Lyapunov Exponents Research Articles

Chaos and regularities in cavity assisted two-channel nonlinear coupler

This paper presents a study of the dynamical behavior in a cavity-assisted two-channel Kerr nonlinear directional coupler. Applying a deterministic periodic laser pump to a two-channel coupled system led to a range of dynamic behaviors, encompassing periodicity, quasiperiodicity, chaoticity and hyperchaoticity. This is confirmed by the exploration of the Lyapunov exponents, Poincaré sections and bifurcation diagrams for resonance and low cavity detuning. For low cavity detuning, chaos arises when the laser pump’s amplitude is large. This is due to higher interactions between photons and the cavity, resulting in more photons inside and higher nonlinearity. On the other hand, the reduced interactions in high cavity detuning displayed regular behavior which is explained qualitatively.

The Phase Transition of Covariant Lyapunov Vector Precisely Locates a Stability Reversal of Quasi-Periodic Response

Abstract Assessing the stability of quasi-periodic (QP) response is crucial, as the bifurcation of QP response is usually accompanied by a stability reversal. The largest Lyapunov exponent (LLE), as an important indicator for chaotic motion, can also be used for the stability analysis of QP response. The precise location of a stability reversal, however, is tough to achieve as a poor convergence rate would be usually encountered when solving the LLE. Herein a straightforward and precise approach is suggested to identify the critical point when a stability reversal happens. Our approach is based on an explicit differential equation that provides the LLE straightforwardly via numerical integration, and the corresponding covariant Lyapunov vector is simultaneously obtained. The major finding consists in the phase transition of the covariant Lyapunov vector, which can happen much early before the LLE reaches a relatively convergent value. More importantly, the phase transition can serve as a strong indicator to locate a stability reversal of the QP response qualitatively. Numerical examples are provided to verify of the effectiveness and wide applicability the presented approach.

Nonlinear dynamic characteristics of smart FG-GPLRC sandwich varying thickness truncated conical shell with internal resonance for first three order modes

This paper examines the 1:1:1 internal resonant nonlinear dynamic characteristic of the simply supported varying thickness functionally graded graphene platelets reinforced composite (FG-GPLRC) smart truncated sandwich conical shell subject to the combined effects of transverse load and in-plane force. The truncated smart sandwich conical shell is composed of an FG-GPLRC varying thickness core and two magneto-electro-elastic face layers, whose material properties and constitutive relations are individually identified by the rule of mixture, improved Halpin-Tsai approach and generalized Hooke's law. Utilizing the first-order shear deformation theory (FSDT), von Karman's geometrical nonlinearity, Hamilton's principle and Galerkin technique, the 3DOF dimensionless nonlinear dynamic formulations for the truncated smart FG-GPLRC conical shell are established. The multiple-scale technique is applied to developing the averaged equations for the truncated smart FG-GPLRC conical shell under combined resonance. The frequency-response and force-response curves, Poincare maps, phase portraits, time history diagrams, bifurcation and maximum Lyapunov exponent diagrams can be portrayed by the nonlinear equation solver and Runge-Kutta approach. The effects of the damping and tuning parameters, transverse and in-plane forces on the 1:1:1 internal resonant nonlinear dynamic characteristic of truncated smart varying thickness FG-GPLRC sandwich conical shell are examined.

Periodic, quasi-periodic, chaotic waves and solitonic structures of coupled Benjamin-Bona-Mahony-KdV system

Abstract In this paper, we mainly focus on studying the dynamical behaviour and soliton solution of the coupled Benjamin-Bona-Mahony-Korteweg-de Vries (BBM-KdV) system, which characterizes the propagation of long waves in weakly nonlinear dispersive media. The paper utilizes different tools to detect chaos, such as time series analysis, bifurcation diagrams, power spectra, phase portraits, Poincare maps, and Lyapunov exponents. This analysis helps in more accurate predictive modeling of the systems. This understanding can aid in the design of control strategies, resulting in enhancements in prediction, control, optimization, and design. Additionally, we construct the system's solitary wave structures using the Jacobi elliptic function (JEF) method. We identify periodic wave solutions expressed in terms of rational, hyperbolic, and trigonometric functions. Certain parameter values can lead to periodic wave solutions, solitary waves (bell-shaped solitons), shock wave solutions (kink-shaped soliton solutions), and double periodic wave solutions.

Dynamical complexity and multifractal analysis of geomagnetic activities at high temporal scales over three solar cycles

Activities in geospace occur at different time scales. Understanding geomagnetic activity at high temporal scales will give insight into fast dynamics in geospace. This study aims to investigate dynamical complexities in geomagnetic activities at a high temporal scale across three solar cycles. Geomagnetic activities, as represented by 5-min SYM-H data, were considered in this study over three solar cycles (22–24) from 1986 to 2019. Chaos analysis using sample entropy, Lyapunov exponent, and correlation dimension indicates that the geomagnetic activities are driven by intrinsic complex and chaotic processes. Positive Lyapunov exponent values between 0.13 and 0.18, 0.15–0.18, and 0.16–0.19 were obtained for solar cycles 22, 23, and 24 respectively. Furthermore, geomagnetic activities were also found to have multifractal structures driven by high fractal exponents with fine structures. A positive relationship was obtained between the annual mean values of SYM-H and the degree of complexity. It is concluded that geomagnetic activities have a short prediction horizon.

Chaos in Bitcoin Cryptocurrency Metrics: Analysis and Forecasts

AbstractCryptocurrencies, particularly Bitcoin have attracted a lot of attention in the last decades of humanity. Analyzing cryptocurrencies algorithmic differences, chaotic behavior and self-similarity in cryptocurrency metrics might give significant insights for identifying risks and opportunities. Determining the degree of chaos in crypto metrics is critical for understanding complexity, improving prediction capabilities, and supporting decision-making. This study focuses on the analysis of chaos and self-similarity in Bitcoin dynamics for predictability perspective. Return, rate of return and volume quantities in different scales are analyzed with using rescaled range method to reveal the degree of self-similarity. Hurst parameter extracts a comprehensive summary providing information on how current values depend on previous ones to reveal any persistence in Bitcoin metrics. Daily rate of return and return give Hurst degree around 0.64 while they are in between 0.52–0.55 for minutely and hourly based prices. However, an increasing persistence is observed with the increasing time window. Although the largest Lyapunov exponents stay in the positive region for prices and returns of Bitcoin, they are approximately zero for inspected statistics. Periodic characteristics of Bitcoin are also investigated to reveal any dependencies on halving mechanism of Bitcoin. Detailed self-similarity analysis on specific periods shows that bull and bear market seasons don’t make any significant effect on the degree of Hurst parameter. Due to nonlinear and unpredictable characteristics of Bitcoin metrics, distribution fittings are applied to characterize BTC return and rate of return. While Wakeby distribution gives best fitting for daily return, Cauchy distribution gives best for hourly returns.

Dynamic analysis of a modified Leslie-Gower model with Michaelis-Menten prey harvesting and additive allee effect on predator population

In the current study, the dynamics of a Leslie-Gower model have been studied considering the Allee effect in the growth of the predator and the Michaelis-Menten type harvesting effect on the prey from biological and theoretical perspectives. Key findings highlight how these effects influence the existence of positive equilibria, with local stability properties and potential bifurcation behaviors examined using the center manifold theorem and linearization. Criteria for saddle-node bifurcation are established through Sotomayor’s theorem, while the direction and stability of Hopf bifurcation are explored via the first Lyapunov exponent. An optimal control model is developed to promote sustainable ecosystem development, utilizing the harvesting rate as a control parameter. Further, a few numerical simulations validate our significant findings using phase portrait diagrams.

Image encryption algorithm based on matrix projective combination-combination synchronization of an 11-dimensional time delayed hyperchaotic system

Abstract Secure image transmission is critical to protect sensitive data from unauthorized access, especially in an era of increasing digital threats. Chaotic systems with their inherent complexity and unpredictability, provide a promising solution for enhancing encryption security. To contribute to this field, we investigate a new 11-dimensional hyperchaotic system by taking advantage of its complex dynamical properties to strengthen security. The high dimensional of the system intensifies chaotic behaviors such as stability, attractors and sensitive to initial conditions, making it particularly suitable for encrypted transmission.
 Time delay is an important factor to be considered affecting the control and synchronization in nonlinear system. Additionally, time delays include the effects of past states, further increasing the unpredictability of the system. To explore these dynamics, we analyze the Lyapunov exponents, stability of equilibrium points, symmetry and dissipation. A matrix projective combination-combination synchronization scheme is proposed to synchronize four identical 11-dimensional hyperchaotic systems with time delay. Nonlinear active controllers designed based on Lyapunov stability theory are used to achieve this synchronization. This work advances an important idea for encryption and decryption algorithms, which is the secure transmission of images using affine encryption. In the affine encryption algorithm, the key is based on the solution of synchronized chaotic delayed systems and the private message of the sender and receiver. This proposed encryption and decryption algorithms have been applied on plain images. Numerical simulations and security analysis including key space, histogram, information entropy and correlation analysis are conducted to validate the theoretical results and encryption algorithm. Experimental analysis and comparisons with existing literature confirm the effectiveness and security of the proposed approach for cryptographic purposes.

Asymmetric transfer matrix analysis of Lyapunov exponents in one-dimensional nonreciprocal quasicrystals

Asymmetric transfer matrix analysis of Lyapunov exponents in one-dimensional nonreciprocal quasicrystals

The impact of the Caputo fractional difference operator on the dynamical behavior of a discrete-time SIR model for influenza A virus

Abstract Since there are few studies that deal with the fractional-order discrete-time epidemic models, this paper presents a new fractional-order discrete-time SIR epidemic model that is constructed based on the Caputo fractional difference operator. The effect of the fractional orders on the global dynamics of the SIR model is analyzed. In particular, the existence and stability of equilibrium points of the model are presented. Furthermore, we investigate the qualitative dynamical properties of the SIR model for both commensurate and incommensurate fractional orders using powerful nonlinear tools such as phase attractors, bifurcation diagrams, maximum Lyapunov exponent, chaos diagrams, and 0-1 test. In addition, the complexity of the discrete model is measured via the spectral entropy complexity algorithm. Further, an active controller is designed to stabilize the chaotic dynamics of the fractional-order SIR model. Finally, the suggested model is fitted with real data to show the accuracy of the current stability study. Our goal was achieved by confirming that the proposed SIR model can display a variety of epidiomologically observed states, including stable, periodic, and chaotic behaviors. The findings suggest that any change in parameter values or fractional orders could lead to unpredictable behavior. As a result, there is a need for additional research on this topic.

Rigidity of Lyapunov exponents for derived from Anosov diffeomorphisms

Abstract For a class of volume-preserving partially hyperbolic diffeomorphisms (or non-uniformly Anosov) $f\colon {\mathbb {T}}^d\rightarrow {\mathbb {T}}^d$ homotopic to linear Anosov automorphism, we show that the sum of the positive (negative) Lyapunov exponents of f is bounded above (respectively below) by the sum of the positive (respectively negative) Lyapunov exponents of its linearization. We show this for some classes of derived from Anosov (DA) and non-uniformly hyperbolic systems with dominated splitting, in particular for examples described by Bonatti and Viana [SRB measures for partially hyperbolic systems whose central direction is mostly contracting. Israel J. Math.115(1) (2000), 157–193]. The results in this paper address a flexibility program by Bochi, Katok and Rodriguez Hertz [Flexibility of Lyapunov exponents. Ergod. Th. & Dynam. Sys.42(2) (2022), 554–591].

Grain storage temperature prediction based on chaos and enhanced RBF neural network

Grain storage has very strict temperature requirements. Aiming at the problems of nonlinear characteristics and poor prediction accuracy of temperature parameters in grain storage, a combination of chaos theory and enhanced radial basis neural network is proposed as a temperature prediction model for grain storage (C-ERBF). The model first determines the embedding dimension and time delay of the grain storage temperature sequence using chaos theory. It then calculates the Lyapunov exponent to confirm its chaotic properties and reconstructs the sequence in the phase space to extract the hidden dynamic information and structure behind the sequence. Furthermore, the q-Normalized Least Mean Square Fourth (qXE-NLMF) algorithm is designed to enhance the radial basis function (RBF) neural network model for weight updating, to improve its prediction accuracy, and to accelerate the training speed of the model. As verified by the simulation experiments of Mackey–Glass chaotic time series prediction, the enhanced RBF (ERBF) network has faster convergence speed and lower steady-state error compared to the traditional RBF network. Finally, the optimized dataset from chaos theory is input into the model to achieve accurate predictions of grain storage temperature series. The experimental results show that the proposed C-ERBF model has higher prediction accuracy compared to other time series prediction methods. It can realize the grain pile temperature in advance, and take control measures in advance. This proactive approach significantly reduces the consumption of stored grain and prevents issues before they arise.

Multi-image encryption scheme using cross-plane coupling permutation and plain-by-plain wave diffusion

Images contain a wealth of visual information, are susceptible to unauthorized access due to their vulnerability and sensitivity. This paper designs a novel multi-image encryption scheme for protecting the privacy of images of different sizes and types. Initially, a 2D memristive hyperchaotic map (2D-MHM) is designed and subjected to various dynamic analyses and randomness evaluations. The results demonstrate that the proposed map possesses an exceptionally large parameter space, high Lyapunov exponent and sample entropy, and has successfully passed the entire suite of NIST test, verifying its feasibility for confidential communication. Then we present a multi-image encryption scheme combining cross-plane coupling permutation and plain-by-plain wave diffusion to realize random exchange and global variation of pixels in different planes. The performance evaluation and numerical analysis demonstrate that the scheme is resilient against multifarious types of attacks, possesses great security while effectively enhancing encryption efficiency. Finally, the proposed scheme is compared with advanced algorithms and its application in healthcare is discussed, exhibiting its superiority in multiple aspects.

Regulation and analysis of multi-scroll attractors in a class of memristive Hopfield neural networks with high Lyapunov exponents

Multi-scroll chaotic systems have gained significant attention in recent years. However, there is little research on the chaotic systems of memristive neural networks with multiple complex structural attractors. A class of multi-scroll chaotic systems are designed in this paper, which are based on Hopfield neural network by using memristors as synapses. Implementing synaptic control with memristors at various coupling positions allows for the creation of multiple chaotic attractors with distinct topological structures. The proposed systems exhibit special and irregular shaped attractors, and can generate a fixed number of scrolls. Through analysis we find that the systems have high Lyapunov exponents, indicating strong sensitivity and randomness. Meanwhile, the systems exhibit extremely rich dynamic behaviors, such as symmetric coexistence phenomena. Especially, we observe signal amplitude control and offset boosting phenomena in the systems. In addition, simulation circuits are designed based on such chaotic systems to verify the physical feasibility of the systems.

A color image encryption algorithm based on a novel 4D hyperchaotic system and bit-level diffusion

As digital images are widely used in social media, medical, military and other fields, ensuring the privacy and security of image data has become a critical concern. Firstly, we propose a novel four-dimensional hyperchaotic system and validate that it exhibits a broad chaotic range, as demonstrated by bifurcation diagrams and Lyapunov exponent experiments. Additionally, simulated circuit diagrams verify the hardware feasibility of the proposed system. Secondly, we design a dynamic iterative scrambling (DIS) scheme that dynamically divides the image into multiple matrices for spatially indexed scrambling. Excellent substitution performance can be ensured by multiple iterations. In the diffusion stage, a multidirectional bit-level L-shaped (MDBL) scheme is proposed. Diffusion is conducted on the bit plane using a designed cross-multiplanar selection algorithm, which fuses the high and low bit planes, thereby enhancing the diffusion performance of MDBL. Thirdly, Based on the above concepts, a novel four-dimensional hyperchaotic system and an encryption algorithm based on bit-level diffusion are proposed. Finally, experimental results and security analyses demonstrate the effectiveness of the novel 4D hyperchaotic system and image encryption scheme. The proposed encryption scheme exhibits robust anti-interference capabilities and effectively safeguards image security.

Instabilities and Pattern Formation in Epidemic Spread Induced by Nonlinear Saturation Effects and Ornstein–Uhlenbeck Noise

Abstract We analytically study the emergence of instabilities and the consequent steady-state pattern formation in a stochastic partial differential equation (PDE) based, compartmental model of spatiotemporal epidemic spread. The model is characterized by: (1) strongly nonlinear forces representing the infection transmission mechanism and (2) random environmental forces represented by the Ornstein–Uhlenbeck (O–U) stochastic process which better approximates real-world uncertainties. Employing second-order perturbation analysis and computing the local Lyapunov exponent, we find the emergence of diffusion-induced instabilities and analyze the effects of O–U noise on these instabilities. We obtain a range of values of the diffusion coefficient and correlation time in parameter space that support the onset of instabilities. Notably, the stability and pattern formation results depend critically on the correlation time of the O–U stochastic process; specifically, we obtain lower values of steady-state infection density for higher correlation times. Also, for lower correlation times the results approach those obtained in the white noise case. The analytical results are valid for lower-order correlation times. In summary, the results provide insights into the onset of noise-induced, and Turing-type instabilities in a stochastic PDE epidemic model in the presence of strongly nonlinear deterministic infection forces and stochastic environmental forces represented by Ornstein–Uhlenbeck noise.

Synchronization of Chaotic Systems with Huygens-Like Coupling

One of the earliest reports on synchronization of inert systems dates back to the time of the Dutch scientist Christiaan Huygens, who discovered that a pair of pendulum clocks coupled through a wooden bar oscillate in harmony. A remarkable feature in Huygens’ experiment is that different synchronous behaviors may be observed by just changing a parameter in the coupling. Motivated by this, in this paper, we propose a novel synchronization scheme for chaotic oscillators, in which the design of the coupling is inspired in Huygens’ experiment. It is demonstrated that the coupled oscillators may exhibit not only complete synchronization, but also mixed synchronization—some states synchronize in anti-phase whereas other states synchronize in-phase—depending on a single parameter of the coupling. Additionally, the stability of the synchronous solution is investigated by using the master stability function approach and the largest transverse Lyapunov exponent. The Lorenz system is considered as particular application example, and the performance of the proposed synchronization scheme is illustrated with computer simulations and validated by means of experiments using electronic circuits.

A new two-parameter controllable multi-scroll 4D Hamiltonian conservative hyperchaotic system with improved nested COS-PWL function

A new two-parameter controllable multi-scroll Hamiltonian conservative hyperchaotic system is proposed in this paper. By utilizing an improved nested cosine-piecewise linear (COS-PWL) saturation function in combination with a 4D Hamiltonian conservative system, a 4D Hamiltonian conservative hyperchaotic system that can regulate multi-scroll according to two parameters is proposed. Compared with the nested Sine-PWL function, the system can produce hyperchaos. And the control parameter is more than just the saturation value E, the gain p is more sensitive to the wider control range of the system’s multi-scroll. The effect of the variation of the saturation value E and gain p of the control parameters on the phase trajectories of the multi-scroll is consistent with the behavior of the Lyapunov exponents (LEs) and bifurcations. Finally, we successfully constructed the system via the DSP hardware platform, demonstrating its practicality.

Nonlinear performance of aerospace face gear-planetary gear transmission system under multi-factors including the effect of temperature

Face gear-planetary gear transmission system combines the advantages of both planetary and face gears. And has enormous advantages in the aerospace field. The meshing of gear teeth generates heat through friction, resulting in thermal deformation and affects nonlinear dynamic characteristics. Exploring the temperature effect on the nonlinear dynamics of the transmission system of face gear-planetary gear is of great significance. Considering the temperature variations effect and time-varying meshing stiffness, tooth side clearance as well as meshing error, a nonlinear dynamic model of the face gear-planetary gear system is established. The nonlinear dynamic characteristics of the system were described by maximum Lyapunov exponent diagram, phase diagram, bifurcation diagram et al. The findings indicate that the effect of temperature variations on the gear system is related to the values of the time-varying mesh stiffness coefficient and mesh damping ratio. When the mesh damping ratio is small, the influence of temperature changes on the bifurcation characteristics of the system is more obvious. As the time-varying mesh stiffness value increases, the temperature rise gradually stabilizes in the range of 15°C to 42°C; Under certain temperature changes, as the meshing damping ratio and time-varying meshing stiffness coefficient increase, the face gear planetary gear system gradually tends to stabilize.

Extreme multi-stability and circuit implementation for a two-ReLU-memristor-based jerk oscillator

Memristor-based oscillation circuits are prone to produce coexisting infinite attractors depending on the initial conditions of memristors, leading to the appearance of extreme multi-stability. In this paper, we propose a novel memristive jerk oscillator by bringing two ReLU-type memristors in a simple jerk oscillator and investigate its dynamical behaviors associated with the coupling parameters using bifurcation plots and Lyapunov exponent plots. Further, we discuss the planar equilibrium state and its stability, and then numerically explore the coexisting infinite attractors driven by the initial conditions of two ReLU-type memristors. Because of the intervention of the two ReLU-type memristors, the memristive jerk oscillator has a planar equilibrium state whose stability closely relies on the initial conditions of two ReLU-type memristors, and different initial conditions cause different attractors to coexist, resulting in bidirectional extreme multi-stability. Finally, the memristive jerk oscillator is implemented by analog circuit and digital hardware platform, and the numerical results are confirmed by circuit simulations and hardware experiments.