both astrocyte and autapse modulate neuron dynamics and they affect memory functions. Considering the idea that some cognitive brain functions occur through chaotic firings, more complex models in which structures such as astrocytes and autaptic synapses are included in the system are needed to better understand the neural system. In this study, the signal detection abilities of astrocyte-dressed Hodgkin–Huxley neurons which have electrical, excitatory, and inhibitory chemical autaptic synapses are examined separately under chaotic environmental conditions. The results show that astrocyte-supported neurons, especially those with electrical and excitatory-chemical autapse, exhibit a strong chaotic resonance for appropriate parameters. After the autaptic synapse, the astrocyte is a second modulator of the neuron’s signal detection success. Additionally, it is observed that autaptic time delay is more effective than autaptic conductance in modulating the signal transmission success of the neuron. For all autapse types, the parameter values at which the supportive effect of the astrocyte in the model is obtained are presented. Furthermore, multiple chaotic resonance is detected in neurons with electrical and excitatory chemical autapse when the autaptic delay is equal to integer multiples of the weak signal period. This is observed for neurons with inhibitory chemical autapse when half of the weak signal period is added to whole multiples of the weak signal period. We hope that our findings will shed light on the chaotic environment behavior of complex nervous systems interacting with astrocytes and autapse.

Chaotic Dynamic Behavior of a Fractional-Order Financial System with Constant Inelastic Demand

The establishment of a financial system should not only consider the current situation, but also need to refer to the past. Due to the memory of the fractional derivative, a fractional-order system can more effectively describe the historical significance of the financial system. Most scholars use the prediction–correction scheme to study fractional-order systems. This paper provides a higher-precision numerical method for the financial system, which more effectively simulate the system. Based on the definition of the Grünwald–Letnikov fractional derivative, the integer-order system with nonconstant demand elasticity is extended to the fractional-order setting, and its dynamic behavior is studied, with some novel chaotic attractors found. The research results are helpful for improving the understanding of the financial system and the financial market and for predicting financial risks.

Synchronization of angular velocities of chaotic leader-follower satellites using a novel integral terminal sliding mode controller

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.

Prediction of dynamical systems from time-delayed measurements with self-intersections

In the context of predicting the behaviour of chaotic systems, Schroer, Sauer, Ott and Yorke conjectured in 1998 that if a dynamical system defined by a smooth diffeomorphism T of a Riemannian manifold X admits an attractor with a natural measure μ of information dimension smaller than k, then k time-delayed measurements of a one-dimensional observable h are generically sufficient for μ-almost sure prediction of future measurements of h. In a previous paper we established this conjecture in the setup of injective Lipschitz transformations T of a compact set X in Euclidean space with an ergodic T-invariant Borel probability measure μ. In this paper we prove the conjecture for all (also non-invertible) Lipschitz systems on compact sets with an arbitrary Borel probability measure, and establish an upper bound for the decay rate of the measure of the set of points where the prediction is subpar. This partially confirms a second conjecture by Schroer, Sauer, Ott and Yorke related to empirical prediction algorithms as well as algorithms estimating the dimension and number of required delayed measurements (the so-called embedding dimension) of an observed system. We also prove general time-delay prediction theorems for locally Lipschitz or Hölder systems on Borel sets in Euclidean space.

Synchronization of non-smooth chaotic systems via an improved reservoir computing

The reservoir computing (RC) is increasingly used to learn the synchronization behavior of chaotic systems as well as the dynamical behavior of complex systems, but it is scarcely applied in studying synchronization of non-smooth chaotic systems likely due to its complexity leading to the unimpressive effect. Here proposes a simulated annealing-based differential evolution (SADE) algorithm for the optimal parameter selection in the reservoir, and constructs an improved RC model for synchronization, which can work well not only for non-smooth chaotic systems but for smooth ones. Extensive simulations show that the trained RC model with optimal parameters has far longer prediction time than those with empirical and random parameters. More importantly, the well-trained RC system can be well synchronized to its original chaotic system as well as its replicate RC system via one shared signal, whereas the traditional RC system with empirical or random parameters fails for some chaotic systems, particularly for some non-smooth chaotic systems.

Machine Learning Approaches for Predicting Chaotic Behavior in Nonlinear Systems

Machine Learning Approaches for Predicting Chaotic Behavior in Nonlinear Systems

Dynamic Analysis of Impulsive Differential Chaotic System and Its Application in Image Encryption

In this paper, we study the dynamic behavior of an impulse differential chaotic system which can be applied to image encryption. Combined with the chaotic characteristics of the high dimensional impulsive differential equations, the plaintext image can be encrypted by using the traditional Henon map and diffusion sequences encryption algorithm. The initial values and control parameters serve as keys for encryption algorithms, and the algorithm has a larger key space. The key is resistant to minor interference and the accuracy can reach 10−12. The simulation results show that the impulsive differential chaotic system has a good application prospect in image encryption.

Utilizing Fractional Artificial Neural Networks for Modeling Cancer Cell Behavior

In this paper, a novel approach involving a fractional recurrent neural network (RNN) is proposed to achieve the observer-based synchronization of a cancer cell model. According to the properties of recurrent neural networks, our proposed framework serves as a predictive method for the behavior of fractional-order chaotic cancer systems with uncertain orders. Through a stability analysis of weight updating laws, we design a fractional-order Nonlinear Autoregressive with Exogenous Inputs (NARX) network, in which its learning algorithm demonstrates admissible and faster convergence. The main contribution of this paper lies in the development of a fractional neural observer for the fractional-order cancer systems, which is robust in the presence of uncertain orders. The proposed fractional-order model for cancer can capture complex and nonlinear behaviors more accurately than traditional integer-order models. This improved accuracy can provide a more realistic representation of cancer dynamics. Simulation results are presented to demonstrate the effectiveness of the proposed method, where mean square errors of synchronization by applying integer and fractional weight matrix laws are calculated. The density of tumor cell, density of healthy host cell and density of effector immune cell errors for the observer-based synchronization of fractional-order (OSFO) cancer system are less than 0.0.0048, 0.0062 and 0.0068, respectively. Comparative tables are provided to validate the improved accuracy achieved by the proposed framework.

Chaotic Behavior of Lorenz-Based Chemical System under the Influence of Fractals

This research examines a chaotic chemical reaction system based on the variation of the Lorenz system. This study demonstrates that although the first phase portraits of the chemical models under consideration and the Lorenz models are comparable, they do not fully follow all the features of the Lorenz system. Questions about the existence of fractals in systems based on chemical reactions are addressed in the current work. Moreover, we have worked on the hidden information inside in each wings of a chaotic system generated through fractal process, for the first time, with the aid of basin for fractals. Additionally, we looked closely at the dynamics of the model across the basin, which revealed additional details regarding the existence of hidden and cyclic attractors inside each wing. We also produced multi-wings for system (1) in the current study, demonstrating in a general manner that the number of cyclic attractors increase in a direct relation to the number of wings. Moreover, Julia approach is used to accomplish the work of multi-wings, whereas for searching cyclic attractors inside each extra wing, we have used fifteen million initial conditions and compiled them as a basin set. The data generated in this work is also provided within this paper for the ease of readers.

Astronomical Forcing of Sea‐Level Changes and the History of the Solar System 1,640 Million Years Ago

AbstractThe Paleoproterozoic was an important stage in the evolution of life and the environment during Earth's history. Understanding astronomical rhythms and solar system behavior in the Paleoproterozoic is often challenging. In this study, ∼190 m high‐resolution magnetic susceptibility (MS) and ∼30 m high‐resolution Ba/Al data are used to conduct cyclostratigraphic analyses of the Chuanlinggou Formation in the Yanliao Rift, North China Craton. Spectral analysis indicates significant peaks at wavelengths of 12.1–7.7, 2.8–1.6, 0.58–0.33, and 0.29–0.22 m, which matches well with the astronomical cycles predicted by Waltham for ∼1,640 Ma. A 9.9‐Myr astronomical time scale is constructed by tuning the MS series to the 405‐kyr‐long eccentricity cycle. The geological records show ∼1.56‐Myr very long eccentricity period and ∼1.0‐Myr very long obliquity period (close to 3(s4 − s3) − 2(g4 − g3) = 0) in the Paleoproterozoic. The sea‐level change curve predicted by dynamic noise models reveals a significant ∼1.0‐Myr period, indicating that sea level fluctuation may have been driven by the very long obliquity period. Bayesian inversion is used to further estimate the evolution of the ancient Earth‐Moon orbital parameters, constraining a precession rate of 87.14 ± 0.25 arcsec/year and an Earth‐Moon distance of 341,530 ± 390 km for the Paleoproterozoic Changcheng System. These results indicate that astronomical orbital forcing may have played an important role in climate and sea‐level changes in the Precambrian, increasing the understanding of the fundamentals of global sea level fluctuations, orbitally driven mechanisms and the chaotic behavior of the Precambrian solar system.

A Class of Discrete Memristor Chaotic Maps Based on the Internal Perturbation

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

Traveling wave structures and analysis of bifurcation and chaos theory for Biswas–Milovic Model in conjunction with Kudryshov’s law of refractive index

The present study investigates the generalized Biswas–Milovic model using Kudryshov’s law of refractive index by employing the G′/(bG′+G+a)-Expansion method. This technique facilitates the extraction of singular exponential, trigonometric and kink soliton solutions for the model under examination. The outcomes are subsequently presented as 2D, 3D, and contour plots. Additionally, the model is transformed into a planner dynamical system, and the bifurcation theory is utilized to explore all potential parameter dependencies of the governing model. The chaotic behavior of dynamical systems is also studied by adding an external periodic force. The phase portraits of the obtained findings are also displayed. The results demonstrate the rich and sophisticated behavior of dynamical systems and emphasized the significance of exact solutions for determining their behavior.

A finite-time sliding mode control technique for synchronization chaotic fractional-order laser systems with application on encryption of color images

In recent years, fractional order Laser chaotic systems have gained popularity in both theory and applications, and various classes of these systems have been introduced. This paper presents a dynamic-free sliding mode control (SMC) methodology to synchronize a class of unknown fractional order (FO) Laser chaotic systems with input saturation. The proposed method uses a defined continuous function instead of the discrete sign function and is based on the FO version of the Lyapunov stability theorem. The result is a novel finite-time SMC (FTSMC) methodology that effectively suppresses chaotic behavior in FO Laser chaotic systems without undesirable chattering. This approach is designed to take advantage of the boundedness feature of the FO chaotic system. The efficacy of the FTSMC is demonstrated by applying the method to a chaotic FO Laser system at two different non-integer orders, and its practical applicability is demonstrated by using it to encrypt/decrypt color pictures. The suggested encryption/decryption methodology uses an adaptation of the pre-diffusion permutation-diffusion structure to increase security. Performance and security analyses, including histogram analysis, neighboring pixel correlation analysis, and information entropy analysis, provide further support for the suggested encryption system's superiority.

Probability in Physics—An Introductory Guide

This book is not only valuable for undergraduate but also for graduate students and lecturers alike. Many physics curricula turn a mostly blind eye to probability; it might only appear in the context of experimental errors or in very specific examples in theoretical physics such as the Fermi-Dirac statistics. Andy Lawrence promises to provide a different approach and instead places probabilistic concepts at the heart of modern physics. After a short introduction to the most basic definitions from probability theory, the book quickly starts to model real physical phenomena and therefore introduces the reader to more advanced probability distributions and probabilistic techniques in a natural, hands-on way that is suitable for most physics students. Thermodynamics, quantum mechanical systems, and astronomy (Lawrence’s own field of expertise) provide numerous examples to motivate the use of more and more advanced techniques and equations. Thus, as early as in the 3rd of 14 chapters, Lawrence presents the first highlight of his book by deriving the Boltzmann distribution purely from this probabilistic point of view. Flexibly switching between the frequentist and Bayesian approach to probability, further chapters cover statistical hypothesis tests, model fitting, and brief glimpses into random processes in time. For curious students and for experienced lecturers, the last four chapters of Lawrence’s book provide thought-provoking discussions of the more philosophical aspects of probability theory, such as the arrow of time, information theory, and the inherent randomness in quantum mechanics. Although he touches on fractals, dynamical systems, and attractors, he unfortunately does not follow up with a chapter on the quasi-random behaviour of chaotic systems with strange attractors.

Chaotic Behavior in Diffusively Coupled Systems

We study emergent oscillatory behavior in networks of diffusively coupled nonlinear ordinary differential equations. Starting from a situation where each isolated node possesses a globally attracting equilibrium point, we give, for an arbitrary network configuration, general conditions for the existence of the diffusive coupling of a homogeneous strength which makes the network dynamics chaotic. The method is based on the theory of local bifurcations we develop for diffusively coupled networks. We, in particular, introduce the class of the so-called versatile network configurations and prove that the Taylor coefficients of the reduction to the center manifold for any versatile network can take any given value.

Chaotic behavior and traveling wave solutions of the fractional stochastic Zakharov system with multiplicative noise in the Stratonovich sense

This study employed the planar dynamics system method to investigate the chaotic behavior and traveling wave solution of the fractional stochastic Zakharov system arising in plasma physics. Through the analysis of the phase portraits, the sensitivity of the dynamic system has been examined and various new exact solutions for traveling waves have been constructed. These obtained solutions are conducive to further understanding the propagation of plasma waves. Furthermore, an analysis of the impact of fractional derivative and random noise on the solutions is done. The chaotic behavior of dynamic systems is examined by time series, Poincaŕe sections, and 2D and 3D phase portraits. Finally, the comparative investigation reveals that the characteristics of the solutions are significantly impacted by both the fractional derivative and random noise. The research methodologies and results are efficacious and can be extended to the exploration of more complex phenomena.

A Novel Approach to the Characterization of Stretching and Folding in Pursuit Tracking with Chaotic and Intermittent Behaviors

Detection of Stretching And Folding (SAF) traits in a time series is still controversial and of great interest. Also, visuo-manual tracking studies did not pay attention to SAF in hand motion trajectories. This research aims to find out the relevance of SAF to the discontinuities in chaotic dynamics of hand motion through target tracking tasks. Specifically, a new method is constructed based on this relation in which SAF can extract accurately trajectories in both time domain and phase space. Consequently, we designed experiments to track sinusoidal and trapezoidal target movements shown on a monitor. In these experiments, fourteen participants were instructed to move the joystick handle by wrist flexion-extension movements. Results confirm intermittency in significant human motor control behavior which results in discontinuities in hand motion trajectories. The relation between SAF and these discontinuities is realized by chaotic and intermittent behaviors of tracking dynamics. Verification of the method’s accuracy is also carried out by taking advantage of the Poincaré section. Our method can provide insight into the dynamical behaviors of chaotic and intermittent systems involving mechanisms in human motor control. It can be applied to general systems with intermittent behavior and other systems with modification.