Effect of Data and Gap Characteristics on the Nonlinear Calculation of Motion During Locomotor Activities.

This study investigated how data series length and gaps in human kinematic data impact the accuracy of Lyapunov exponents (LyE) calculations with and without cubic spline interpolation. Kinematic time series were manipulated to create various data series lengths (28% and 100% of original) and gap durations (0.05-0.20 s). Longer gaps generally resulted in significantly higher LyE% error values in each plane in noninterpolated data. During cubic spline interpolation, only the 0.20-second gap in frontal plane data resulted in a significantly higher LyE% error. Data series length did not significantly affect LyE% error in noninterpolated data. During cubic spline interpolation, sagittal plane LyE% errors were significantly higher at shorter versus longer data series lengths. These findings suggest that not interpolating gaps in data could lead to erroneously high LyE values and mischaracterization of movement variability. When applying cubic spline, a long gap length (0.20 s) in the frontal plane or a short sagittal plane data series length (1000 data points) could also lead to erroneously high LyE values and mischaracterization of movement variability. These insights emphasize the necessity of detailed reporting on gap durations, data series lengths, and interpolation techniques when characterizing human movement variability using LyE values.

Stability analysis of chaotic systems from data.

The prediction of the temporal dynamics of chaotic systems is challenging because infinitesimal perturbations grow exponentially. The analysis of the dynamics of infinitesimal perturbations is the subject of stability analysis. In stability analysis, we linearize the equations of the dynamical system around a reference point and compute the properties of the tangent space (i.e. the Jacobian). The main goal of this paper is to propose a method that infers the Jacobian, thus, the stability properties, from observables (data). First, we propose the echo state network (ESN) with the Recycle validation as a tool to accurately infer the chaotic dynamics from data. Second, we mathematically derive the Jacobian of the echo state network, which provides the evolution of infinitesimal perturbations. Third, we analyse the stability properties of the Jacobian inferred from the ESN and compare them with the benchmark results obtained by linearizing the equations. The ESN correctly infers the nonlinear solution and its tangent space with negligible numerical errors. In detail, we compute from data only (i) the long-term statistics of the chaotic state; (ii) the covariant Lyapunov vectors; (iii) the Lyapunov spectrum; (iv) the finite-time Lyapunov exponents; (v) and the angles between the stable, neutral, and unstable splittings of the tangent space (the degree of hyperbolicity of the attractor). This work opens up new opportunities for the computation of stability properties of nonlinear systems from data, instead of equations. Supplementary Information: The online version contains supplementary material available at 10.1007/s11071-023-08285-1.

Quantum Bounds on the Generalized Lyapunov Exponents.

We discuss the generalized quantum Lyapunov exponents Lq, defined from the growth rate of the powers of the square commutator. They may be related to an appropriately defined thermodynamic limit of the spectrum of the commutator, which plays the role of a large deviation function, obtained from the exponents Lq via a Legendre transform. We show that such exponents obey a generalized bound to chaos due to the fluctuation-dissipation theorem, as already discussed in the literature. The bounds for larger q are actually stronger, placing a limit on the large deviations of chaotic properties. Our findings at infinite temperature are exemplified by a numerical study of the kicked top, a paradigmatic model of quantum chaos.

Permanence via invasion graphs: incorporating community assembly into modern coexistence theory.

To understand the mechanisms underlying species coexistence, ecologists often study invasion growth rates of theoretical and data-driven models. These growth rates correspond to average per-capita growth rates of one species with respect to an ergodic measure supporting other species. In the ecological literature, coexistence often is equated with the invasion growth rates being positive. Intuitively, positive invasion growth rates ensure that species recover from being rare. To provide a mathematically rigorous framework for this approach, we prove theorems that answer two questions: (i) When do the signs of the invasion growth rates determine coexistence? (ii) When signs are sufficient, which invasion growth rates need to be positive? We focus on deterministic models and equate coexistence with permanence, i.e., a global attractor bounded away from extinction. For models satisfying certain technical assumptions, we introduce invasion graphs where vertices correspond to proper subsets of species (communities) supporting an ergodic measure and directed edges correspond to potential transitions between communities due to invasions by missing species. These directed edges are determined by the signs of invasion growth rates. When the invasion graph is acyclic (i.e. there is no sequence of invasions starting and ending at the same community), we show that permanence is determined by the signs of the invasion growth rates. In this case, permanence is characterized by the invasibility of all [Formula: see text] communities, i.e., communities without species i where all other missing species have negative invasion growth rates. To illustrate the applicability of the results, we show that dissipative Lotka-Volterra models generically satisfy our technical assumptions and computing their invasion graphs reduces to solving systems of linear equations. We also apply our results to models of competing species with pulsed resources or sharing a predator that exhibits switching behavior. Open problems for both deterministic and stochastic models are discussed. Our results highlight the importance of using concepts about community assembly to study coexistence.

Chaotic dynamics in a novel COVID-19 pandemic model described by commensurate and incommensurate fractional-order derivatives.

Mathematical models based on fractional-order differential equations have recently gained interesting insights into epidemiological phenomena, by virtue of their memory effect and nonlocal nature. This paper investigates the nonlinear dynamic behavior of a novel COVID-19 pandemic model described by commensurate and incommensurate fractional-order derivatives. The model is based on the Caputo operator and takes into account the daily new cases, the daily additional severe cases, and the daily deaths. By analyzing the stability of the equilibrium points and by continuously varying the values of the fractional order, the paper shows that the conceived COVID-19 pandemic model exhibits chaotic behaviors. The system dynamics are investigated via bifurcation diagrams, Lyapunov exponents, time series, and phase portraits. A comparison between integer-order and fractional-order COVID-19 pandemic models highlights that the latter is more accurate in predicting the daily new cases. Simulation results, besides to confirming that the novel fractional model well fit the real pandemic data, also indicate that the numbers of new cases, severe cases, and deaths undertake chaotic behaviors without any useful attempt to control the disease. Supplementary Information: The online version contains supplementary material available at 10.1007/s11071-021-06867-5.

A Novel Measure Inspired by Lyapunov Exponents for the Characterization of Dynamics in State-Transition Networks.

The combination of network sciences, nonlinear dynamics and time series analysis provides novel insights and analogies between the different approaches to complex systems. By combining the considerations behind the Lyapunov exponent of dynamical systems and the average entropy of transition probabilities for Markov chains, we introduce a network measure for characterizing the dynamics on state-transition networks with special focus on differentiating between chaotic and cyclic modes. One important property of this Lyapunov measure consists of its non-monotonous dependence on the cylicity of the dynamics. Motivated by providing proper use cases for studying the new measure, we also lay out a method for mapping time series to state transition networks by phase space coarse graining. Using both discrete time and continuous time dynamical systems the Lyapunov measure extracted from the corresponding state-transition networks exhibits similar behavior to that of the Lyapunov exponent. In addition, it demonstrates a strong sensitivity to boundary crisis suggesting applicability in predicting the collapse of chaos.

Taming the Chaos in Neural Network Time Series Predictions.

Machine learning methods, such as Long Short-Term Memory (LSTM) neural networks can predict real-life time series data. Here, we present a new approach to predict time series data combining interpolation techniques, randomly parameterized LSTM neural networks and measures of signal complexity, which we will refer to as complexity measures throughout this research. First, we interpolate the time series data under study. Next, we predict the time series data using an ensemble of randomly parameterized LSTM neural networks. Finally, we filter the ensemble prediction based on the original data complexity to improve the predictability, i.e., we keep only predictions with a complexity close to that of the training data. We test the proposed approach on five different univariate time series data. We use linear and fractal interpolation to increase the amount of data. We tested five different complexity measures for the ensemble filters for time series data, i.e., the Hurst exponent, Shannon's entropy, Fisher's information, SVD entropy, and the spectrum of Lyapunov exponents. Our results show that the interpolated predictions consistently outperformed the non-interpolated ones. The best ensemble predictions always beat a baseline prediction based on a neural network with only a single hidden LSTM, gated recurrent unit (GRU) or simple recurrent neural network (RNN) layer. The complexity filters can reduce the error of a random ensemble prediction by a factor of 10. Further, because we use randomly parameterized neural networks, no hyperparameter tuning is required. We prove this method useful for real-time time series prediction because the optimization of hyperparameters, which is usually very costly and time-intensive, can be circumvented with the presented approach.

Combining Measures of Signal Complexity and Machine Learning for Time Series Analyis: A Review.

Measures of signal complexity, such as the Hurst exponent, the fractal dimension, and the Spectrum of Lyapunov exponents, are used in time series analysis to give estimates on persistency, anti-persistency, fluctuations and predictability of the data under study. They have proven beneficial when doing time series prediction using machine and deep learning and tell what features may be relevant for predicting time-series and establishing complexity features. Further, the performance of machine learning approaches can be improved, taking into account the complexity of the data under study, e.g., adapting the employed algorithm to the inherent long-term memory of the data. In this article, we provide a review of complexity and entropy measures in combination with machine learning approaches. We give a comprehensive review of relevant publications, suggesting the use of fractal or complexity-measure concepts to improve existing machine or deep learning approaches. Additionally, we evaluate applications of these concepts and examine if they can be helpful in predicting and analyzing time series using machine and deep learning. Finally, we give a list of a total of six ways to combine machine learning and measures of signal complexity as found in the literature.

Novel fractional order SIDARTHE mathematical model of COVID-19 pandemic.

Nowadays, COVID-19 has put a significant responsibility on all of us around the world from its detection to its remediation. The globe suffer from lockdown due to COVID-19 pandemic. The researchers are doing their best to discover the nature of this pandemic and try to produce the possible plans to control it. One of the most effective method to understand and control the evolution of this pandemic is to model it via an efficient mathematical model. In this paper, we propose to model COVID-19 pandemic by fractional order SIDARTHE model which did not appear in the literature before. The existence of a stable solution of the fractional order COVID-19 SIDARTHE model is proved and the fractional order necessary conditions of four proposed control strategies are produced. The sensitivity of the fractional order COVID-19 SIDARTHE model to the fractional order and the infection rate parameters are displayed. All studies are numerically simulated using MATLAB software via fractional order differential equation solver.

Application of the nonlinear methods in pneumocardiogram signals.

In this work, the pneumocardiogram signals of nine rats were analysed by scale index, Boltzmann Gibbs entropy and maximum Lyapunov exponents. The scale index method, based on wavelet transform, was proposed for determining the degree of aperiodicity and chaos. It means that the scale index parameter is close to zero when the signal is periodic and has a value between zero and one when the signal is aperiodic. A new entropy calculation method by normalized inner scalogram was suggested very recently. In this work, we also used this method for the first time in an empirical data. We compared the both methods with maximum Lyapunov exponents and observed that using together the scale index and the entropy calculation method by normalized inner scalogram increases the reliability of the pneumocardiogram signal analysis. Thus, the analysis of the pneumocardiogram signals by those methods enables to compare periodical and/or nonlinear aspects for further understanding of dynamics of cardiorespiratory system.

Local dynamic stability in temporal pattern of intersegmental coordination during various stride time and stride length combinations.

For the regulation of walking speed, the central nervous system must select appropriate combinations of stride time and stride length (stride time-length combinations) and coordinate many joints or segments in the whole body. However, humans achieve both appropriate selection of stride time-length combinations and effortless coordination of joints or segments. Although this selection of stride time-length combination has been explained by minimized energy cost, it may also be explained by the stability of kinematic coordination. Therefore, we investigated the stability of kinematic coordination during walking across various stride time-length combinations. Whole body kinematic coordination was quantified as the kinematic synergies that represents the groups of simultaneously move segments (intersegmental coordination) and their activation patterns (temporal coordination). In addition, the maximum Lyapunov exponents were utilized to evaluate local dynamic stability. We calculated the maximum Lyapunov exponents in temporal coordination of kinematic synergies across various stride time-length combinations. The results showed that the maximum Lyapunov exponents of temporal coordination depended on stride time-length combinations. Moreover, the maximum Lyapunov exponents were high at fast walking speeds and very short stride length conditions. This result implies that fast walking speeds and very short stride length were associated with lower local dynamic stability of temporal coordination. We concluded that fast walking is associated with lower local dynamic stability of temporal coordination of kinematic synergies.

d-Dimensional Classical Heisenberg Model with Arbitrarily-Ranged Interactions: Lyapunov Exponents and Distributions of Momenta and Energies.

We numerically study the first-principle dynamics and thermostatistics of a d-dimensional classical inertial Heisenberg ferromagnetic model ( d = 1 , 2 , 3 ) with interactions decaying with the distance r i j as 1 / r i j α ( α ≥ 0 ), where the limit α = 0 ( α → ∞ ) corresponds to infinite-range (nearest-neighbour) interactions, and the ratio α / d > 1 ( 0 ≤ α / d ≤ 1 ) characterizes the short-ranged (long-ranged) regime. By means of first-principle molecular dynamics we study: (i) The scaling with the system size N of the maximum Lyapunov exponent λ in the form λ ∼ N - κ , where κ ( α / d ) depends only on the ratio α / d ; (ii) The time-averaged single-particle angular momenta probability distributions for a typical case in the long-range regime 0 ≤ α / d ≤ 1 (which turns out to be well fitted by q-Gaussians), and (iii) The time-averaged single-particle energies probability distributions for a typical case in the long-range regime 0 ≤ α / d ≤ 1 (which turns out to be well fitted by q-exponentials). Through the Lyapunov exponents we observe an intriguing, and possibly size-dependent, persistence of the non-Boltzmannian behavior even in the α / d > 1 regime. The universality that we observe for the probability distributions with regard to the ratio α / d makes this model similar to the α -XY and α -Fermi-Pasta-Ulam Hamiltonian models as well as to asymptotically scale-invariant growing networks.

Short- and long-term effects of altered point of ground reaction force application on human running energetics.

The current study investigates the effect of altering the point of force application (PFA) from the rearfoot towards the fore of the foot on the metabolic energy consumption and the influence of transitioning to this technique over a short or a longer timeframe. The participants were randomly assigned into two experimental and one control group: a short-term intervention group (STI, N=17; two training sessions), a long-term intervention group (LTI, N=10; 14-week gradual transition) and a control group (CG, N=11). Data were collected at two running velocities (2.5 and 3.0 m s-1). The cost coefficient (i.e. energy required per unit of vertical ground reaction force; J N-1) decreased (P<0.001) after both interventions due to a more anterior PFA during running (STI: 12%, LTI: 11%), but led to a higher (P<0.001) rate of force generation (STI: 17%, LTI: 15.2%). Dynamic stability of running showed a significant (P<0.001) decrease in the STI (2.1%), but no differences (P=0.673) in the LTI. The rate of metabolic energy consumption increased in the STI (P=0.038), but remained unchanged in the LTI (P=0.660). The CG had no changes. These results demonstrate that the cost coefficient was successfully decreased following an alteration in the running technique towards a more anterior PFA. However, the energy consumption remained unchanged because of a simultaneous increase in rate of force generation due to a decreased contact time per step. The increased instability found during the short-term intervention and its neutralization after the long-term intervention indicates a role of motor control errors in the economy of running after acute alterations in habitual running execution.

Modeling the odor-landscape resulting from the pumping behavior of bivalve clams in the presence of predators.

Motivated by experimental findings, a computational fluid dynamics (CFD) model was used to investigate whether the clam Mercenaria mercenaria may alter its cue downstream variability by an exhalant random pumping behavior. This behavior was hypothesized to occur in the presence of predator chemical signals in order to prevent successful tracking by the predator. Simulated downstream flow and mixing conditions derived from the random nature of the clam exhalant jet in a crossflow were analyzed by computing an intermittency factor, determining the field of finite-time Lyapunov exponents (FTLEs) and identifying the resulting Lagrangian coherent structures (LCSs). Numerical simulations illustrate that the effectiveness of a fluctuating exhalant jet to prevent downstream tracking by a crab, depends on the ratio of the exhalant jet to the crossflow. Specifically, the clam could effectively enhance the downstream dispersion to prevent tracking, but only in the range of parameters where LCSs are generated (jet-to-crossflow ratio ≥ 1). Then, the probability of detection is reduced with respect to the case of a less fluctuating exhalant jet.

Strange Attractors Generated by Multiple-Valued Static Memory Cell with Polynomial Approximation of Resonant Tunneling Diodes.

This paper brings analysis of the multiple-valued memory system (MVMS) composed by a pair of the resonant tunneling diodes (RTD). Ampere-voltage characteristic (AVC) of both diodes is approximated in operational voltage range as common in practice: by polynomial scalar function. Mathematical model of MVMS represents autonomous deterministic dynamical system with three degrees of freedom and smooth vector field. Based on the very recent results achieved for piecewise-linear MVMS numerical values of the parameters are calculated such that funnel and double spiral chaotic attractor is observed. Existence of such types of strange attractors is proved both numerically by using concept of the largest Lyapunov exponents (LLE) and experimentally by computer-aided simulation of designed lumped circuit using only commercially available active elements.

Quantifying Chaos by Various Computational Methods. Part 2: Vibrations of the Bernoulli-Euler Beam Subjected to Periodic and Colored Noise.

In this part of the paper, the theory of nonlinear dynamics of flexible Euler-Bernoulli beams (the kinematic model of the first-order approximation) under transverse harmonic load and colored noise has been proposed. It has been shown that the introduced concept of phase transition allows for further generalization of the problem. The concept has been extended to a so-called noise-induced transition, which is a novel transition type exhibited by nonequilibrium systems embedded in a stochastic fluctuated medium, the properties of which depend on time and are influenced by external noise. Colored noise excitation of a structural system treated as a system with an infinite number of degrees of freedom has been studied.

Quantifying Chaos by Various Computational Methods. Part 1: Simple Systems.

The aim of the paper was to analyze the given nonlinear problem by different methods of computation of the Lyapunov exponents (Wolf method, Rosenstein method, Kantz method, the method based on the modification of a neural network, and the synchronization method) for the classical problems governed by difference and differential equations (Hénon map, hyperchaotic Hénon map, logistic map, Rössler attractor, Lorenz attractor) and with the use of both Fourier spectra and Gauss wavelets. It has been shown that a modification of the neural network method makes it possible to compute a spectrum of Lyapunov exponents, and then to detect a transition of the system regular dynamics into chaos, hyperchaos, and others. The aim of the comparison was to evaluate the considered algorithms, study their convergence, and also identify the most suitable algorithms for specific system types and objectives. Moreover, an algorithm of calculation of the spectrum of Lyapunov exponents based on a trained neural network has been proposed. It has been proven that the developed method yields good results for different types of systems and does not require a priori knowledge of the system equations.

Automated Diagnosis of Heart Sounds Using Rule-Based Classification Tree.

In order to assist the diagnosis procedure of heart sound signals, this paper presents a new automated method for classifying the heart status using a rule-based classification tree into normal and three abnormal cases; namely the aortic valve stenosis, aortic insufficient, and ventricular septum defect. The developed method includes three main steps as follows. First, one cycle of the heart sound signals is automatically detected and segmented based on time properties of the heart signals. Second, the segmented cycle is preprocessed with the discrete wavelet transform and then largest Lyapunov exponents are calculated to generate the dynamical features of heart sound time series. Finally, a rule-based classification tree is fed by these Lyapunov exponents to give the final decision of the heart health status. The developed method has been tested successfully on twenty-two datasets of normal heart sounds and murmurs with success rate of 95.5%. The resulting error can be easily corrected by modifying the classification rules; consequently, the accuracy of automated heart sounds diagnosis is further improved.

Relationship between dynamical entropy and energy dissipation far from thermodynamic equilibrium.

Connections between microscopic dynamical observables and macroscopic nonequilibrium (NE) properties have been pursued in statistical physics since Boltzmann, Gibbs, and Maxwell. The simulations we describe here establish a relationship between the Kolmogorov-Sinai entropy and the energy dissipated as heat from a NE system to its environment. First, we show that the Kolmogorov-Sinai or dynamical entropy can be separated into system and bath components and that the entropy of the system characterizes the dynamics of energy dissipation. Second, we find that the average change in the system dynamical entropy is linearly related to the average change in the energy dissipated to the bath. The constant energy and time scales of the bath fix the dynamical relationship between these two quantities. These results provide a link between microscopic dynamical variables and the macroscopic energetics of NE processes.

A dynamical systems analysis of assisted and unassisted anterior and posterior hand-held load carriage.

Load carriage is recognised as a primary occupational factor leading to slip and fall injuries, and therefore assessing balance maintenance during such tasks is critical in assessing injury risk. Ten males completed 55 strides under five carriage conditions: (1) unassisted anterior, (2) unassisted posterior, (3) assisted anterior, (4) assisted posterior and (5) unloaded gait (UG). Kinematic data were recorded from markers affixed to landmarks on the right side of each participant, in order to calculate segment angles for the foot, shank, thigh and pelvis. Continuous relative phase (CRP) variability was calculated for each segment pair and local dynamic stability was calculated for each segment in all three movement planes. In general, irrespective of the assistive device or movement plane, anterior load carriage was most stable (lower CRP variability and maximum finite-time Lyapunov exponents). Moreover, load carriage was less dynamically stable than UG, displaying the importance of objectively investigating safe load carriage practices. PRACTITIONER SUMMARY: Dynamical systems analyses were used to comprehensively evaluate the stability of various handheld load carriage methods. In general, anterior load carriage was significantly more stable than posterior load carriage,Mover's assistive device had small but beneficial effects on stability, and load carriage was less stable than UG.