Analysis Of Dynamical Systems Research Articles

A systematic evaluation of the influence of macrophage phenotype descriptions on inflammatory dynamics.

Macrophages play a wide range of roles in resolving the inflammatory damage that underlies many medical conditions and have the ability to adopt different phenotypes in response to different environmental stimuli. Categorising macrophage phenotypes exactly is a difficult task, and there is disparity in the literature around the optimal nomenclature to describe these phenotypes; however, what is clear is that macrophages can exhibit both pro- and anti-inflammatory behaviours dependent upon their phenotype, rendering mathematical models of the inflammatory response potentially sensitive to their description of the macrophage populations that they incorporate. Many previous models of inflammation include a single macrophage population with both pro- and anti-inflammatory functions. Here, we build upon these existing models to include explicit descriptions of distinct macrophage phenotypes and examine the extent to which this influences the inflammatory dynamics that the models emit. We analyse our models via numerical simulation in MATLAB and dynamical systems analysis in XPPAUT, and show that models that account for distinct macrophage phenotypes separately can offer more realistic steady state solutions than precursor models do (better capturing the anti-inflammatory activity of tissue resident macrophages), as well as oscillatory dynamics not previously observed. Finally, we reflect on the conclusions of our analysis in the context of the ongoing hunt for potential new therapies for inflammatory conditions, highlighting manipulation of macrophage polarisation states as a potential therapeutic target.

Molecular modelling of DNA cytosine methylation

Epigenetic modifications are significant processes controlling gene expression. Since their discovery by C. H. Waddington, several multisubject research proceeded and allowed investigation of how biochemical regulations can influence gene transcription. The main epigenetic modification is DNA methylation. This catalytic process of attaching methyl groups to nucleobases (mostly cytosine) is fundamental for the regulation of gene expression and chromatin accessibility. Methylated parts of the genome, which are mostly gene promoters and enhancers, are silenced due to specific biological mechanisms. Scientists postulate that aberrant DNA methylation and other epigenetic modifications can contribute to the development of diverse pathologies such as Alzheimer's disease, epilepsy, Schizophrenia, tumorigenesis, and atherosclerosis. Due to these discoveries, it is important to characterize in detail the mechanism of epigenetic modifications to develop new possible ways of treatment. In this work, we presented complex studies of DNA methylation mechanism including dynamic and energetic analysis of the reaction system. A cluster chemical approach was used to model the active site of enzyme catalyzing methylation - DNMT1 (DNA methyltransferase 1). The created model was subjected to quantum chemical calculations. Geometry optimization was performed with ADF software, providing an energetic characterization of the studied process. Dynamic properties of the enzyme model were described due to molecular dynamics (MD) procedures performed with CP2K software. MD calculations were focused on the observation and description of proton transfer between glutamate acid and cytosine, which is considered the main enzyme-substrate interaction leading to the creation of 5-methylcytosine 5mC (product of the studied reaction) and covalent bonding between enzymes cysteine and nucleobase. This research provides a comprehensive understanding of the DNA methylation mechanism and may contribute to the invention of new drugs and modern ways of treatment.

Dynamical system analysis of scalar field cosmology in coincident f(Q) gravity

In this article, we investigate scalar field cosmology in the coincident f(Q) gravity formalism. We calculate the motion equations of f(Q) gravity under the flat Friedmann-Lemaître-Robertson-Walker background in the presence of a scalar field. We consider a non-linear f(Q) model, particularly f(Q) = − Q + α Q n , which is nothing but a polynomial correction to the Symmetric Teleparallel Equivalent to General Relativity (STEGR) case. Further, we assumed two well-known specific forms of the potential function, specifically the exponential from V(ϕ) = V 0 e −β ϕ and the power-law form V(ϕ) = V 0 ϕ −k . We employ some phase-space variables and transform the cosmological field equations into an autonomous system. We calculate the critical points of the corresponding autonomous systems and examine their stability behaviors. We discuss the physical significance corresponding to the exponential case for parameter values n = 2 and n = − 1 with β = 1, and n = − 1 with β=3 . Moreover, we discuss the same corresponding to the power-law case for the parameter value n = − 2 and k = 0.16. We also analyze the behavior of corresponding cosmological parameters such as scalar field and dark energy density, deceleration, and the effective equation of state parameter. Corresponding to the exponential case, we find that the results obtained for the parameter constraints in Case III is better among all three cases, and that represents the evolution of the Universe from a decelerated stiff era to an accelerated de-Sitter era via matter-dominated epoch. Further, in the power-law case, we find that all trajectories exhibit identical behavior, representing the evolution of the Universe from a decelerated stiff era to an accelerated de-Sitter era. Lastly, we conclude that the exponential case shows better evolution as compared to the power-law case.

Research of the dynamic regulatory mechanism of Compound Danshen Dripping Pills on myocardial infarction based on metabolic trajectory analysis

BackgroundMyocardial infarction (MI) is a serious cardiovascular disease, which presents different pathophysiological changes with the prolongation of the disease. Compound danshen dripping pills (CDDP) has obvious advantages in MI treatment and widely used in the clinic. However, the current studies were mostly focused on the endpoint of CDDP intervention, lacking the dynamic attention to the disease process. It is of great value to establish a dynamic research strategy focused on the changes in pharmacodynamic substances for guiding clinical medication more precisely. PurposeIt is aimed to explore the dynamic regulating pattern of CDDP on MI based on metabolic trajectory analysis, and then clarify the variation characteristic biomarkers and pharmacodynamic substances in the intervention process. MethodsThe MI model was successfully prepared by coronary artery left anterior descending branch ligation, and then CDDP intervention was given for 28 days. Endogenous metabolites and the components of CDDP in serum were measured by LC/MS technique simultaneously to identify dynamic the metabolic trajectory and screen the characteristic pharmacodynamic substances at different points. Finally, network pharmacology and molecular docking techniques were used to simulate the core pharmacodynamic substances and core target binding, then validated at the genetic and protein level by Q-PCR and western blotting technology. ResultsCDDP performed typical dynamic regulation features on metabolite distribution, biological processes, and pharmacodynamic substances. During 1–7 days, it mainly regulated lipid metabolism and inflammation, the Phosphatidylcholine (PC(18:1(9Z/18:1(9Z)) and Sphingomyelin (SM(d18:1/23:1(9Z)), SM(d18:1/24:1(15Z)), SM(d18:0/16:1(9Z))) were the main characteristic biomarkers. Lipid metabolism was the mainly regulation pathway during 14–21 days, and the characteristic biomarkers were the Lysophosphatidylethanolamine (LysoPE(0:0/20:0), PE-NMe2(22:1(13Z)/15:0)) and Sphingomyelin (SM(d18:1/23:1(9Z))). At 28 days, in addition to inflammatory response and lipid metabolism, fatty acid metabolism also played the most important role. Correspondingly, Lysophosphatidylcholine (LysoPC(20:0/0:0)), Lysophosphatidylserine (LPS(18:0/0:0)) and Fatty acids (Linoelaidic acid) were the characteristic biomarkers. Based on the results of metabolite distribution and biological process, the characteristic pharmacodynamic substances during the intervention were further identified. The results showed that various kinds of Saponins and Tanshinones as the important active ingredients performed a long-range regulating effect on MI. And the other components, such as Tanshinol and Salvianolic acid B affected Phosphatidylcholine and Sphingomyelin through Relaxin Signaling pathway during the early intervention. Protocatechualdehyde and Rosmarinic acid affected Lysophosphatidylethanolamine and Sphingomyelin through EGFR Tyrosine kinase inhibitor resistance during the late intervention. Tanshinone IIB and Isocryptotanshinone via PPAR signaling pathway affected Lysophosphatidylcholine, Lysophosphatidylserine, and Fatty acids. ConclusionThe dynamic regulating pattern was taken as the entry point and constructs the dynamic network based on metabolic trajectory analysis, establishes the dynamic correlation between the drug-derived components and the endogenous metabolites, and elucidates the characteristic biomarkers affecting the changes of the pharmacodynamic indexes, systematically and deeply elucidate the pharmacodynamic substance and mechanism of CDDP on MI. It also enriched the understanding of CDDP and provided a methodological reference for the dynamic analysis of complex systems of TCM.

Kinematic and Dynamic Analysis of Eccentric Balanced Positive Torque Pumping Unit

Beam pumping units have numerous energy transfer links, significant net torque fluctuations and negative net torque in gear boxes, which lead to a high installed power and low system efficiency. In order to improve the efficiency of the pumping unit, in this paper, an eccentric balanced positive torque pump unit was designed based on the principle of eccentric balance that is small in size and adopts a flexible energy transfer system instead of a rigid structure to significantly reduce the energy transfer link. The kinematics and dynamics of the eccentric balanced positive torque pumping unit were analyzed and the ability to operate with positive torque over the entire cycle was verified using theoretical computations. And the results of the theoretical calculation were verified by using the virtual prototype simulation and analysis software ADAMS (Automatic Dynamic Analysis of Mechanical Systems); the pumping unit was optimized for relevant parameters such as the balance crank angle λ and the counterweight center of mass offset angle δ. Indoor testing and comparative field application have demonstrated that the installed power has been reduced by 80%, the weight of the entire prototype has decreased by 25% and the system efficiency has reached 24.3%. Additionally, electricity savings of over 50% have been achieved. These results indicate that the pumping unit has been designed correctly, has high reliability and has significantly improved system efficiency while also providing an obvious energy-saving effect.

Theory and practice for assessing structural integrity and dynamical integrity of high-speed trains

PurposeThe safety and reliability of high-speed trains rely on the structural integrity of their components and the dynamic performance of the entire vehicle system. This paper aims to define and substantiate the assessment of the structural integrity and dynamical integrity of high-speed trains in both theory and practice. The key principles and approaches will be proposed, and their applications to high-speed trains in China will be presented.Design/methodology/approachFirst, the structural integrity and dynamical integrity of high-speed trains are defined, and their relationship is introduced. Then, the principles for assessing the structural integrity of structural and dynamical components are presented and practical examples of gearboxes and dampers are provided. Finally, the principles and approaches for assessing the dynamical integrity of high-speed trains are presented and a novel operational assessment method is further presented.FindingsVehicle system dynamics is the core of the proposed framework that provides the loads and vibrations on train components and the dynamic performance of the entire vehicle system. For assessing the structural integrity of structural components, an open-loop analysis considering both normal and abnormal vehicle conditions is needed. For assessing the structural integrity of dynamical components, a closed-loop analysis involving the influence of wear and degradation on vehicle system dynamics is needed. The analysis of vehicle system dynamics should follow the principles of complete objects, conditions and indices. Numerical, experimental and operational approaches should be combined to achieve effective assessments.Originality/valueThe practical applications demonstrate that assessing the structural integrity and dynamical integrity of high-speed trains can support better control of critical defects, better lifespan management of train components and better maintenance decision-making for high-speed trains.

Time Transfer Link fusion algorithm based on wavelet multi-resolution analysis

To raise the precision and stability of time transfer links, a wavelet-based multi-resolution data fusion technique is offered. This approach fixes the “Day-Boundary discontinuities” in global positioning system (GPS) precise point positioning (PPP) link and lessens the diurnal in Two-way satellite time and frequency transfer (TWSTFT) link by integrating the model-based dynamic system analysis method with the statistically based multi-scale signal transformation method. It can also handle data from GPS PPP links and TWSTFT links at different sampling rates. More specifically, after applying wavelet transformation to the raw data and Kalman filtering estimation on various scales, this method uses the Mallat fast reconstruction algorithm of wavelet transform to achieve the fused results. The National Time Service Center (NTSC) and the Physikalisch-Technische Bundesanstalt (PTB) handle time transfer results using this method. Experiments show that this strategy greatly improves the link's short-term stability and can manage atypical interruptions while also boosting the link's robustness and reliability, which has a wide range of possible applications.

Estimating mixed-effects state-space models via particle filters and the EM algorithm

In this paper, we focus on studying the Mixed-Effects State-Space (MESS) models previously introduced by Liu et al. [Liu D, Lu T, Niu X-F, et al. Mixed-effects state-space models for analysis of longitudinal dynamic systems. Biometrics. 2011;67(2):476–485]. We propose an estimation method by combining the auxiliary particle learning and smoothing approach with the Expectation Maximization (EM) algorithm. First, we describe the technical details of the algorithm steps. Then, we evaluate their effectiveness and goodness of fit through a simulation study. Our method requires expressing the posterior distribution for the random effects using a sufficient statistic that can be updated recursively, thus enabling its application to various model formulations including non-Gaussian and nonlinear cases. Finally, we demonstrate the usefulness of our method and its capability to handle the missing data problem through an application to a real dataset.

Strategies for Mitigating Urban Residential Carbon Emissions: A System Dynamics Analysis of Kunming, China

To effectively combat environmental challenges, it is necessary to evaluate urban residential building carbon emissions and implement energy-efficient, emission-reducing strategies. The lack of a specialized carbon emission monitoring system complicates merging macro- and micro-level analyses to forecast urban residential emissions accurately. This study employs a system dynamics (SD) model to examine the influence of social, economic, energy, and environmental factors on carbon emissions in urban residences in Kunming, China. The SD model forecasts household carbon emissions from 2022 to 2030 and establishes three scenarios: a low-carbon scenario (LCS), a medium low-carbon scenario (MLCS), and a high low-carbon scenario (HLCS) to assess emission reduction potentials. It predicts emissions will climb to 4.108 million tons by 2030, significantly surpassing the 2014 baseline, with economic growth, urbanization, residential energy consumption, and housing investment as key drivers. To curb emissions, the study suggests enhancing low-carbon awareness, altering energy sources, promoting research and development investment, and expanding green areas. The scenarios indicate a 5.1% to 16.1% emission reduction by 2030 compared to the baseline. The study recommends an 8.3% to 11.4% reduction in MLCS as a practical short-term target for managing urban residential emissions, offering a valuable SD approach for optimizing carbon strategies and aiding low-carbon development.

Impedance Circle Modeling and Dynamic Characteristics Analysis of Contactless Energy Transfer System Under Load Variations in Rotary Ultrasonic Machining

In rotary ultrasonic machining, contactless energy transfer (CET) is developed to transmit energy to the transducer at the series resonant frequency or resonant frequency of the transducer. Specifically, during energy transmission, variations in working conditions such as variable axial forces will change the load of the transducer and introduce undesired dynamic effects into the contactless energy transfer system (CETS). In order to explore the connections between these variations and the working performance more directly and theoretically, a new theoretical model—CETS impedance circle is proposed in this paper to reflect the CETS dynamic characteristics, including the CETS frequency characteristics and the relationships between CETS transfer performances at different frequencies under load variations. With the developed theoretical model, the effect of load variations on the trends of CETS impedance circle is investigated. Moreover, based on the impedance circle and its trends, the CETS frequency characteristics, the effect of primary series compensation, and a comparison of the transfer performances at the two working frequencies are theoretically studied. Finally, the CETS impedance circle theories are experimentally verified, and the experimental results are consistent with the analysis of the theoretical model.

Dynamic Modelling Analysis on The Effectiveness of Coastal Land Resources for Aquaculture Activities Utilization

Coastal areas are considered biodiversity zones. This study aimed to analyze the effectiveness of land use on aquaculture productivity rate in coastal areas using a dynamic modeling analysis system. This study was conducted in the Pekalongan Province. Water quality parameters and qualitative research data were obtained from an in-depth survey. Furthermore, the data were analyzed using dynamic modeling analysis. The results of this study indicate that the level of coastal land use has a variable relationship with the productivity rate of aquaculture. Based on the causal loop model, it could be analyzed that the social, environmental conditions in coastal areas were described as continuing to experience quantification of population increase and environmental utilization rates. Overall, the productivity level of aquaculture cultivation in coastal areas tended to decrease along with the increasing value of productive land use accompanied by increased inputs and the number of human resource populations in coastal areas. The optimal inclusion of this study was that the optimal land use level would impact the productivity management level of aquaculture cultivation in coastal areas. Based on the results of this study, it can be concluded that the optimal utilization of coastal land greatly impacted the management of the productivity level of aquaculture cultivation, which was carried out in an integrated manner. Furthermore, the analysis of the dynamic modeling system also illustrated the increasing social-environmental conditions in coastal land utilization.

Stability Analysis of Switched Linear Systems with Neural Lyapunov Functions

Neural-based, data-driven analysis and control of dynamical systems have been recently investigated and have shown great promise, e.g. for safety verification or stability analysis. Indeed, not only do neural networks allow for an entirely model-free, data-driven approach, but also for handling arbitrary complex functions via their power of representation (as opposed to, e.g. algebraic optimization techniques that are restricted to polynomial functions). Whilst classical Lyapunov techniques allow to provide a formal and robust guarantee of stability of a switched dynamical system, very little is yet known about correctness guarantees for Neural Lyapunov functions, nor about their performance (amount of data needed for a certain accuracy). We formally introduce Neural Lyapunov functions for the stability analysis of switched linear systems: we benchmark them on this paradigmatic problem, which is notoriously difficult (and in general Turing-undecidable), but which admits existing recently-developed technologies and theoretical results. Inspired by switched systems theory, we provide theoretical guarantees on the representative power of neural networks, leveraging recent results from the ML community. We additionally experimentally display how Neural Lyapunov functions compete with state-of-the-art results and techniques, while admitting a wide range of improvement, both in theory and in practice. This study intends to improve our understanding of the opportunities and current limitations of neural-based data-driven analysis and control of complex dynamical systems.

Automated State Estimation for Summarizing the Dynamics of Complex Urban Systems Using Representation Learning

Complex urban systems can be difficult to monitor, diagnose and manage because the complete states of such systems are only partially observable with sensors. State estimation techniques can be used to determine the underlying dynamic behavior of such complex systems with their highly non-linear processes and external time-variant influences. States can be estimated by clustering observed sensor readings. However, clustering performance degrades as the number of sensors and readings (i.e. feature dimension) increases. To address this problem, we propose a framework that learns a feature-centric lower dimensional representation of data for clustering to support analysis of system dynamics. We propose Unsupervised Feature Attention with Compact Representation (UFACR) to rank features contributing to a cluster assignment. These weighted features are then used to learn a reduced-dimension temporal representation of the data with a deep-learning model. The resulting low-dimensional representation can be effectively clustered into states. UFACR is evaluated on real-world and synthetic wastewater treatment plant data sets, and feature ranking outcomes were validated by Wastewater treatment domain experts. Our quantitative and qualitative experimental analyses demonstrate the effectiveness of UFACR for uncovering system dynamics in an automated and unsupervised manner to offer guidance to wastewater engineers to enhance industrial productivity and treatment efficiency.

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

This 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.

Can Recurrence Quantification Analysis Be Useful in the Interpretation of Airborne Turbulence Measurements?

AbstractIn airborne data or model outputs, clouds are often defined using information about Liquid Water Content (LWC). Unfortunately LWC is not enough to retrieve information about the dynamical boundary of the cloud, that is, volume of turbulent air around the cloud. In this work, we propose an algorithmic approach to this problem based on a method used in time series analysis of dynamical systems, namely Recurrence Plot (RP) and Recurrence Quantification Analysis (RQA). We construct RPs using time series of turbulence kinetic energy, vertical velocity and temperature fluctuations as variables important for cloud dynamics. Then, by studying time series of laminarity (LAM), a variable which is calculated using RPs, we distinguish between turbulent and non‐turbulent segments along a horizontal flight leg. By selecting a single threshold of this quantity, we are able to reduce the number of subjective variables and their thresholds used in the definition of the dynamical cloud boundary.

System dynamics analysis of Retailer's emergency strategies when facing irrational demand and supply disruption

Emergency events, such as epidemics, typhoons, and floods, can lead to increased demand instability, resulting in a higher risk of supply–demand imbalances in supply chains. Companies often face challenges in managing irrational demand and supply disruptions due to their limited experience. In this paper, we utilize system dynamics (SD) and susceptible-infected-recovered (SIR) model to examine the phenomenon of irrational consumer demand within the context of supply disruption. Specifically, we focus on analyzing such demand patterns across three external supply scenarios: sufficient, stable, and learning types. Furthermore, we consider emergency strategies (pricing, transshipment, and hybrid) for the retailer based on the consideration of either delayed or non-delayed delivery mechanisms. The results show that the upsurge of irrational demand following an emergency outbreak is usually short term and enormous. A higher level of panic effect leads to increased irrational demand, whereas a stronger conformity effect results in faster increases and declines of such demand during early and later stages, respectively. Furthermore, the emergency strategies should aim to mitigate the negative impacts of the disruption retailer operations and improve the retailer's emergency capabilities. Numerical analysis indicates that the retailer can increase its profit in the sufficient-type external supply scenario by adopting a pricing strategy that incorporates a delayed delivery mechanism. When facing a stable-type scenario, the recommendation is the hybrid strategy. In a learning-type scenario, the retailer should consider using the hybrid strategy with non-delayed delivery as the default option, but use a pricing strategy if delayed delivery is necessary.

Cosmology of f(Q) gravity in non-flat Universe

We investigate the cosmological implications of f(Q) gravity, which is a modified theory of gravity based on non-metricity, in non-flat geometry. We perform a detailed dynamical-system analysis keeping the f(Q) function completely arbitrary. As we show, the cosmological scenario admits a dark-matter dominated point, as well as a dark-energy dominated de Sitter solution which can attract the Universe at late times. However, the main result of the present work is that there are additional critical points which exist solely due to curvature. In particular, we find that there are curvature-dominated accelerating points which are unstable and thus can describe the inflationary epoch. Additionally, there is a point in which the dark-matter and dark-energy density parameters are both between zero and one, and thus it can alleviate the coincidence problem. Finally, there is a saddle point which is completely dominated by curvature. In order to provide a specific example, we apply our general analysis to the power-law case, showing that we can obtain the thermal history of the Universe, in which the curvature density parameter may exhibit a peak at intermediate times. These features, alongside possible indications that non-zero curvature could alleviate the cosmological tensions, may serve as advantages for f(Q) gravity in non-flat geometry.

Optimum design of HNBR and PCR-based isolation system to protect a sensitive article stored in a naval vessel container subjected to shock load

In naval engineering, protecting sensitive articles from shock loads is a major concern, as it may lead to adverse consequences. This paper focuses on the design of a novel Hydrogenated Nitrile Butadiene Rubber (HNBR) and Polychloroprene Rubber (PCR)-based shock isolation system to protect a sensitive article stored in a hollow cylindrical naval vessel container (NVC). Firstly, NVC is modeled as a two-degree-of-freedom lumped parameter model (LPM), wherein isolators made of HNBR and PCR materials are modeled with 5-order polynomial nonlinear stiffness and linear damping. The optimal values of the isolator parameters are obtained using a multi-objective genetic algorithm, and corresponding responses to a shock load are found using MATLAB. Furthermore, a three-dimensional (3D) finite element model (FEM) of the NVC is generated in ABAQUS and subjected to shock loading in a transverse direction to conduct a comprehensive dynamic analysis of the isolation system. In 3D simulation, HNBR and PCR materials are modeled using a hyper-viscoelastic model developed based on experimental test data. The 3D simulation results of the NVC for the optimized design of HNBR and PCR isolators agree with the optimized results of the LPM. Moreover, the proposed HNBR and PCR isolators are very effective in shock reduction.

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.

Hermitian Dynamic Mode Decomposition - Numerical Analysis and Software Solution

The Dynamic Mode Decomposition (DMD) is a versatile and increasingly popular method for data driven analysis of dynamical systems that arise in a variety of applications in, e.g., computational fluid dynamics, robotics or machine learning. In the framework of numerical linear algebra, it is a data driven Rayleigh-Ritz procedure applied to a DMD matrix that is derived from the supplied data. In some applications, the physics of the underlying problem implies hermiticity of the DMD matrix, so the general DMD procedure is not computationally optimal. Furthermore, it does not guarantee important structural properties of the Hermitian eigenvalue problem and may return non-physical solutions. This paper proposes a software solution to the Hermitian (including the real symmetric) DMD matrices, accompanied with a numerical analysis that contains several fine and instructive numerical details. The eigenpairs are computed together with their residuals, and perturbation theory provides error bounds for the eigenvalues and eigenvectors. The software is developed and tested using the LAPACK package.