Lyapunov Exponents Research Articles

Applications of Finite-Time Lyapunov Exponent in detecting Lagrangian Coherent Structures for coastal ocean processes: a review

Addressing the threats of climate change, pollution, and overfishing to marine ecosystems necessitates a deeper understanding of coastal and oceanic fluid dynamics. Within this context, Lagrangian Coherent Structures (LCS) emerge as essential tools for elucidating the complexities of marine fluid dynamics. Methods used to detect LCS include geometric, probabilistic, cluster-based and braid-based approaches. Advancements have been made to employ Finite-time Lyapunov Exponents (FTLE) to detect LCS due to its high efficacy, reliability and simplicity. It has been proven that the FTLE approach has provided invaluable insights into complex oceanic phenomena like shear, confluence, eddy formations, and oceanic fronts, which also enhanced the understanding of tidal-/wind-driven processes. Additionally, FTLE-based LCS were crucial in identifying barriers to contaminant dispersion and assessing pollutant distribution, aiding environmental protection and marine pollution management. FTLE-based LCS has also contributed significantly to understanding ecological interactions and biodiversity in response to environmental issues. This review identifies pressing challenges and future directions of FTLE-based LCS. Among these are the influences of external factors such as river discharges, ice formations, and human activities on ocean currents, which complicate the analysis of ocean fluid dynamics. While 2D FTLE methods have proven effective, their limitations in capturing the full scope of oceanic phenomena, especially in 3D environments, are evident. The advent of 3D LCS analysis has marked progress, yet computational demands and data quality requirements pose significant hurdles. Moreover, LCS extracted from FTLE fields involves establishing an empirical threshold that introduces considerable variability due to human judgement. Future efforts should focus on enhancing computational techniques for 3D analyses, integrating FTLE and LCS into broader environmental models, and leveraging machine learning to standardize LCS detection.

Effects of metronome walking on long-term attractor divergence and correlation structure of gait: a validation study in older people

This study investigates the effects of metronome walking on gait dynamics in older adults, focusing on long-range correlation structures and long-range attractor divergence (assessed by maximum Lyapunov exponents). Sixty older adults participated in indoor walking tests with and without metronome cues. Gait parameters were recorded using two triaxial accelerometers attached to the lumbar region and to the foot. We analyzed logarithmic divergence of lumbar acceleration using Rosenstein’s algorithm and scaling exponents for stride intervals from foot accelerometers using detrended fluctuation analysis (DFA). Results indicated a concomitant reduction in long-term divergence exponents and scaling exponents during metronome walking, while short-term divergence remained largely unchanged. Furthermore, long-term divergence exponents and scaling exponents were significantly correlated. Reliability analysis revealed moderate intrasession consistency for long-term divergence exponents, but poor reliability for scaling exponents. Our results suggest that long-term divergence exponents could effectively replace scaling exponents for unsupervised gait quality assessment in older adults. This approach may improve the assessment of attentional involvement in gait control and enhance fall risk assessment.

Lyapunov stability of suspension bridges in turbulent flow

In the era of sleek, super slender suspension bridges, facing the issue of stability against dynamic wind actions represents an increasingly complex challenge. Despite the significant progress over the last decades, the impact of atmospheric turbulence on bridge stability remains partially not understood, evoking the need for innovative research approaches. This study aims to address a gap in current research by investigating the random flutter stability associated with variations in the angle of attack due to turbulence, which has not formally been addressed yet. The present investigation employs the 2D rational function approximation model to express self-excited forces in a turbulent flow. The application of this type of models to bridge dynamics yields a viscoelastic coupled dynamic system characterized by memory effects and driven by broad-band long-time-scale noise, described here by a linear homogeneous time-variant differential equation, which shows apparent nonlinear features, and which has rarely been matter of research. Utilizing a Monte Carlo methodology, this work innovates in applying the largest Lyapunov exponent (LE) and the moment Lyapunov exponents (MLE) to study bridge random flutter stability. The calculation of LE and MLE under diverse turbulent wind conditions uncovers lower flutter stability than without turbulence effects. In most cases, sample and low-order p-th moment stability thresholds closely align with the bridge dynamic response pattern; therefore, the flutter critical wind speed is unequivocal. However, under certain turbulence scenarios, it is necessary to resort to MLE for a complete description of stability, evoking some additional consideration of which statistical moments should be considered for the engineering assessment of the flutter limit. Finally, this work provides a qualitative insight into the instability mechanisms by approximating the random parametric excitation with a sinusoidal gust and evaluating the time-periodic system stability via Floquet theory.

Delayed feedback active suspension control for chaos in quarter car model with jumping nonlinearity

Vehicles that operate on off-road terrain, such as rough and unpaved roads, sometimes suffer from severe vibrations. This vibration causes vehicle tires to lose contact with the supporting ground and subsequently collide with it. This jumping leads to nonlinear impact dynamics similar to those of a bouncing ball. In this study, a quarter car model with jumping nonlinearity is newly developed to represent the impact dynamics of an off-road vehicle under harsh terrain conditions. Subsequently, complicated vibrations, such as quasi-periodic and chaotic vibrations in the developed model, are identified using a bifurcation diagram and Lyapunov exponents. Additionally, delayed feedback (DF) control of the active suspension is designed for the developed model. The numerical results show that DF control is effective at eliminating chaos and reducing vibration levels. The noise robustness of the DF control is validated for sinus excitation and small random road noise. The results indicate that DF is effective even when road noise exists. Thus, this study demonstrates the feasibility of DF active suspension control for future suspension control design of the off-road vehicle.

Chaotic Transport of Solutes in Unsaturated Porous Media.

Unsaturated porous media, characterized by the combined presence of several immiscible fluid phases in the pore space, are highly relevant systems in nature, because they control the fate of contaminants and the availability of nutrients in the subsoil. However, a full understanding of the mechanisms controlling solute mixing in such systems is still missing. In particular, the role of saturation in the development of chaotic solute mixing has remained unexplored. Using three-dimensional numerical simulations of flow and transport at the pore scale, built upon X-ray tomograms of a porous medium at different degrees of liquid (wetting)-phase saturation, we show the occurrence of chaotic dynamics in both the deformation of the solute plume, as characterized by computed chaos metrics (Lyapunov exponents), and the mixing of the injected solute. Our results show an enhancement of these chaotic dynamics at lower saturation and their occurrence even under diffusion-relevant conditions over the medium's length, also being strengthened by larger flow velocities. These findings highlight the dominant role of the pore-scale spatial heterogeneity of the system, enhanced by the presence of an immiscible phase (e.g., air), on the mixing efficiency. This represents a stepping stone for the assessment of mixing and reactions in unsaturated porous media.

The design, circuit realization and applications of chaotic system with offset-boosting and multistability

Based on the foundation of a three-dimensional Jerk chaotic system, a novel four-dimensional chaotic system based on a memristor with cosine magneto-memristor is proposed. The system’s dynamical behavior is analyzed using bifurcation diagrams, Lyapunov exponents, multiple bifurcation diagrams, two-parameter Lyapunov exponents, complexity, and attractor basins. It is found that this new chaotic system exhibits characteristics such as multistability, transient chaos, intermittent chaos, spike discharge, controllable multi-vortex, and offset-boosting. The chaotic system is designed and simulated in analog circuit and realized in a digital circuit using a Field-Programmable Gate Array (FPGA). The consistency between simulation results and implementation results demonstrates the existence and feasibility of the system. Finally, the significance and superiority of the control scheme are verified based on the adaptive anti-synchronization control strategy, and the synchronization strategy is applied to image encryption, and the results show that the encryption effect is excellent and the anti-jamming ability is strong.

Black hole bulk-cone singularities

Lorentzian correlators of local operators exhibit surprising singularities in theories with gravity duals. These are associated with null geodesics in an emergent bulk geometry. We analyze singularities of the thermal response function dual to propagation of waves on the AdS Schwarzschild black hole background. We derive the analytic form of the leading singularity dual to a bulk geodesic that winds around the black hole. Remarkably, it exhibits a boundary group velocity larger than the speed of light, whose dual is the angular velocity of null geodesics at the photon sphere. The strength of the singularity is controlled by the classical Lyapunov exponent associated with the instability of nearly bound photon orbits. In this sense, the bulk-cone singularity can be identified as the universal feature that encodes the ubiquitous black hole photon sphere in a dual holographic CFT. To perform the computation analytically, we express the two-point correlator as an infinite sum over Regge poles, and then evaluate this sum using WKB methods. We also compute the smeared correlator numerically, which in particular allows us to check and support our analytic predictions. We comment on the resolution of black hole bulk-cone singularities by stringy and gravitational effects into black hole bulk-cone “bumps”. We conclude that these bumps are robust, and could serve as a target for simulations of black hole-like geometries in table-top experiments.

Symmetry-Optimized Dynamical Analysis of Optical Soliton Patterns in the Flexibly Supported Euler–Bernoulli Beam Equation: A Semi-Analytical Solution Approach

This study employs spatial optimization principles to investigate the nonlinear vibration of a flexibly supported Euler–Bernoulli beam, a (1 + 1)-dimensional system subjected to axial loads. The modified Khater method, a crucial tool in mechanical engineering, is utilized to analyze analytical solutions, which include a symmetric spatial representation of the waveform as an integral part of each solution. Notably, periodic soliton solutions for the nonlinear model closely align with numerical and approximate analytical solutions, demonstrating the accuracy of our modeling approach. Density diagrams, contour diagrams, and Poincaré maps depicting the obtained analytical solutions are presented to elucidate their accuracy and provide visual confirmation of the optimized engineering model’s physical significance. The planar dynamical system is derived through the Galilean transformation by employing mathematical models and appropriate parameter values, thereby further refining problem understanding. Sensitivity analysis is conducted, and phase portraits with equilibrium points are illustrated by analyzing a special case of the investigated dynamical system, emphasizing its symmetrical properties. Lastly, we perform a global analysis to identify periodic, quasi-periodic, and chaotic behaviors, with an extra weak forcing term confirmed by Poincaré maps and a two-dimensional symmetric basin of the largest Lyapunov exponent.

Analysis of Multistability of Financial Risk Chaos Systems and Its Application to Voice Cryptography

In the chaos literature, the application of modeling and control of dynamic systems in chaos theory arising in several fields is investigated. In this article we analyze complex financial chaos systems with countries as interest rates, investment demand, and price indices. The proposed chaotic flow's dynamic behavior is examined using phase portraits, eigenvalues, bifurcation diagrams, and Lyapunov exponent spectra. A significant quantity of research on secure communication systems has been published in recent years as a result of the major advancements in communications equipment and encryption techniques. A new voice encryption algorithm design is given using a financial chaos model. An application for voice encryption is conducted using the suggested algorithm, and the outcomes are described.

On New Symmetric Fractional Discrete-Time Systems: Chaos, Complexity, and Control

This paper introduces a new symmetric fractional-order discrete system. The dynamics and symmetry of the suggested model are studied under two initial conditions, mainly a comparison of the commensurate order and incommensurate order maps, which highlights their effect on symmetry-breaking bifurcations. In addition, a theoretical analysis examines the stability of the zero equilibrium point. It proves that the map generates typical nonlinear features, including chaos, which is confirmed numerically: phase attractors are plotted in a two-dimensional (2D) and three-dimensional (3D) space, bifurcation diagrams are drawn with variations in the derivative fractional values and in the system parameters, and we calculate the Maximum Lyapunov Exponents (MLEs) associated with the bifurcation diagram. Additionally, we use the C0 algorithm and entropy approach to measure the complexity of the chaotic symmetric fractional map. Finally, nonlinear 3D controllers are revealed to stabilize the symmetric fractional order map’s states in commensurate and incommensurate cases.

Stability and bifurcations of a host–parasitoid model with general host escape function and general stocking upon parasitoids

This paper analyzes the generalization of a model presented in J. Bektešević, V. Hadžiabdić, S. Kalabušić, M. Mehuljić and E. Pilav [Dynamics of a class of host–parasitoid models with external stocking upon parasitoids, Adv. Differ. Equ. 2021(31) (2021)]. The study explores the behavior of the solution near equilibrium points when the system has different outcomes, such as extinction, infinitely many exclusion points or unique exclusion and coexistence. We prove global stability for the extinction and host-exclusion equilibrium. We also investigate the non-hyperbolic case of parasitoid-exclusion equilibrium and delve deeper into the 1:1 resonance. The transcritical bifurcation occurs at the host-exclusion equilibrium, indicating a threshold for host population invasion through transcritical bifurcation. Moreover, the local dynamics around the coexisting equilibrium can be highly complex due to the appearance of the Neimark–Sacker and period-doubling bifurcations. We provide the explicit form of the first Lyapunov exponent for the Neimark–Sacker bifurcation. Through numerical examples, we illustrate the theoretical findings.

On fractional discrete financial system: Bifurcation, chaos, and control

Abstract The dynamic analysis of financial systems is a developing field that combines mathematics and economics to understand and explain fluctuations in financial markets. This paper introduces a new three-dimensional (3D) fractional financial map and we dissect its nonlinear dynamics system under commensurate and incommensurate orders. As such, we evaluate when the equilibrium points are stable or unstable at various fractional orders. We use many numerical methods, phase plots in 2D and 3D projections, bifurcation diagrams and the maximum Lyapunov exponent. These techniques reveal that financial maps exhibit chaotic attractor behavior. This study is grounded on the Caputo-like discrete operator, which is specifically influenced by the variance of the commensurate and incommensurate orders. Furthermore, we confirm the presence and measure the complexity of chaos in financial maps by the 0–1 test and the approximate entropy algorithm. Additionally, we offer nonlinear-type controllers to stabilize the fractional financial map. The numerical results of this study are obtained using MATLAB.

Pseudo-random number generation based on spatial chaotic map of Logistic type and its cryptographic application

The pseudo-random number generator (PRNG) based on chaos has been widely used in the fields of digital communication, cryptography and computer simulation. In this paper, we study a new PRNG based on spatial surface chaotic system (SSCS), which is constructed based on coupling mapping lattice (CML) and one-dimensional Logistic chaotic map. To verify the performance of this generator, we study its nonlinear dynamic properties, including the Lyapunov exponent, ergodicity and bifurcation phenomena. Moreover, we analyze the cryptographic properties such as key space, key sensitivity, correlation, histogram and information entropy, while performing NIST SP800-22 test and TestU01 test for the randomness of this system. Theoretical analysis and numerical simulation results show that the PRNG proposed in this paper has good complexity, ergodicity, sensitivity and randomness. Moreover, the key space can increase dynamically with the increase of encrypted data, especially suitable for the protection of big data such as multimedia, which is more advantageous than most existing chaotic mapping. The results of this paper will provide a new idea for the study of chaotic cryptography, and motivate the study of new discrete chaotic models with desirable statistical properties.

Lyapunov exponents and phase structure of Lifshitz and hyperscaling violating black holes

We study the phase structure of Lifshitz and hyperscaling violating (HSV) black holes using Lyapunov exponents. For describing hyperscaling violating system, we chose a particular gravity model constructed from generalized Einstein-Maxwell-Dilaton action which includes the Lifshitz cases at appropriate limits. We study the relationship between Lyapunov exponents and black hole phase transitions considering both the timelike and null geodesics. We observe that, the black hole phase transiton properties are reflected in Lyapunov exponent where its multiple branches correspond to the distinct phases of the black hole. The discontinuos change of the Lyapunov exponent during the phase transition serve as an order parameter with critical exponent 1/2 near the critical point. Our numerical study reveals that the correlation between the Lyapunov exponent and black hole thermodynamic properties can be generalised beyond the AdS spacetime. We find that it is independent of the HSV parameter as well as the Lifshitz exponent.

Bifurcations of phase portraits and chaotic behaviors of the (2+1)-dimensional double‐chain DNA system with beta derivative: A qualitative approach

Qualitative analysis in mathematical modeling has become an important research area within the broad domain of nonlinear sciences. In the realm of qualitative analysis, the bifurcation method is one of the significant approaches for studying the structure of orbits in nonlinear dynamical systems. To apply the bifurcation method to the (2 + 1)-dimensional double-chain Deoxyribonucleic Acid system with beta derivative, the bifurcations of phase portraits and chaotic behaviors, combined with sensitivity and multi-stability analysis of this system, are examined. Initially, the bifurcations of phase portraits are visually identified at the obtained equilibrium points of a planar dynamical system via both Hamiltonian and Jacobian algorithms. The obtained results indicate Jacobian algorithm is more efficient in identifying the stability of bifurcations than the Hamiltonian algorithm for this system. Subsequently, by introducing an external perturbation term into the planar dynamical system, the chaotic behavior is effectively identified by using a variety of tools, such as two- and three-dimensional phase portraits, time series, Lyapunov exponents, and Poincaré maps. The findings suggest that the perturbed dynamical system deviates from regular patterns and exhibits behavior ranging from periodic to quasi-periodic and from quasi-periodic to chaotic. Finally, the sensitivity and multi-stability of the system are examined using the Runge-Kutta method to assess the model's response to minor variations in initial conditions through numerical solutions, revealing that the model is sensitive and multi-stable. The outcomes of this study will enhance a relationship between applied mathematicians and experimental biologists, helping to explore hidden features of Deoxyribonucleic Acid through the studied model.

Discontinuous transition to chaos in a canonical random neural network.

We study a paradigmatic random recurrent neural network introduced by Sompolinsky, Crisanti, and Sommers (SCS). In the infinite size limit, this system exhibits a direct transition from a homogeneous rest state to chaotic behavior, with the Lyapunov exponent gradually increasing from zero. We generalize the SCS model considering odd saturating nonlinear transfer functions, beyond the usual choice ϕ(x)=tanhx. A discontinuous transition to chaos occurs whenever the slope of ϕ at 0 is a local minimum [i.e., for ϕ^{'''}(0)>0]. Chaos appears out of the blue, by an attractor-repeller fold. Accordingly, the Lyapunov exponent stays away from zero at the birth of chaos.

Accounting for observation noise in equation‐free forecasting: The hidden‐Markov S‐map

Abstract Ecosystems contain numerous species interacting with each other and the environment. However, data on all relevant state variables is rarely available, hampering inference and prediction. Empirical dynamic modelling (EDM) is a valuable tool for prediction, inference and control in such partially observed systems. However, EDM typically assumes that the available time series are observed without error. Failing to account for observation noise strongly biases estimates of Lyapunov exponents and reduces forecast accuracy. To address this limitation, we propose incorporating EDM into a hidden Markov framework and using an iterative scheme based on the expectation maximization (EM) algorithm to obtain filtered state and parameter estimates. We evaluate the performance of this approach on several simulated dynamical systems with a range of additive noise levels, as well as on insect population time series. Accounting for observation noise improved accuracy of population forecasts and estimates of Lyapunov exponents (LE) over a wide range of noise levels relevant to ecological time series.

Discrete superior dynamics of a generalized chaotic system

In the past few decades, the discrete dynamics of difference maps have attained the remarkable attention of researchers owing to their incredible applications in different domains, like cryptography, secure communications, weather forecasting, traffic flow models, neural network models, and population biology. In this article, a generalized chaotic system is proposed, and superior dynamics is disclosed through fixed point analysis, time-series evolution, cobweb representation, period-doubling, period-3 window, and Lyapunov exponent properties. The comparative bifurcation and Lyapunov plots report the superior stability and chaos performance of the generalized system. It is interesting to notice that the generalized system exhibits superior dynamics due to an additional control parameter β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta $$\\end{document}. Analytical and numerical simulations are used to explore the superior dynamical characteristics of the generalized system for some specific values of parameter β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta $$\\end{document}. Further, it is inferred that the superiority in dynamics of the generalized system may be efficiently used for better future applications.

Coastal current convergence structures in the Bay of Biscay from optimized high-frequency radar and satellite data

The southeastern Bay of Biscay has been described as a “dead end” for floating marine litter, often accumulating along small-scale linear streaks. Coastal Current Convergence Structures (CCS), often associated with vertical motions at river plume edges, estuarine fronts, or other physical processes, can be at the origin of the accumulation. Understanding the formation of CCS and their role in the transport of marine litter is essential to better quantify and to help mitigate marine litter pollution. The Lagrangian framework, used to estimate the absolute dispersion, and the finite-size Lyapunov exponents (FSLE), have proved very effective for identifying CCS in the current velocity field. However, the quality of CCS identification depends strongly on the Eulerian fields. Two surface current velocity data sets were used in the analysis: the remotely sensed velocities from the EuskOOS High-Frequency Radar (HFR) network and velocities from three-dimensional model outputs. They were complemented by drifting buoy velocity measurements. An optimization method, involving the fusion of drifting buoys and HFR velocities is proposed to better reconstruct the fine-scale structure of the current velocity field. Merging these two sources of velocity data reduced the mean Lagrangian error and the Root Mean Square Error (RMSE) by 50 % and 30 % respectively, significantly improving velocity reconstruction. FSLE ridgelines obtained from the Lagrangian analysis of optimized velocities were compared with remotely sensed concentrations of Chlorophyll-a. It was shown that ridgelines control the spatial distribution of phytoplankton. They fundamentally represent the CCS which can potentially affect marine litter aggregation. Analysis of the absolute dispersion revealed large stirring in the alongshore direction which was also confirmed by spatial distribution of FSLE ridgelines. The alignment between FSLE ridgelines and patterns of high Chlorophyll-a concentration was observed, often determining the limits of river plume expansion in the study area.

Complex dynamic behaviors in a small network of three ring coupled Rayleigh-Duffing oscillators: Theoretical study and circuit simulation

This work focuses on the dynamics of a small network of three ring-coupled unidirectional Rayleigh-Duffing oscillators. The equations governing the Rayleigh-Duffing oscillator, containing a cubic term, make this study a more interesting and complex case to analyze. Coupling is achieved by perturbing the amplitude of each oscillator with a signal proportional to the amplitude of the other. The sixth-order self-driven nonlinear system obtained after coupling is analyzed, and presents up to twenty seven equilibrium points. Amongst these equilibrium points, we determined which can present the Hopf bifurcation. Also, the effects of the coupling coefficients and damping coefficients are analyzed. It is shown that varying these different coefficients leads to the appearance of extremely complex dynamic phenomena such as: instability and bifurcations (i.e coexistence of bifurcation branches), coexistence of up to fifteen attractors (heterogeneous multistability) and eight spiral chaotic attractor. The investigation of the coupled system is carried out by using to both analytical and numerical tools such as Hopf bifurcation theorem, the phase portraits, bifurcation diagrams, Lyapunov exponent diagram, frequency spectrum, to name but a few. The Routh-Hurwitz criterion is also used to analyze the stability of equilibrium points. We compute basins of attraction to highlight different zones corresponding to coexisting attractors. The implementation of an analog circuit of coupled Rayleigh-Duffing oscillators has enabled us to confirm the analytical and numerical results.