Computation Of Lyapunov Exponents Research Articles

Analysis of the Dynamical Properties of Discrete Predator-Prey Systems with Fear Effects and Refuges

This paper examines the dynamic behavior of a particular category of discrete predator-prey system that feature both fear effect and refuge, using both analytical and numerical methods. The critical coefficients and properties of bifurcating periodic solutions for Flip and Hopf bifurcations are computed using the center manifold theorem and bifurcation theory. Additionally, numerical simulations are employed to illustrate the bifurcation phenomenon and chaos characteristics. The results demonstrate that period-doubling and Hopf bifurcations are two typical routes to generate chaos, as evidenced by the calculation of the maximum Lyapunov exponents near the critical bifurcation points. Finally, a feedback control method is suggested, utilizing feedback of system states and perturbation of feedback parameters, to efficiently manage the bifurcations and chaotic attractors of the discrete predator-prey model.

Secure Multiple-Image Transfer by Hybrid Chaos System: Encryption and Visually Meaningful Images

The secure transmission of information is one of the most important topics in the field of information technology. Considering that images contain important visual information, it is crucial to create a safe platform for image transfer. One commonly employed tool to enhance the complexity and randomness in image encryption methods is the chaos system. The logistic and sine maps are utilized in encryption algorithms but these systems have some weaknesses, notably chaotic behavior in a confined area. In this study, to address these weaknesses, a hybrid system based on the Atangana–Baleanu fractional derivative is proposed. The various tests employed to evaluate the behavior of the new system, including the NIST test, histogram analysis, Lyapunov exponent calculation, and bifurcation diagram, demonstrate the efficiency of the proposed system. Furthermore, in comparison to the logistic and sine maps, the proposed hybrid exhibits chaotic behavior over a broader range. This system is utilized to establish a secure environment for the transmission of multiple images within an encryption algorithm, subsequently concealing them within a meaningful image. Various tools employed to assess the security of the proposed algorithm, including histogram analysis, NPCR, UACI, and correlation values, indicate that the proposed hybrid system has application value in encryption.

Efficient numerical calculation of Lyapunov-exponents and stability assessment for quasi-periodic motions in nonlinear systems

Investigating the stability of stationary motions is a highly relevant aspect when characterizing dynamical systems. For equilibria and periodic motions, well established theories and approaches exist to assess their stability: in both cases stability may be assessed using eigenvalue analyses of small perturbations. When it comes to quasi-periodic motions, such eigenvalue analyses are not applicable, since these motions can not be parameterized on finite time intervals. However, quasi-periodic motions can be densely embedded on finite invariant manifolds with periodic boundaries. In this contribution, a new approach is presented, which exploits this embedding in order to derive a sequence of finite mappings. Based on these mappings, the spectrum of 1st order Lyapunov-exponents is efficiently calculated. If the linearization of the problem is regular in the sense of Lyapunov, these exponents may be used to assess stability of the investigated solution. Beyond the numerical calculation of Lyapunov-exponents, an approach is presented which allows to check Lyapunov-regularity numerically. Together, both methods allow for an efficient numerical stability assessment of quasi-periodic motions. To demonstrate, verify and validate the developed approach, it is applied to quasi-periodic motions of two coupled van-der-Pol oscillators as well as a quasi-periodically forced Duffing equation. Additionally, a “step-by-step application instruction” is provided to increase comprehensibility and to discuss the required implementation steps in an applied context.

Processing alcoholism eeg signals using neural networks

Alcoholism is one of the most common diseases, which creates problems for contemporary society. To date, there are still no standardized methods and tests to detect this disease. This study presents a new methodology based on an integrated approach to the study of Electroencephalography (EEG) signals, using a combination of nonlinear dynamics and machine learning methods. We applied signal encoding in the form of a 16-channel image, which was fed into the ResNet20 and GoogLeNet models for training. Wavelet transforms and fast Fourier transforms were considered for data preprocessing. In addition to the application of neural networks, the study of EEG signals was carried out by computing Lyapunov exponents and multiscale entropy. The question of choosing the most effective maternal wavelet was previously investigated. The wavelet transformations of Morlet, Meyer, Mexican Hat, Daubechies, and Gauss were studied. A spectrum of five Lyapunov exponents was calculated using the Sano – Savada method. To confirm the reliability of the results, the senior Lyapunov exponent was previously calculated by three more methods: Wolf, Kantz, Rosenstein. Previously, the methods were tested on such classical problems as the hyperchaotic Hénon map, logistic map, Rössler attractor and Lorenz attractor. The use of several approaches for the analysis of EEG signals confirmed the reliability of the results. The study demonstrates that the GoogLeNet model is more efficient than ResNet50. The experiment results showed that, based on the developed methodology, the GoogLeNet model is more efficient than ResNet50. The results of the experiment show that the wavelet transform, compared to the Fourier transform, gives higher accuracy for the analysis of EEG signals of patients with alcoholism.

A trio of maps: chaos on an ellipse, a shark’s fin, and Sluze’s pearls

Three one-dimensional maps are presented. The maps are analysed with a combination of Feigenbaum diagrams, the calculation of global Lyapunov exponents, and the evaluation of fixed points. It is shown that in certain cases period two orbits can be evaluated by careful observation of the form of the related pairs of equations. A range of classroom exercises are provided to enable students to investigate the maps further.

Stability, bifurcation, and control: Modeling interaction of the predator-prey system with Alles effect

In this study, we investigate the resilience and bifurcation properties of a discrete-time predator-prey model incorporating an Allee effect, the transfer of prey to predator energy, and a necessary threshold for prey consumption prior to predator reproduction. The discrete-time formulation involves iteratively adjusting the predator and prey populations at distinct time steps. The model encompasses equations that depict the fluctuations in population sizes, considering processes like breeding, predation, death, and the influence of the Allee effects. The investigation results indicate that the system's boundary equilibrium goes through a period-doubling bifurcation. Moreover, when the system is in close proximity to the exclusive positive equilibrium influenced by the prey-predator population, it exhibits Neimark-Sacker bifurcations. It's worth noting that our observations suggest that the predator population might approach extinction or achieve a stable equilibrium when chaotic dynamics appear in the prey population. The validity of this phenomenon is confirmed using the Lyapunov exponents calculation, which verifies the range of diverse dynamic behaviors. The outcomes and analysis of this research offer intriguing insights in both mathematical and biological realms. Implementing methods for chaos control could be beneficial in stabilizing or mitigating the occurrence of chaotic dynamics. Additionally, the theoretical results are supported by numerical simulations, phase portraits, bifurcation diagrams, graphs depicting chaos control, and maximum Lyapunov exponent graphs.

A relative investigation of one-dimensional chaotic maps intended for light-weight cryptography in smart grid

Technology is quite important in today's contemporary society, especially when it comes to information transmission. The free and open interchange of data via networks is essential for the development of communication technologies. As a result, it is essential to ensure the security of transmitted data, especially digital data like readings from smart meters in smart grids. As a result, the study of chaos-based cryptography has received a great deal of interest in the realm of data security. Due to its favourable characteristics, including ergodicity, pseudo-randomness, non-periodicity, and sensitivity to beginning circumstances, chaotic maps have become a popular choice for data encryption. Researchers have concentrated on factors like the highest Lyapunov exponent and sensitivity to beginning circumstances while evaluating these maps. The MATLAB platform has been used to run simulations that produce bifurcation diagrams and calculate Lyapunov exponents in order to analyse and compare various chaotic maps. Further research is required since these experiments have shown considerable improvements in the chaotic behaviour of one-dimensional maps over time. There are two key areas covered by the review. On the basis of their bifurcation diagrams and Lyapunov exponents, it compares newly presented maps in the first place. It then looks at the techniques used to combine one-dimensional chaotic maps. The work shows the potential of chaotic maps for cryptography applications and emphasises the growing significance of data security in the digital age.

Hoeffding’s independence test for an ion dynamics characterization in the octupole trap

In the present work, we propose a rapid analysis of the trapped ion dynamics regimes. The Hoeffding’s Independence Test (HIT) mapping technique has been considered as a promising approach for the detection of dynamic regimes. As a model problem, we have applied the proposed method to ion dynamics in an octupole ion trap. We compare the results of HIT mapping with the results of direct calculation of the non-trivial Lyapunov exponent. The HIT mapping result well agree with a Lyapunov exponent mapping. The computational time for plotting the HIT maps was significantly less than the computational time for plotting the Lyapunov exponent maps.

Deep Hierarchical Super Resolution for Scientific Data.

We present a novel technique for hierarchical super resolution (SR) with neural networks (NNs), which upscales volumetric data represented with an octree data structure to a high-resolution uniform grid with minimal seam artifacts on octree node boundaries. Our method uses existing state-of-the-art SR models and adds flexibility to upscale input data with varying levels of detail across the domain, instead of only uniform grid data that are supported in previous approaches. The key is to use a hierarchy of SR NNs, each trained to perform 2× SR between two levels of detail, with a hierarchical SR algorithm that minimizes seam artifacts by starting from the coarsest level of detail and working up. We show that our hierarchical approach outperforms baseline interpolation and hierarchical upscaling methods, and demonstrate the usefulness of our proposed approach across three use cases including data reduction using hierarchical downsampling+SR instead of uniform downsampling+SR, computation savings for hierarchical finite-time Lyapunov exponent field calculation, and super-resolving low-resolution simulation results for a high-resolution approximation visualization.

Low-dimensional behavior of a Kuramoto model with inertia and Hebbian learning

We study low-dimensional dynamics in a Kuramoto model with inertia and Hebbian learning. In this model, the coupling strength between oscillators depends on the phase differences between the oscillators and changes according to a Hebbian learning rule. We analyze the special case of two coupled oscillators, which yields a five-dimensional dynamical system that decouples into a two-dimensional longitudinal system and a three-dimensional transverse system. We readily write an exact solution of the longitudinal system, and we then focus our attention on the transverse system. We classify the stability of the transverse system’s equilibrium points using linear stability analysis. We show that the transverse system is dissipative and that all of its trajectories are eventually confined to a bounded region. We compute Lyapunov exponents to infer the transverse system’s possible limiting behaviors, and we demarcate the parameter regions of three qualitatively different behaviors. Using insights from our analysis of the low-dimensional dynamics, we examine the original high-dimensional system in a situation in which we draw the intrinsic frequencies of the oscillators from Gaussian distributions with different variances.

Parameters Analysis of a Fractional Order Chaotic Chua System

The problem of analyzing the parameters of a chaotic system of fractional order is considered. As an example, a chaotic system based on a Chua circuit of an integer order was chosen. To obtain a Chua circuit of fractional order, it is proposed to replace the capacitive elements in the original circuit with elements with fractal impedance. The purpose of the work is to create a mathematical model of a fractional order Chua circuit with real values of the circuit elements, on the basis of which to evaluate existing software products in the Matlab environment suitable for modeling chaotic fractional order systems, as well as to search for software products suitable for calculating the spectrum of Lyapunov exponents. An analysis of ready-made solutions in the Matlab environment, from the literature known to us, showed that the most suitable option for modeling a system of differential equations of a fractional order is the «fde_pi12_pc» program. The selected program allows you to set all the system parameters of interest and has a relatively high accuracy due to the use of the predictor-corrector algorithm. Based on this program, a domain for determining fractional indicators of the Chua system was constructed in which the presence of a characteristic attractor is visually observed. Existing software products in the Matlab environment are analyzed, allowing the calculation of Lyapunov exponents for chaotic systems described by fractional order differential equations. Of the solutions considered, the best, in our opinion, is the «FO_NC_Lyapunov» program. Using the selected program, a spatial graph of the spectrum of Lyapunov exponents was constructed when changing the fractional exponents of the Chua system. Based on the spatial graph of the spectrum of Lyapunov exponents and the criteria for the chaotic behavior of the system based on the values of Lyapunov exponents, a definition area was constructed with visualization of zones of chaotic behavior of the Chua system. The domains of definition obtained by visual analysis of the presence of an attractor and calculating the spectrum of Lyapunov exponents for the same mathematical model did not coincide; therefore, there are currently no software products for reliably assessing the chaos of fractional order systems by calculating the spectrum of Lyapunov exponents.

Phase-space Analysis of Ordered and Disordered Nonthermal Ion Energization during Magnetic Reconnection

Anomalous ion heating is frequently observed to accompany magnetic reconnection, yet there is little consensus on its origin. Instead of the usual velocity-space analysis, we use phase-space analysis to exhaustively explain how ions are nonthermally energized during collisionless, antiparallel magnetic reconnection. There are both ordered and disordered aspects in the process; the former is explained in terms of conservative quantities, and the latter is explained by demonstrating chaos through a direct calculation of Lyapunov exponents. The former induces “multibeam-like heating” in all three directions, whereas the latter induces stochastic bulk heating. Profiles of the ion temperature tensor components during reconnection can be easily understood by the phase-space distributions of ions in different motional stages.

Instability of the optimal edge trajectory in the Blasius boundary layer

In the context of linear stability analysis, considering unsteady base flows is notoriously difficult. A generalisation of modal linear stability analysis, allowing for arbitrarily unsteady base flows over a finite time, is therefore required. The recently developed optimally time-dependent (OTD) modes form a projection basis for the tangent space. They capture the leading amplification directions in state space under the constraint that they form an orthonormal basis at all times. The present numerical study illustrates the possibility to describe a complex flow case using the leading OTD modes. The flow under investigation is an unsteady case of the Blasius boundary layer, featuring streamwise streaks of finite length and relevant to bypass transition. It corresponds to the state space trajectory initiated by the minimal seed; such a trajectory is unsteady, free from any spatial symmetry and shadows the laminar–turbulent separatrix for a finite time only. The finite-time instability of this unsteady base flow is investigated using the 8 leading OTD modes. The analysis includes the computation of finite-time Lyapunov exponents as well as instantaneous eigenvalues, and of the associated flow structures. The reconstructed instantaneous eigenmodes are all of outer type. They map unambiguously the spatial regions of largest instantaneous growth. Other flow structures, previously reported as secondary, are identified with this method as relevant to streak switching and to streamwise vortical ejections. The dynamics inside the tangent space features both modal and non-modal amplification. Non-normality within the reduced tangent subspace, quantified by the instantaneous numerical abscissa, emerges only as the unsteadiness of the base flow is reduced.

Exploring a family of Bernoulli-like shift chaotic maps and its amplitude control

Chaos theory is a branch of mathematics that studies systems highly sensitive to initial conditions. Various methods for analyzing and controlling these systems, such as bifurcation theory and amplitude control, have been developed and applied in diverse fields, including cryptography, physics, and engineering.In this study, we introduce a new class of Bernoulli-Like Shift (BLS) maps. These maps are generated by using a specific class of Piecewise Linear (PWL) maps combined with a modulo operation. We conduct a fixed point analysis by establishing propositions and corollaries, which outline the conditions that guarantee the presence or absence of fixed points. Additionally, we provide numerical examples to validate these theoretical results.Utilizing bifurcation diagrams and Lyapunov exponent calculations, we demonstrate that the range of chaotic behavior in the BLS maps is broader than those of the logistic and Bernoulli Shift maps. Moreover, we show that the BLS maps possess higher values of Lyapunov exponents and a better distribution than both the Bernoulli shift and logistic maps.Lastly, we introduce a transformation that generates a modified BLS map, allowing for amplitude control. This modification enables the regulation of amplitude while maintaining constant Lyapunov exponents.

Marine ecosystem models of realistic complexity rarely exhibits significant endogenous non-stationary dynamics

Despite the observation of cyclic and chaotic dynamics in nature, it is still not clear whether this behaviour is inherent to ecological systems or caused by external forcings. Here we explored a set of approximately 210,000 simulations to assess how often a model of realistic complexity exhibits non-stationary dynamics when external perturbations are excluded. Remarkably, less than one third of the population shown non-stationary dynamics and, even when present, fluctuations were rather small. The inherent stability of plankton communities showed to be related to the presence of multiple feedbacks in the food web structure, omnivory, low centre of gravity, and supports the conclusion that food webs of realistic complexity rarely exhibit significant endogenous non-stationary dynamics. Finally, we computed Lyapunov exponents for the non-stationary trajectories, in order to assess in which proportion they were periodic or chaotic, and we concluded that less than 10% of the non-stationary trajectories (3% of the total) showed sensitivities to initial conditions. This further supports the conclusion that complex topology mainly damps endogenous fluctuations in the food web.

Characterization of Aeroelastic Behavior in a High Aspect Ratio Wing Using Computational and Wind Tunnel Experiments

The objective of this article is to characterize an aeroelastic system in terms of its dynamical behavior, which could be either chaotic or periodic before, during, and after achieving the flutter velocity. The aeroelastic system shown here is a wing with a high aspect ratio, which leads to a very flexible behavior subjected to unsteady flow. This paper compares the computational and experimental dynamical behavior of an aeroelastic system at the flutter velocity for the different dynamic stall models proposed. To understand the nonlinear behavior of this system, the traditional attractor reconstruction and Lyapunov exponent calculation are compared with the 0–1 test. In addition to this comparison, two dynamic stall semi-empirical models are applied directly to the time history. All these comparisons show that the computational and wind tunnel experiments are in good agreement, and the dynamic behavior usually gives close results for the 0–1 test and Lyapunov exponent. It is concluded that the system presents chaotic behavior when no dynamic stall correction is applied or when Gangwani’s correction is applied. However, Boeing–Vertol’s correction postpones the chaotic behavior, meaning that the chaotic behavior is only observed for velocities above the flutter.

Sustainability and robust decision-support strategy for multi-echelon supply chain system against disruptions

ABSTRACT Due to the higher nonlinearity and parametric sensitivity of multi-echelon supply chains, minor changes may result in a completely different evolution of supply chain capabilities. Realizing resilient and sustainable development goals for enterprise logistics requires great insights into management schemes in a volatile market. This paper deals with dynamic characteristics and resilience issues of managing multi-echelon supply chain systems against disruptions. Discrete-time nonlinear systems with time delay are extensively evaluated via computation of Lyapunov exponent, bifurcation diagram, permutation entropy, and Poincaré map. For time-series evolution, forecasting and predictability are assessed by the Hurst exponent. An echo state network (ESN) will be utilized to predict the future states of the supply chain dynamics. Based on a decision-support strategy with an ESN model, a resilience strategy concerning disturbing factors is implemented to ensure a sustainable supply chain network. Numerical simulations are demonstrated to validate dynamical behaviors and verify the efficiency of control theory with machine learning, guaranteeing the resilience and sustainability of a supply chain against market volatility. The test results demonstrate how machine learning algorithms can be integrated with management schemes. Finally, the presented methodology can ensure more supply chain visibility and traceability under highly uncertain markets.

A Novel Conservative Chaotic System Involved in Hyperbolic Functions and Its Application to Design an Efficient Colour Image Encryption Scheme

It is well known that, compared to low-dimension chaotic systems, three-dimensional chaotic systems have a wider parameter range, more complicated behaviour, and better unpredictability. This fact motivated us to introduce a novel image encryption method that employs a three-dimensional chaotic system. We proposed a novel three-dimensional conservative system that can exhibit chaotic behaviour involving hyperbolic functions. The dynamical behaviours of the proposed system are discovered by calculating Lyapunov exponents and bifurcation diagrams. Thereafter, we designed an image encryption method based on the proposed system and a 4×4 self-invertible matrix. A modified Diffie–Hellman key exchange protocol was utilised to generate the self-invertible key matrix Km employed in the diffusion stage. Our approach has three main stages. In the first stage, the proposed three-dimensional system utilises the original image to create three sequences, two of which are chosen for confusion and diffusion processes. The next stage involves confusing the image’s pixels by changing the positions of pixels using these sequences. In the third stage, the confused image is split into sub-blocks of size 4×4, and each block is encrypted by multiplying it with Km. Simulation findings demonstrated that the proposed image scheme has a high level of security and is resistant to statistical analysis, noise, and other attacks.

A GPU-Accelerated Particle Advection Methodology for 3D Lagrangian Coherent Structures in High-Speed Turbulent Boundary Layers

In this work, we introduce a scalable and efficient GPU-accelerated methodology for volumetric particle advection and finite-time Lyapunov exponent (FTLE) calculation, focusing on the analysis of Lagrangian coherent structures (LCS) in large-scale direct numerical simulation (DNS) datasets across incompressible, supersonic, and hypersonic flow regimes. LCS play a significant role in turbulent boundary layer analysis, and our proposed methodology offers valuable insights into their behavior in various flow conditions. Our novel owning-cell locator method enables efficient constant-time cell search, and the algorithm draws inspiration from classical search algorithms and modern multi-level approaches in numerical linear algebra. The proposed method is implemented for both multi-core CPUs and Nvidia GPUs, demonstrating strong scaling up to 32,768 CPU cores and up to 62 Nvidia V100 GPUs. By decoupling particle advection from other problems, we achieve modularity and extensibility, resulting in consistent parallel efficiency across different architectures. Our methodology was applied to calculate and visualize the FTLE on four turbulent boundary layers at different Reynolds and Mach numbers, revealing that coherent structures grow more isotropic proportional to the Mach number, and their inclination angle varies along the streamwise direction. We also observed increased anisotropy and FTLE organization at lower Reynolds numbers, with structures retaining coherency along both spanwise and streamwise directions. Additionally, we demonstrated the impact of lower temporal frequency sampling by upscaling with an efficient linear upsampler, preserving general trends with only 10% of the required storage. In summary, we present a particle search scheme for particle advection workloads in the context of visualizing LCS via FTLE that exhibits strong scaling performance and efficiency at scale. Our proposed algorithm is applicable across various domains, requiring efficient search algorithms in large, structured domains. While this article focuses on the methodology and its application to LCS, an in-depth study of the physics and compressibility effects in LCS candidates will be explored in a future publication.

Homoclinic orbit and the violation of the chaos bound around a black hole with anisotropic matter fields

We study the homoclinic orbit and the violation of chaos bound, which are obtained by particle motions around a black hole that coexist with anisotropic matter fields. The homoclinic one is associated with an unstable local maximum of the effective potential. By perturbing a particle located slightly away from the homoclinic one, we numerically compute Lyapunov exponents indicating the sensitivity of the initial value. Our results demonstrate that the violation of the chaos bound increases with higher angular momentum, and the anisotropic matter gives rise to violating the chaos bound further, even in the case of the nonextremal black hole. We utilize the Hamiltonian-Jacobi formalism to explicitly illustrate how the geodesic motion of a particle can be integrable in the procedure of obtaining our findings.