Lyapunov Exponents Research Articles

Связанные квазипериодические генераторы с разными типами связи (дискретная модель)

In the paper we examine the regimes of two coupled radio-physical generators that exhibit quasi-periodic oscillations, with reactive, combined, and active types of coupling. In order to simplify the computer calculations, a discrete version of the system being investigated is introduced. Lyapunov exponent charts illustrating various regimes, including quasi-periodic oscillations with a different number of incommensurable frequencies, are presented. For a purely reactive coupling, in comparison to a dissipative one, there is no stable equilibrium regime, and the areas of two-frequency regimes are very small in size. For dissipative coupling, with the addition of a reactive component, the characteristic regimes at strong coupling are preserved. When the coupling is weak, five-frequency regimes disappear; they are replaced first by four-frequency, and then by three-frequency, chaotic and hyperchaotic ones. For reactive coupling with the addition of dissipative, the tongues of the three-frequency regimes do not have peaks, but occupy certain intervals on the frequency disorder axis. There is also no stable equilibrium for an active (repelling) coupling. A three-frequency area with a built-in set of resonant two-frequency tongues becomes typical, the overlap of which leads to chaos.

Pole-skipping and chaos in D3-D7 brane systems

In this paper, we analyze the pole-skipping phenomena of finite temperature Yang-Mills theory with quark flavors, which is dual to D3-D7 brane systems in bulk. We also consider the external electric field in the boundary field theory, which is dual to the world volume electric field on the D7 brane. We will work in the probe limit where the D7 branes do not backreact to the D3 brane background. In this scenario, we decode the characteristic parameters of the chaos, namely, Lyapunov exponent λL and butterfly velocity vb from the pole-skipping points by performing the near effective horizon analysis of the linearized Einstein equations. Unlike pure Yang-Mills, once charged quarks with a background electric field are added into the system, the characteristic parameters of the chaos show nontrivial dependence on the quark mass and external electric field. We have observed that λL and vb decreases with increasing electric field. We further perform the pole-skipping analysis for the gauge invariant sound, shear, and tensor modes of the perturbation in the bulk and discuss their physical importance in the holographic context. Published by the American Physical Society 2024

Effects of the kinematic variable, time delay and data length on test-retest reliability of the maximal Lyapunov exponent of human walking.

The maximal Lyapunov exponent (MLE) has been used to quantify the dynamic stability of human locomotion. The method for estimating MLE requires selecting a proper time series of kinematic variables and reconstructing phase space using proper time delay. The data length also affects the reliability of the measured MLE. However, there has been no criterion for the choice of the time series, time delay or data length. Here, we quantified the effect of these factors on the test-retest reliability of MLE estimations. We recruited 15 young and healthy adults and let them walk on a treadmill three times. We calculated MLE employing various lengths of time series of 18 frequently used kinematic variables and two typical choices of time delay: fixed delay and delay selected by average mutual information algorithm. Then, we measured the intraclass correlation coefficient (ICC) of the measured MLE under each condition. Our results show that the choice of time delay does not affect reliability. Five among the 18 kinematic variables enabled excellent reliability with ICC above 0.9 within 450 strides and also enabled ICC above 0.75 even with 60 or less strides. These findings can contribute to establishing the criteria for measuring the dynamic stability of human walking.

Flow topology and mixing in alveolar edema: Unsteady flow in interconnected cavities with moving walls

The study of alveolar fluid mechanics is critical for comprehending respiratory function and lung diseases, particularly in cases of alveolar lesions that result in significant structural and fluid dynamic changes. This study investigates the flow topology and chaotic mixing within both normal and edematous alveoli, where the alveoli in the edematous model are interconnected by pores. To numerically simulate alveolar flow, a mathematical model is developed to ascertain the key parameters of Reynolds number (Re) and alveolar expansion ratio. Subsequently, the flow fields are analyzed to determine wall shear stress (WSS) and to identify WSS critical points and critical points of velocity vector, with a thorough presentation of the various flow topologies corresponding to these critical points. Moreover, a dynamic mode decomposition-based method is introduced to track particle trajectories, and the exploration of chaotic mixing is conducted through tracer advection, Poincare map, and the calculation of finite-time Lyapunov exponents. Results indicate that the edematous model exhibits a higher Re and higher WSS due to the fluid properties. Within the alveoli, high WSS is usually localized at the pores. The pores increase critical points and alter flow topologies, significantly changing chaotic mixing. Additionally, Re and alveolar locations also affect mixing patterns. These findings are crucial for understanding alveolar physiology and designing inhaled drugs for lung diseases, considering the role of chaos in particle transport in the lung acini.

A Central Limit Theorem with Explicit Lyapunov Exponent and Variance for Products of $$2\times 2$$ Random Non-invertible Matrices

A Central Limit Theorem with Explicit Lyapunov Exponent and Variance for Products of $$2\times 2$$ Random Non-invertible Matrices

Expansiveness, generators and Lyapunov exponents for random bundle transformations

We generalize Fathi's results by showing that a compact metrizable space admits an fiber expansive homeomorphism if and only if it has a compatible hyperbolic metric. Moreover, we prove that a compact metrizable space admits an fiber expansive homeomorphism if and only if it has a generator in detail. Furthermore, we show that a fiber expansive homeomorphism has finite fiber topological entropy. Finally, we show that fiber Lyapunov exponents for a fiber expansive system are different from zero, indicating that the system presents a chaotic system. Meanwhile, we also prove that negative fiber Lyapunov exponents for compact invariant sets of a dynamical system imply that the compact set is a fiber attractor.

A modified neural network method for computing the Lyapunov exponent spectrum in the nonlinear analysis of dynamical systems

This study presents a modified neural network method for computing the Lyapunov exponent spectrum in non-linear dynamical systems. Its mathematical description is an introduction. The proposed modified neural network method allows the addition of bias and constant neurons to the neural network topology and the use of different numbers of activation functions to suit different cases. Various algorithms for computing Lyapunov exponents, such as the Benettin method, the Wolf method, the Rosenstein method, the Kantz method, the Synchronisation method, the Sano-Sawada algorithm and the proposed modification of the neural network method, are used for classical problems in nonlinear dynamics. These problems include the generalised Hénon map, the chaotic attractor of the Baier-Klein map, and the vibrations of mechanical systems such as the flexible Bernoulli-Euler beam and the flexible functionally graded porous closed cylindrical shell under alternating load. The comparative analyses presented in this study are aimed at validating the accuracy and effectiveness of the methods, and at identifying the most relevant approaches for different types of systems and classes of problems. The proposed method is demonstrated to be superior to existing methods based on time-series evaluation in terms of sample size and accuracy. Furthermore, it does not require the initial system equations.

Floating particles transport through the free surface vortex technique: A novel numerical study to assess the interaction among different scales of vortex structures

Fat, oil, and grease (FOG) floating particles in the sump of sewage pumping stations will accumulate together to form rigid layers, resulting in failure for pump device. To overcome this, the free surface vortex (FSV) technique has been considered and applied to transport floating particles toward the submerged suction pump inlet. This paper investigates the potential of vortices as a means of downward motion of FOG. The entrainment capacity of FSV is investigated by numerical simulations using a coupled level-set and volume-of-fluid method. Two coherent structures are decomposed by proper orthogonal decomposition: FSV represented by the first two orders with high energy content and spiral vortex bands represented by low energy and high order models. The extracted ridges of the finite-time Lyapunov exponent (FTLE) delineate different regions of the flow field and effectively capture the evolution of Lagrangian coherent structures. The floating particles in the sump are first caught by the dividing line formed by the FTLE ridges, mixed in the entrainment zone, and then merged into the vortex. The enstrophy production term dominates the development of vorticity. Subject to the influence of flow velocity gradients, both radial and tangential vortices undergo a transition into axial vortices. This transformation enhances the vortex's capacity to entrain particles within the vortex core area, leading to their rapid inward spiraling toward the vortex center and eventual expulsion due to the vortex's entrainment effect.

ON THE CHAOTIC NATURE OF A CAPUTO FRACTIONAL MATHEMATICAL MODEL OF CANCER AND ITS CROSSOVER BEHAVIORS

In this paper, a three-dimensional mathematical model of cancer, incorporating tumor cells, host normal cells, and effector immune cells, is examined. The chaotic nature of this cancer model is explored within the framework of Caputo fractional calculus. The analysis employs local stability assessment using the Matignon criteria, Lyapunov exponents (LEs) for various fractional orders of derivatives, bifurcation plots reflecting changes in the fractional derivative order, simulation outcomes of a novel chaotic attractor, Kaplan–Yorke dimension, and sensitivity to initial conditions. The results demonstrate that the Caputo fractional cancer model exhibits chaotic behavior for selected parameters. Moreover, the fractional derivative orders significantly influence the extent of chaotic behavior. Specifically, as the fractional derivative order increases from zero to one, the intensity of chaos diminishes, evidenced by a decrease in both the LE and the Kaplan–Yorke dimension. A piecewise modeling approach is applied to the cancer model, incorporating deterministic, stochastic, and power-law behaviors. Three distinct scenarios are developed within this framework, revealing several novel crossover behaviors through simulation. Notably, it is observed that when cancer dynamics initially follow a power-law behavior and subsequently transition into a stochastic process, the disease dynamics can stabilize, suggesting a positive outlook for disease control. Conversely, if the cancer dynamics begin with a deterministic process and then shift to a stochastic process, the system may enter a chaotic state, complicating disease management efforts.

Automated recognition of mental cognitive workload through nonlinear EEG analysis

Nowadays, with the remarkable advancements in detection instruments and artificial intelligence, there has been extensive utilization of human mental state monitoring in various domains. Few studies have explored how nonlinear analysis methods can detect cognitive workload despite the complex nature of EEG signals and advancements in signal processing techniques. In addition, the fuzziness of human mental conditions makes the need to use fuzzy engineering tools tangible in this field. Therefore, this investigation aimed to develop a decision support algorithm to improve previous efforts for the classification of task EEG and resting through machine learning algorithms. Various nonlinear features were calculated from all 19 EEG channels: Hurst exponent, Lempel–Ziv complexity, detrended fluctuation analysis, Higuchi fractal dimension, Katz fractal dimension, permutation entropy, singular value decomposition entropy, Petrosian fractal dimension, sample entropy, and Lyapunov exponent. During the classification step, a newly developed EPC-FC (Expert per Class Fuzzy Classifier) is introduced, utilizing an ensemble framework with specialized sub-classifiers for identifying a particular condition. By training sub-classifiers with the negative correlation learning (NCL) approach, the EPC-FC is designed to be exceptionally adaptable. Additionally, the separation of sub-classifiers within each class provides versatility and clarity to the system’s design. The proposed approach based on fuzzy systems and nonlinear analyses was applied to EEG data for mental workload recognition, which provides an excellent accuracy of 98.50% and an F1-score of 98.56% which is much higher than previous findings in this field. Also, the obtained results indicate that utilizing the proposed EPC-FC classifier maintains a consistently high accuracy exceeding 90% across various levels of SNRs. The obtained results proved the high potential of nonlinear analysis to detect cognitive states of the brain, which is consistent with the nonlinear and fuzzy nature of EEG data. Other nonlinear approaches should be considered for future studies to improve the current results.

Image compression encryption algorithm combining two-dimensional modular hyperchaotic map and compressed sensing

Recently, to offer better ensure for image privacy security, numerous new image encryption algorithms have been proposed. However, these algorithms still suffer from the problems of chaotic performance scarcity, low encryption effect, and high consumption of computational resources. To solve the above issues, we first construct a two-dimensional modular hyperchaotic map (2D-MHM). Then, we further develop an image encryption algorithm based on 2D-MHM and compressed sensing (CS). Several chaotic metrics verify the randomness and validity of 2D-MHM. These metrics include bifurcation diagram, Lyapunov exponent, initial value sensitivity, 0–1 test, and NIST test. Specifically, CS significantly reduces the ciphertext image size thereby reducing its resource consumption during transmission. Reality-preserving fractional DCT (RP-Fdct) diffusion is utilized to transform pixels into the frequency domain to enhance the encryption effect. Subsequently, lightweight index confusion and XOR diffusion further improve the algorithm security. The security of the algorithm is verified through various experiments. It is able to encrypt grayscale and color images of different sizes with good results. Notably, this algorithm also implements the encryption requirements for binary images. Due to our designs, it outperforms recently reported encryption algorithms in several areas, especially in reconstruction performance.

Chaotic responses across the potential barrier of bistable vortex-induced vibration energy harvesters

Large-amplitude interwell limit cycle oscillations (LCOs) are regarded as an ideal high-output state of multistable vibration energy harvesters. However, chaotic responses, which result in lower output voltages than interwell LCOs, can be observed in bistable vortex-induced vibration energy harvesters (VIVEHs). In this paper, the chaos in bistable VIVEHs is investigated through numerical simulation, theoretical analyses, and experimental validation. The distribution of the periodic solutions obtained by the incremental harmonic balance (IHB) method and chaos judgment using Lyapunov exponents are elucidated. It is found that the bistable VIVEH shows intrawell periodic solutions at high reduced wind speeds (RWSs) and period-doubling bifurcation appears with the periodic solutions turning into chaos with decreasing RWS. Analyses of the influence of nonlinear parameters on the chaotic responses identify the nonlinear stiffness as the dominant factor contributing to the chaos. Meanwhile, the chaotic responses and interwell LCOs would exist at the same RWS with different initial states. Finally, wind tunnel experiments confirm the occurrence of homologous chaotic responses of the bistable VIVEH which feature aperiodic jumping back and forth motion between two potential wells, in agreement with numerical simulations. Overall, this paper provides a framework for analyzing the chaotic responses of bistable VIVEHs, offering valuable insights for complex response mechanism understanding.

Probing the effect of time conformal factor on the particle dynamics around AdS Reissner–Nordström black hole

This study entails the effect of time on particle dynamics (both charged and uncharged) in the surrounding of the Reissner–Nordström AdS black hole in light of a general time conformal factor by acquiring Noether symmetry relation using the Lagrangian of the considered black hole spacetime. Upon acquiring the required time conformal spacetime as well as the Noether and approximated Noether symmetries along with their respective laws of conservation, the study further aims at investigating different parameters, like effective potential, effective force, and trajectories of escape velocity for the particle dynamics of the specified black hole. Additionally, in order to investigate orbit stability, the study employs the Lyapunov exponent.

Probing the thermodynamics of charged Gauss Bonnet AdS black holes with the Lyapunov exponent

In this paper, we investigate the thermodynamic properties of the charged AdS Gauss–Bonnet black holes and their associations with the Lyapunov exponent. The chaotic features of the black holes and the isobaric heat capacity characterized by the Lyapunov exponent are studied to reveal the thermodynamic stability of the black hole phases. By considering both the timelike and null geodesics, we find that the relationship between the Lyapunov exponent and the Hawking temperature can accurately represent the features of the Small/Large phase transition and even the triple point. We also reveal the properties of the difference in the Lyapunov exponent as an order parameter. It is demonstrated that there is a negative correlation between the Lyapunov exponent and the size of the black hole shadow, which can be used to bridge the thermodynamic properties and the shadow of black holes.

Dynamics and Adaptive Control of a Novel 5D Hyperchaotic System: Either Hidden Attractor or Self-excited with Unusual Nature of Unstable Equilibria

In 2020, J. Sprott proposed a new three dimensional chaotic system with special features such like 1) dissipative and time-reversible; 2) no equilibrium point; 3) lien of initial conditions goes to the attractor. We observed that an extension of the so-called Sprott's 2020 system to four dimensional system with complex dynamics showed either chaotic or hyperchaotic with unbounded orbits. In this paper, a novel five dimensional hyperchaotic system based on Sprott's 2020 system has been proposed. The proposed system shows complex dynamics like hyperchaotic. The proposed system can be classified as a hidden attractor where no equilibrium point appeared or self-excited where an unusual nature of unstable equilibrium points connected to a very complicated function called Lambert W appeared. The dynamical properties of such system are discovered by computing the Lyapunov exponents and bifurcation diagram. Adaptive control to the proposed system was provided.

Research on the influence of labyrinth ring configuration on runaway characteristics of pump-turbines

The labyrinth ring can reduce the volume loss, mitigate the influence of force characteristics when the unit is operating, and ensure the efficient, safe, and stable operation of the unit. It is usually installed in the upper crown and lower ring of the pump-turbines runner. When the pump-turbines unit is running in runaway condition, the internal flow field structure is more complicated due to the different forms of labyrinth ring, so a new method needs to be adopted to analyze it from a new perspective. In this paper, a high head pump-turbines was selected as the research object. Combined with the experimental and numerical simulation results, the flow field structure corresponding to different upper labyrinth ring forms under runaway condition was analyzed and visualized in many aspects. The results show that the mixed type labyrinth ring can improve the undesirable flow regime in the runner and draft tube. The guide vanes under the serrated type labyrinth ring are more likely to be damaged due to greater force. The influence of the mixed type labyrinth ring on the radial force of the runner is only half that of the ladder type, which has a great advantage in maintaining the radial stability of the runner. The ladder and serrated type labyrinth ring can make the unit obtain less axial water thrust. In addition, based on the Finite-Time Lyapunov Exponent (FTLE) method, the flow field structure in the vaneless region was analyzed, and it was found that the labyrinth ring may affect the flow in the flow path of the guide vane and the vaneless region, thus causing the change of axial water thrust. This study reveals the internal flow and force characteristics of pump-turbines with different types of labyrinth rings under runaway condition and provides a technical reference for further understanding and studying the influence of labyrinth ring forms on pump-turbines performance and force characteristics.

Chaotic behavior in Josephson junction for high-quality random-number generation

The configuration of a high-quality security system or computer using a Monte Carlo simulation high-quality random-number generator has been expected. The high-quality random-number generators are obtained using physical randomness phenomena. In particular, chaos is an appropriate random-number generator. First, we investigate the chaotic behavior in the Josephson junction (JJ) under irradiation with external radio frequency (RF) at 77 K using an equivalent circuit of the JJ. JJ is assumed to be an intrinsic JJ in a Bi2Sr2CaCu2O8+δ single crystal. The output voltages of the JJ behave chaotic under some conditions and are evaluated using the Lyapunov exponent, Poincaré section, and attractors. Next, random numbers were generated from the chaotic output voltage in rgw JJ under irradiation with RF and evaluated by a verification test. Thus, random-number generators can be applied using the output voltage of theJJ under irradiation with external RF.

A Rectified Linear Unit-Based Memristor-Enhanced Morris–Lecar Neuron Model

This paper introduces a modified Morris–Lecar neuron model that incorporates a memristor with a ReLU-based activation function. The impact of the memristor on the dynamics of the ML neuron model is analyzed using bifurcation diagrams and Lyapunov exponents. The findings reveal chaotic behavior within specific parameter ranges, while increased magnetic strength tends to maintain periodic dynamics. The emergence of various firing patterns, including periodic and chaotic spiking as well as square-wave and triangle-wave bursting is also evident. The modified model also demonstrates multistability across certain parameter ranges. Additionally, the dynamics of a network of these modified models are explored. This study shows that synchronization depends on the strength of the magnetic flux, with synchronization occurring at lower coupling strengths as the magnetic flux increases. The network patterns also reveal the formation of different chimera states, such as traveling and non-stationary chimera states.

Multiscale revelation of asphalt morphology and adhesion performance evolution during stress relaxation process

The objective of this study is to investigate the evolution of asphalt morphology and adhesion properties during stress relaxation using atomic force microscopy (AFM) and molecular dynamics (MD) simulations. Changes in surface roughness, Derjaguin-Muller-Toporov (DMT) modulus, adhesion force, and force-distance curves were analyzed. Morphological and adhesion property evolution was assessed using correlation length and Lyapunov exponent. MD simulations provided insights into the internal stress-time relationship and free volume fraction of asphalt during relaxation. Results indicate that under constant tensile strain, the “bee structure” on the asphalt surface elongates, dark pits transform into light protrusions, microcracks heal, and surface roughness stabilizes after 10 h. Under compressive strain, similar changes occur but are more pronounced during tensile relaxation. The combined use of AFM and MD simulations offers a comprehensive understanding of the microscopic evolution mechanisms of asphalt under stress relaxation, providing valuable insights for optimizing asphalt pavement design and maintenance.

Quasinormal modes and universality of the Penrose limit of black hole photon rings

We study the physics of photon rings in a wide range of axisymmetric black holes admitting a separable Hamilton-Jacobi equation for the geodesics. Utilizing the Killing-Yano tensor, we derive the Penrose limit of the black holes, which describes the physics near the photon ring. The obtained plane wave geometry is directly linked to the frequency matrix of the massless wave equation, as well as the instabilities and Lyapunov exponents of the null geodesics. Consequently, the Lyapunov exponents and frequencies of the photon geodesics, along with the quasinormal modes, can be all extracted from a Hamiltonian in the Penrose limit plane wave metric. Additionally, we explore potential bounds on the Lyapunov exponent, the orbital and precession frequencies, in connection with the corresponding inverted harmonic oscillators and we discuss the possibility of photon rings serving as effective holographic horizons in a holographic duality framework for astrophysical black holes. Our formalism is applicable to spacetimes encompassing various types of black holes, including stationary ones like Kerr, Kerr-Newman, as well as static black holes such as Schwarzschild, Reissner-Nordström, among others.