Analysis Of Nonlinear Dynamical Systems Research Articles

Proper generalized decomposition-based iterative enrichment process combined with shooting method for steady-state forced response analysis of nonlinear dynamical systems

AbstractThis paper presents a novel framework combining proper generalized decomposition (PGD) with the shooting method to determine the steady-state response of nonlinear dynamical systems upon a general periodic input. The proposed PGD approximates the response as a low-rank separated representation of the spatial and temporal dimensions. The Galerkin projection is employed to formulate the subproblem for each dimension, then the fixed-point iteration is applied. The subproblem for the spatial vector can be regarded as computing a set of reduced-order basis vectors, and the shooting problem projected onto the subspace spanned by these basis vectors is defined to obtain the temporal coefficients. From this procedure, the proposed framework replaces the complex nonlinear time integration of the full-order model with the series of solving simple iterative subproblems. The proposed framework is validated through two descriptive numerical examples considering the conventional linear normal mode method for comparison. The results show that the proposed shooting method based on PGD can accurately capture nonlinear characteristics within 10 modes, whereas linear modes cannot easily approximate these behaviors. In terms of computational efficiency, the proposed method enables CPU time savings of about one order of magnitude compared with the conventional shooting methods.

A LAPACK Implementation of the Dynamic Mode Decomposition

The Dynamic Mode Decomposition (DMD) is a method for computational analysis of nonlinear dynamical systems in data driven scenarios. Based on high fidelity numerical simulations or experimental data, the DMD can be used to reveal the latent structures in the dynamics or as a forecasting or a model order reduction tool. The theoretical underpinning of the DMD is the Koopman operator on a Hilbert space of observables of the dynamics under study. This paper describes a numerically robust and versatile variant of the DMD and its implementation using the state-of-the-art dense numerical linear algebra software package LAPACK . The features of the proposed software solution include residual bounds for the computed eigenpairs of the DMD matrix, eigenvectors refinements and computation of the eigenvectors of the Exact DMD, compressed DMD for efficient analysis of high dimensional problems that can be easily adapted for fast updates in a streaming DMD. Numerical analysis is the bedrock of numerical robustness and reliability of the software, that is tested following the highest standards and practices of LAPACK . Important numerical topics are discussed in detail and illustrated using numerous numerical examples.

Representing model uncertainties in brittle fracture simulations

This work focuses on the representation of model-form uncertainties in phase-field models of brittle fracture. Such uncertainties can arise from the choice of the degradation function for instance, and their consideration has been unaddressed to date. The stochastic modeling framework leverages recent developments related to the analysis of nonlinear dynamical systems and relies on the construction of a stochastic reduced-order model. In the latter, a POD-based reduced-order basis is randomized using Riemannian projection and retraction operators, as well as an information-theoretic formulation enabling proper concentration in the convex hull defined by a set of model proposals. The model thus obtained is mathematically admissible in the almost sure sense and involves a low-dimensional hyperparameter, the calibration of which is facilitated through the formulation of a quadratic programming problem. The relevance of the modeling approach is further assessed on one- and two-dimensional applications. It is shown that model uncertainties can be efficiently captured and propagated to macroscopic quantities of interest. An extension based on localized randomization is also proposed to handle the case where the forward simulation is highly sensitive to sample localization. This work constitutes a methodological development allowing phase-field predictions to be endowed with statistical measures of confidence, accounting for the variability induced by modeling choices.

Multiscale entropy analysis of single lead ECG and ECG derived respiration for AI based prediction of sleep apnea events

Sleep apnea (SA) is a prevalent sleep disorder that affects a significant portion of the adult population. The proposed method involves coarse graining a signal at different scales, using the popular multiscale entropy algorithms to detect apnea and normal events from single lead Electrocardiography (ECG), Instantaneous Heart Rate and ECG derived respiration. The idea of nonlinear dynamical systems analysis for feature extraction along with machine learning approach, the authors present here a moderately accurate sleep apnea detection model with an accuracy ranging from 70% to 100% based on two different probabilistic thresholds of 50 % and 70% with reduction in false positive rate from 28% to 14 % for applications in development of AI based IoT connected smart wearable devices. The two thresholding modes offer a choice to do a trade-off between high training accuracy with high false positive rate and low accuracy with low false positive rate. The performance metrics of each model have been reported and choice is made for best suitable model with prospectives of implementation for diagnostic purposes. The effectiveness of the proposed method in accurate detection of sleep apnea requires further validation of the models with out of distribution dataset.

Application of a Psychosocial Approach to the Identification and Strengthening of Adaptation Mechanisms of Humans and a Small Social Group during the Isolation Experiment “SIRIUS 2017–2023”

TOPIC: The task of the 21st century is the implementation of manned flights in Earth’s orbit with the view to building orbital and planetary bases. This requires addressing the impacts on people and small social groups in terms of psychological, psychosocial, physiological and health. The author presents her own comprehensive research and intervention approach to exploring and supporting the operation of the space crew in the four-month isolation period of “SIRIUS-18/19”, which can be used in the future for manned flights into deep space. GOAL: The main goal is to present three main areas, within the implementation of social research, designed to analyze the operation of the crew in a simulated space flight: 1. WORKING CONDITIONS, WORKING ENVIRONMENT AND SOCIAL ATMOSPHERE; 2. the STRUCTURE AND DYNAMICS OF RELATIONSHIPS and TIES; 3. a set of other specific areas. The key outputs of the comprehensive analysis of the “SIRIUS-19” crew operations concerning the level of satisfaction with the working environment and conditions, the structure and dynamics of relationships and other specific areas are presented. The suitability of the implementation of intervention activities for isolated crews is pointed out. The purpose is to contribute to the preparation of human crews for manned flights in deep space and to reduce the risks of damage to human biopsychosocial health. METHODS: For a comprehensive analysis, a set of the author’s own questionnaire methods, verified over 25 years in the normal and extremely demanding conditions of specific professions, was used. The diagnostic and intervention method sociomapping, based on fuzzy theory and the mathematical modeling of outputs, was used for the analysis of the structure and dynamics of relationships as it is a technique suitable for the analysis of nonlinear dynamical systems. The methodology enabled the author to obtain a comprehensive view of the experimental situation from a psychosocial and sociological point of view. RESULTS: The model of the author’s analytical approach confirmed the legitimacy of its implementation in the case of isolation experiments. A comprehensive analysis of the “SIRIUS-18/19” crew’s work environment yielded outputs from the 10 main and 48 sub-areas analyzed. The analysis of the six-member, gender-mixed, multicultural crew in the area of structure and dynamics of relationships focused on 35 areas; a total of 344 sociomaps were created. The files were analyzed qualitatively and quantitatively using control diagrams. CONCLUSIONS: Outputs have the potential to be used in other isolation experiments as sociotechnical measures for project organizers and as verification of the need to introduce work with the crew in the form of development workshops using the sociomapping method.

The mpEDMD Algorithm for Data-Driven Computations of Measure-Preserving Dynamical Systems

Koopman operators globally linearize nonlinear dynamical systems and their spectral information is a powerful tool for the analysis and decomposition of nonlinear dynamical systems. However, Koopman operators are infinite dimensional, and computing their spectral information is a considerable challenge. We introduce measure-preserving extended dynamic mode decomposition (mpEDMD), the first Galerkin method whose eigendecomposition converges to the spectral quantities of Koopman operators for general measure-preserving dynamical systems. mpEDMD is a data-driven algorithm based on an orthogonal Procrustes problem that enforces measure-preserving truncations of Koopman operators using a general dictionary of observables. It is flexible and easy to use with any preexisting dynamic mode decomposition (DMD)-type method, and with different types of data. We prove convergence of mpEDMD for projection-valued and scalar-valued spectral measures, spectra, and Koopman mode decompositions. For the case of delay embedding (Krylov subspaces), our results include the first convergence rates of the approximation of spectral measures as the size of the dictionary increases. We demonstrate mpEDMD on a range of challenging examples, its increased robustness to noise compared to other DMD-type methods, and its ability to capture the energy conservation and cascade of a turbulent boundary layer flow with Reynolds number and state-space dimension .

Novel Girsanov correction based Milstein schemes for analysis of nonlinear multi-dimensional stochastic dynamical systems

This work proposes Girsanov corrected explicit, semi-implicit, and implicit Milstein approximations for the solution of nonlinear stochastic differential equations. The solution trajectories provided by the Milstein schemes are corrected by employing the change of measures, aimed at removing the error associated with the diffusion process incurred due to the transformation between two probability measures. The change of measures invoked in the Milstein schemes ensures that the solution from the mapping is measurable with respect to the filtration generated by the error process. The proposed scheme incorporates the error between the approximated mapping and the exact representation as an innovation that is accounted in the Milstein trajectories as an additive term. Numerical demonstrations using parametrically and non-parametrically excited stochastic oscillators, a practical problem involving ring type gyroscope, and a 7-degree-of-freedom nonlinear structural system subjected to both stationary and non-stationary excitation demonstrate the improvement in the solution accuracy for the proposed schemes with much coarser time steps when compared with the classical Milstein approximation with finer time steps.

Effects of climate change on vegetation dynamics of the Qinghai-Tibet Plateau, a causality analysis using empirical dynamic modeling

Given the vital role of the Qinghai-Tibet Plateau (QTP) as water tower in Asia and regulator for regional and even global climate, the relationship between climate change and vegetation dynamics on it has received considerable focused attention. Climate change may influence the vegetation growth on the plateau, but clear empirical evidence of such causal linkages is sparse. Herein, using datasets CRU-TS v4.04 and AVHHR NDVI from 1981 to 2019, we quantify causal effects of climate factors on vegetation dynamics with an empirical dynamical model (EDM) -- a nonlinear dynamical systems analysis approach based on state-space reconstruction rather than correlation. Results showed the following: (1) climate change promotes the growth of vegetation on the QTP, and specifically, this favorable influence of temperature is stronger than precipitation's; (2) the direction and strength of climate effects on vegetation varied over time, and the effects are seasonally different; (3) a significant increase in temperature and a slight increase in precipitation are beneficial to vegetation growth, specifically, NDVI will increase within 2% in the next 40 years with the climate trend of warming and humidity. Besides the above results, another interesting finding is that the two seasons in which precipitation strongly influence vegetation in the Three-River Source region (part of the QTP) are spring and winter. This study provides insights into the mechanisms by which climate change affects vegetation growth on the QTP, aiding in the modeling of vegetation dynamics in future scenarios.

Entropy for team communication pattern recognition

This work aims to demonstrate entropy as a method that can be used to analyze team coordination using naturalistic team communication content. Much of team coordination happens through communication; understanding team communication is vital to better formulate and train teams for successful performance outcomes. Several decades of team communication research have led to various methods for analyzing team communication patterns. Many existing methods for team communication analysis have not been tested with naturalistic communication or only consider the frequency or flow of communication. Sliding-window entropy is used to analyze team coordination dynamics with team communication as a proxy. The resulting time series are evaluated using nonlinear dynamical systems analysis and clustering. Various team coordination patterns are identified using communication entropy at the team level. Entropy can be used to identify team communication patterns and their relation to team performance. While team coordination happens at the team level, a posteriori analysis indicates members' individual characteristics affect the overall team coordination patterns. In teams where the contribution is unequal, some members disproportionately affect the overall team coordination, which may undermine the team impact and affect team performance.

Fuzzy Graph Modeling and Clustering Analysis of Nonlinear Dynamical Systems

Analyzing stability and designing control strategies for interconnected nonlinear dynamical systems poses mathematical challenges due to combinatorial growth in complexity. This paper develops a methodology integrating fuzzy graph theory with spectral clustering techniques to enable tractable certification of stability properties. Fuzzy similarity relations between state variables model imprecise couplings within the nonlinear dynamics. Representing these relations through graph adjacency matrices facilitates partitioning strongly connected states using spectral algorithms. Stability of the overall fuzzy state graph is inferred from the spectral radii of decoupled fuzzy subgraphs. The graph-theoretic abstraction provides a coarse-grained lens into nonlinear stability properties while circumventing computational barriers. The fuzzy graph modeling and spectral partitioning pipeline ultimately streamlines control synthesis targeting local clustered subdynamics. Case studies on power network stabilization, swarm navigation, and process operations showcase scalable applications to high-dimensional complex systems. The integrated fuzzy graph and spectral clustering approach provides a systematic toolkit for analysis and control of heavily interconnected nonlinear dynamical systems across engineering domains.

Sequential application of denoising autoencoder and long-short recurrent convolutional network for noise-robust remaining-useful-life prediction framework of lithium-ion batteries

Accurate prediction of the lithium-ion battery lifetime is important to maintain the performance of the battery system. Because data-driven methods are extensively used in the analysis of nonlinear dynamical systems in research, the existing literature is largely focused on the application of these methods for the prediction of battery state. Data-driven methods that are highly dependent on data are sensitive to the noise of the measurement data. Distorted data diminish the performance of data-driven models for lifetime prediction. In this study, we propose an artificial neural network-based framework, which is robust to noise, to increase the prediction accuracy of the remaining-useful-life (RUL) of lithium-ion batteries. A denoising autoencoder trained using a distorted dataset with Gaussian noise and dropout is presented to improve the robustness of the model to noise. Artificial neural network models predict RUL based on the state-of-health estimated using measurement data in the initial step. The proposed prediction model is compared with the base model and the training-noise model using distorted data to validate the robustness to noise. The results demonstrate that the proposed framework is robust to noise and has lower cycle errors compared to other methods.

Recovery and adjustment trajectories among Hurricane Florence survivors: Analysis utilizing nonlinear dynamic system modeling.

Self-regulation shift theory (SRST) argues that most individuals are able to successfully recover from trauma via engagement in self-regulation processes as well as the effective utilization of internal and environmental resources. However, a minority of individuals may instead experience a self-determination violation as a result of their self-regulatory capacity being overwhelmed. This self-determination violation is marked by chaotic and shifting adjustment, maladaptive regulation attempts, and, ultimately, a shift to an impaired self-state and the development of persistent psychopathology, such as posttraumatic stress disorder (PTSD). The current study utilized nonlinear dynamic system (NDS) analysis to identify adjustment trajectory dynamics among rural hurricane survivors in North Carolina (N = 131) who completed daily ecological momentary assessments (EMAs) regarding their distress (i.e., negative mood and PTSD symptoms), regulation efforts (e.g., coping), and appraisals (e.g., coping self-efficacy) over a 6-week period. Four adjustment trajectories were identified, including two largely adaptive trajectories (69.0% and 5.7%), a less stable adjustment trajectory (6.9%), and a fourth trajectory (18.4%) marked by shifting adjustment states and more frequent maladaptive regulation and negative appraisals, suggesting possible self-determination violation. Consistent with this possibility, this final trajectory was also associated with more severe PTSD symptoms relative to the other three trajectories at enrollment and 6-month follow-up. Future work should utilize NDS to model posttrauma adjustment dynamics from within a SRST framework to identify patterns of positive and negative adjustment dynamics at different time points in the trauma recovery process.

A review on the numerical and experimental modeling of the floatover installations

Floatover is a large deck installation method that the integrated deck is slowly transferred from the floatover vessel to the pre-installed substructure by falling tide or ballasting vessel. Compared with other installation methods, the floatover method has the advantages of high efficiency, economy and large lifting capacity, which has been widely used in both fixed and floating platforms. The inherently complex problems of floatover systems involve complex hydrodynamic interactions and nonlinear multi-body coupled dynamics. Establishing an accurate numerical or experimental model to investigate the dynamic responses of the vessel, topside, and substructure, the tension of the mooring system, and the impact loads of load buffer devices induced by environmental loads are essential to the design and field operation of floatover installations. In this review, we introduce the state-of-the-art progress of numerical and experimental modeling and analysis of floatover systems made by industrial and academic researchers, including the modeling and analysis of the load buffer devices, hydrodynamic analysis, and the analysis of the nonstationary nonlinear dynamical system. Finally, the major challenges and future research prospects of floatover are provided.

Nonlinear dynamical systems to study epileptic seizures and extract average amount of mutual information from encephalographs – Part I

In this study, we apply the non-linear dynamical systems theory for the assessment built on recurrence-quantification analysis technique for characterizing—differentiating non-linear electro encephalograph (EEG) signals dynamics. The technique offers convenient quantifiable data plus information over normal, tumultuous, or probability and statistical stochastic properties of inherent systems dynamics theory. The R.Q.A-established processes as the quantifiable mathematical features of non-linear electroencephalograph signals dynamics. Average amount of mutual information (AAMI) applied to compute highly applicable feature-manifestation sub-sets out of R.Q.A-built centered-features. The chosen features were then fed into the computer using artificial intelligence based neural net-works for clustering the data of encephalograph-signals to identify ictic(i.e.,ictal), inter ictal, followed by state of controls. The study is implemented by validating R.Q.A with a data base for various issues of categorization. Results showed that the combination of five selected features created on AAMI attained the precision of100% and proves dominance of R.Q.A. Nonlinear dynamical control systems theory and analysis techniques centered on R.Q.A can be used as an appropriate methodology for distinguishing the non-linear systems dynamics of encephalograph signals data also epileptic seizures tracing.

Robustness Analysis of Nonlinear Systems Along Uncertain Trajectories

The paper presents a novel approach for robustness analysis of nonlinear dynamic systems in the vicinity of a reference trajectory. The approach linearizes the system with respect to a nominal trajectory and calculates a guaranteed upper bound on the worst-case gain. In contrast to existing methods rooted in linear time-varying systems analysis, the approach accurately includes perturbations that drive the system away from the reference trajectory. The approach further includes a bound for the error associated with the time-varying linearization. Hence, the results obtained in the linear framework provide a valid upper bound for the worst-case performance of the nonlinear system. The calculation of the upper bound relies on the dissipation inequalities formulated in the framework of integral quadratic constraints. It is therefore computationally much cheaper than sample-based methods such as Monte Carlo simulation. The feasibility of the approach is demonstrated on a numerical example.

Parallel Subdomain Synthesis of Cell Mapping for Capturing Global Invariant Sets in Higher-Dimensional Dynamical Systems

Unaffordable computational cost and memory storage induced by the curse of dimensionality has become the bottleneck of numerical methods in different fields. In the global analysis of nonlinear dynamical systems, the capability of numerical methods, like cell mapping methods, are mostly feasible only to a system dimension less than four. Although cell mappings are naturally parallelizable that may be used to greatly enhance the computational efficiency, it is still not enough to release the computational burden on a higher-dimensional system of greater than seven, not to mention the memory in dealing with millions of billion cells. In this paper, the subdomain synthesis method, which partitions the chosen region in state space into subdomains suitable for operating in a computational unit and then synthetizes the so-called virtual invariant sets to get the underlying global invariant sets, is promoted to be parallelizable on the subdomains so as to build a two-layer massively parallel architecture in both cell and subdomain levels. The proposed approach can be implemented by GPU Cluster that can maximize the powerful computation capability of hardwares. Examples with global invariant sets in very fine distances of a Jerk system and in bifurcations of a twelve-dimensional nonsmooth rotor system are presented for the first time to demonstrate then the feasibility of the proposed approach.

A non-intrusive frequency normalisation approach for uncertain response analysis of nonlinear dynamic systems

This paper proposes a non-intrusive technique based on the frequency normalisation for non-intrusive propagations of parameter uncertainties in nonlinear mechanical systems. The multi-solution dilemma found in resonance areas, which prohibits successful applications of uncertainty quantification methods, is alleviated by the additional normalised frequency measure. The spurious peaks around the nonlinear peaks and discontinuous points, known as the Gibbs phenomenon, due to parameter variabilities are resolved as well. The proposed technique coupled with the non-intrusive surrogate modelling is applied to two deliberately constructed nonlinear dynamic systems, i.e., a mass-spring system with interactive cubic nonlinearity and a piece-wise rotor/stator contact problem in rotating machines. The two systems exhibit complex amplitude-frequency response characteristics featured by the softening and hardening effects. Numerous case investigations show the effectiveness of the proposed normalisation method for uncertainty analysis of complex nonlinear vibration systems and its working principle is demonstrated in detail via examples. Accuracy tests against the traditional sampling methods are carried out in both systems. The proposed technique can be easily generalised to other nonlinear systems for random and non-random uncertainty propagations because it works non-intrusively and permits users to choose arbitrary nonlinear tools.

A unique cardiac electrocardiographic 3D model. Toward interpretable AI diagnosis.

Mathematical models of cardiac electrical activity are one of the most important tools for elucidating information about heart diagnostics. In this paper, we present an efficient mathematical formulation for this modeling simple enough to be easily parameterized and rich enough to provide realistic signals. It relies on a five dipole representation of the cardiac electric source, each one associated with the well-known waves of the electrocardiogram signal. Beyond the physical basis of the model, the parameters are physiologically interpretable as they characterize the wave shape, similar to what a physician would look for in signals, thus making them very useful in diagnosis. The model accurately reproduces the electrocardiogram signals of any diseased or healthy heart. This new discovery represents a significant advance in electrocardiography research. It is especially useful for diagnosis, patient follow-up or decision-making on new therapies; is also a promising tool for well-performing, transparent and interpretable AI approaches.

Integrating PZT layer with tuned mass damper for simultaneous vibration suppression and energy harvesting considering exciter dynamics: An analytical and experimental study

This paper analytically and experimentally investigates the potential of a tuned mass damper (TMD) with an integrated piezoelectric layer for simultaneous energy harvesting and vibration suppression. The system investigated is composed of a main beam that is excited by a shaker. A TMD comprising a unimorph piezoelectric beam is attached to the main beam. The shaker force drops at the system resonance frequencies means that a constant force sine sweep is difficult in practice, and does not allow model validation and empirical assessment of the TMD. Also, the force drop off phenomenon and an accurate validated shaker model are vital for the modelling, analysis and validation of nonlinear dynamic systems. This paper validates and analyses a linear model, in preparation for future studies of a nonlinear vibration absorber and energy harvester. Therefore, the force drop off phenomenon is examined comprehensively, and a constant shaker input voltage sweep is proposed instead of a constant force sweep. The continuous electromechanical equations governing the coupled system of the main beam-TMD-shaker are developed. The proposed model is then utilized to examine the shaker force drop off, TMD efficiency, and shaker electrical characteristics. The results show that the constant voltage sweep works well to investigate the performance of the TMD and capture the shaker force drop off phenomenon. The TMD suppresses the host structure vibration and harvests energy efficiently, and is demonstrated by extensive experimental investigations. The model is validated by the experiments, including the force drop off. The system parameters have significant effects on the dynamical response of the coupled system and the energy harvested; the effects on the harvested power, reduced vibration, applied force, and force drop off frequencies are demonstrated. Overall, the TMD is shown to be efficient and applicable in reducing vibration and harvesting energy.

Koopman operator for time-dependent reliability analysis

Time-dependent structural reliability analysis of nonlinear dynamical systems is non-trivial; subsequently, scope of most of the structural reliability analysis methods is limited to time-independent reliability analysis only. In this work, we propose a Koopman operator based approach for time-dependent reliability analysis of nonlinear dynamical systems. Since the Koopman representations can transform any nonlinear dynamical system into a linear dynamical system, the time evolution of dynamical systems can be obtained by Koopman operators seamlessly regardless of the nonlinear or chaotic behavior. Despite the fact that the Koopman theory has been in vogue a long time back, identifying intrinsic coordinates is a challenging task; to address this, we propose an end-to-end deep learning architecture that learns the Koopman observables and then use it for time marching the dynamical response. Unlike purely data-driven approaches, the proposed approach is robust even in the presence of uncertainties; this renders the proposed approach suitable for time-dependent reliability analysis. We propose two architectures; one suitable for time-dependent reliability analysis when the system is subjected to random initial condition and the other suitable when the underlying system have uncertainties in system parameters. The proposed approach is robust and generalizes to unseen environment (out-of-distribution prediction). Efficacy of the proposed approached is illustrated using four numerical examples. Results obtained indicate supremacy of the proposed approach as compared to purely data-driven auto-regressive neural network, long short- term memory network, and DeepONet.