Fast Variables Research Articles

Asymptotic analysis of self-excited and forced vibrations of a self-regulating pressure control valve

Pressure vibrations in hydraulic systems are a widespread problem and can be caused by external excitation or self-exciting mechanisms. Although vibrations cannot be completely avoided in most cases, at least their frequencies must be known in order to prevent resonant excitation of adjacent components. While external excitation frequencies are known in most cases, the estimation of self-excited vibration amplitudes and frequencies is often difficult. Usually, numerical studies have to be executed in order to elaborate parameter influences, which is computationally expensive. The same holds true for the prediction of forced oscillation amplitudes. This contribution proposes asymptotic approximations of forced and self-excited oscillations in a simple hydraulic circuit consisting of a pump, an ideal consumer and a pressure control valve. Two excitation mechanisms of practical interest, namely pump pulsations (forced vibrations) and valve instability (self-excited vibrations), are analyzed. The system dynamics are described by a singularly perturbed third-order differential equation. By separating slow and fast variables in the system without external excitation, a first-order approximation of the slow manifold is computed. The flow on the slow manifold is approximated by an averaging procedure, whose piecewise defined zero-order solution maps the valve’s switching property. A modification of the procedure allows for the asymptotic approximation of the system’s forced response to an external excitation. The approximate solutions are validated within a realistic parameter range by comparison with numerical solutions of the full system equations.

A dynamical study of fusion hindrance with the Nakajima–Zwanzig projection method

Abstract A new framework is proposed for the study of collisions between very heavy ions which lead to the synthesis of Super-Heavy Elements (SHE), to address the fusion hindrance phenomenon. The dynamics of the reaction is studied in terms of collective degrees of freedom undergoing relaxation processes with different time scales. The Nakajima–Zwanzig projection operator method is employed to eliminate fast variables and derive a dynamical equation for the reduced system with only slow variables. There, the time evolution operator is renormalised and an inhomogeneous term appears, which represents a propagation of the given initial distribution. The term results in a slip to the initial values of the slow variables. We expect that gives a dynamical origin for the so-called “injection point $s$” introduced by Swiatecki et al. in order to reproduce absolute values of measured cross sections for SHE. A formula for the slip is given in terms of physical parameters of the system, which confirms the results recently obtained with a Langevin equation, and permits us to compare various incident channels.

Rapid spectral variability of a giant flare from a magnetar in NGC253.

Magnetars are neutron stars with extremely strong magnetic fields (1013 to 1015 gauss)1,2, which episodically emit X-ray bursts approximately 100 milliseconds long and with energies of 1040 to 1041 erg. Occasionally, they also produce extremely bright and energetic giant flares, which begin with a short (roughly 0.2 seconds), intense flash, followed by fainter, longer-lasting emission that is modulated by the spin period of the magnetar3,4 (typically 2 to 12 seconds). Over the past 40 years, only three such flares have been observed in our local group of galaxies3-6, and in all cases the extreme intensity of the flares caused the detectors to saturate. It has been proposed that extragalactic giant flares are probably a subset7-11 of short γ-ray bursts, given that the sensitivity of current instrumentation prevents us from detecting the pulsating tail, whereas the initial bright flash is readily observable out to distances of around 10 to 20 million parsecs. Here we report X-ray and γ-ray observations of the γ-ray burst GRB200415A, which has a rapid onset, very fast time variability, flat spectra and substantial sub-millisecond spectral evolution. These attributes match well with those expected for a giant flare from an extragalactic magnetar12, given that GRB200415A is directionally associated13 with the galaxy NGC253 (roughly 3.5 million parsecs away). The detection of three-megaelectronvolt photons provides evidence for the relativistic motion of the emitting plasma. Radiation from such rapidly moving gas around a rotating magnetar may have generated the rapid spectral evolution that we observe.

Регуляризованные асимптотические решения интегродифференциальных уравнений с быстрыми и медленными переменными

The article considers a nonlinear integro-differential system of equations with fast and slow variables. Such systems were not considered previously from the point of view of constructing regularized (according to Lomov) asymptotic solutions. The known studies were mainly devoted to construction of the asymptotics of the Butuzov-Vasil'eva boundary layer type, which, as is known, can be applied only if the spectrum of the first variation matrix (on the degenerate solution) is located strictly in the open left-half plane of a complex variable. If the spectrum of this matrix falls on the imaginary axis, the S.A. Lomov regularization method is commonly used. However, this method was mainly developed for singularly perturbed differential systems that do not contain integral terms, or for integro-differential problems without slow variables. In this article, the regularization method is generalized for two-dimensional integro-differential equations with fast and slow variables.

Relaxation Oscillations and the Entry-Exit Function in Multidimensional Slow-Fast Systems

For a slow-fast system of the form $\dot{p}=\epsilon f(p,z,\epsilon)+h(p,z,\epsilon)$, $\dot{z}=g(p,z,\epsilon)$ for $(p,z)\in \mathbb R^n\times \mathbb R^m$, we consider the scenario that the system has invariant sets $M_i=\{(p,z): z=z_i\}$, $1\le i\le N$, linked by a singular closed orbit formed by trajectories of the limiting slow and fast systems. Assuming that the stability of $M_i$ changes along the slow trajectories at certain turning points, we derive criteria for the existence and stability of relaxation oscillations for the slow-fast system. Our approach is based on a generalization of the entry-exit relation to systems with multi-dimensional fast variables. We then apply our criteria to several predator-prey systems with rapid ecological evolutionary dynamics to show the existence of relaxation oscillations in these models.

Data-Informed Decomposition for Localized Uncertainty Quantification of Dynamical Systems

Industrial dynamical systems often exhibit multi-scale responses due to material heterogeneity and complex operation conditions. The smallest length-scale of the systems dynamics controls the numerical resolution required to resolve the embedded physics. In practice however, high numerical resolution is only required in a confined region of the domain where fast dynamics or localized material variability is exhibited, whereas a coarser discretization can be sufficient in the rest majority of the domain. Partitioning the complex dynamical system into smaller easier-to-solve problems based on the localized dynamics and material variability can reduce the overall computational cost. The region of interest can be specified based on the localized features of the solution, user interest, and correlation length of the material properties. For problems where a region of interest is not evident, Bayesian inference can provide a feasible solution. In this work, we employ a Bayesian framework to update the prior knowledge of the localized region of interest using measurements of the system response. Once, the region of interest is identified, the localized uncertainty is propagate forward through the computational domain. We demonstrate our framework using numerical experiments on a three-dimensional elastodynamic problem.

Eyring equation and fluctuation-dissipation far away from equilibrium.

Understanding and managing the influence that either external forces or non-equilibrated environments may have on chemical processes is essential for the current and future development of theoretical chemistry. One of the central questions to solve is how to generalize the transition state theory in order to make it applicable in far from equilibrium situations. In this sense, here we propose a way to generalize Eyring's equation based on the definition of an effective thermal energy (temperature) emerging from the coupling of both fast and slow dynamic variables analyzed within the generalized Langevin dynamics scheme. This coupling makes the energy distribution of the fast degrees of freedom not equilibrate because they have been enslaved to the dynamics of the corresponding slow degrees. However, the introduction of the effective thermal energy enables us to restore an effective adiabatic separation of timescales leading to a renormalization of the generalized fluctuation-dissipation theorem. Hence, this procedure opens the possibility to deal with systems far away from equilibrium. A significant consequence of our results is that Eyring's equation is generalized to treat systems under the influence of strong external forces.

A two-component Comptonization model for the type-B QPO in MAXI J1348−630

ABSTRACT Spectral-timing analysis of the fast variability observed in X-rays is a powerful tool to study the physical and geometrical properties of the accretion/ejection flows in black hole (BH) binaries. The origin of type-B quasi-periodic oscillations (QPO), predominantly observed in BH candidates in the soft-intermediate state, has been linked to emission arising from the relativistic jet. In this state, the X-ray spectrum is characterized by a soft-thermal blackbody-like emission due to the accretion disc, an iron emission line (in the 6–7 keV range), and a power-law-like hard component due to inverse-Compton scattering of the soft-photon source by hot electrons in a corona or the relativistic jet itself. The spectral-timing properties of MAXI J1348−630 have been recently studied using observations obtained with the NICER observatory. The data show a strong type-B QPO at ∼4.5 Hz with increasing fractional rms amplitude with energy and positive lags with respect to a reference band at 2–2.5 keV. We use a variable-Comptonization model that assumes a sinusoidal coherent oscillation of the Comptonized X-ray flux and the physical parameters of the corona at the QPO frequency, to fit simultaneously the energy-dependent fractional rms amplitude and phase lags of this QPO. We show that two physically connected Comptonization regions can successfully explain the radiative properties of the QPO in the full 0.8–10 keV energy range.

Data-based reduced-order modeling of nonlinear two-time-scale processes

This work focuses on data-based reduced-order modeling of nonlinear processes that exhibit time-scale multiplicity. Using time-series data from all the state variables of a nonlinear process, an approach that involves nonlinear principal component analysis and neural network function approximators is employed to identify the fast and slow process state variables as well as compute a nonlinear slow manifold function approximation in which the fast variables are “slaved” in terms of the slow variables. Subsequently, a nonlinear sparse identification approach is employed to calculate a dynamic model of nonlinear first-order ordinary differential equations that describe the temporal evolution of the slow process states. The method is applied to two chemical process examples, and the advantages of the proposed method over using only sparse identification for the original two-time-scale process are discussed. The approach can be thought of as a data-based analogue of the classical singular perturbation modeling of two-time-scale processes where explicit time-scale separation is assumed (and expressed in terms of a small positive parameter ε multiplying the time derivative of the fast states), and the reduced-order slow subsystem is calculated analytically.

Relevence and Effectiveness of Mathematical Models Dealing with Covid-19 Pandemic

How does COVID-19 pandemic affect the India? How many people can be hit in a state and how many of them will succumb to the disease? When is it going to peak? How long should the government continue with the lockdown? What is the damage to the economy and what is its impact on each sector? These are some of the questions that haunt not only the decision makers but every sensible people in India. Mathematical models developed by mathematicians and epidemiologists has come to assist decision makers in evaluating the effects of countermeasures to an epidemic before they actually deploy them. The model could give political and beuricatic person's critical insights into the best steps they could take to counter the spread of disease in the face of pandemics. Mathematicians use modeling to represent, analyze and make predictions or otherwise provide insight into real world phenomena. Real world scenarios can be designed into a mathematical model to bring clarity to big messy questions amid fast changing variables. These models aim to make simplifying assumptions in order to arrive at tractable equations. Dealing with the novel coronavirus is an unprecedented situation which the world could not have foreseen. In order to track the COVID-19 pandemic, make predictions about the disease's progression and take decisions, as of now, the government is solely dependent on data from doctors and health workers.

Dynamic behavior analysis of a mechanical system with two unstable modes coupled to a single nonlinear energy sink

This paper investigates a problem of passive mitigation of vibratory instabilities caused by two unstable modes by means of a single nonlinear energy sink (NES). For this purpose, a linear four-degree-of-freedom (DOF) primary structure having two unstable modes (reproducing the typical dynamic behavior of a friction system) and undergoing, as it is linear, unbounded motions when it is unstable, is coupled to a NES. In this work, the NES involves an essentially cubic restoring force and a linear damping force. We are interested in characterizing analytically the response regimes resulting from the coupling of the two unstable linear modes of the primary structure and the nonlinear mode of the NES. To this end, from a suitable rescaling of the governing equations of the coupled system in which the dynamics of the primary structure is reduced to its unstable modal coordinates, the complexification-averaging method is applied. The resulting averaged system appears to be a fast-slow system with four fast variables and two slow ones related to the two unstable modes of the primary structure. The critical manifold of the averaged dynamics is obtained through the geometric singular perturbation theory and appears as a two-dimensional parametric surface (with respect to two of the four fast variables) which evolves in the whole six-dimensional variable space. The asymptotic analysis reveals that the NES attachment can produce some bounded responses and suggests that the system may have simultaneous stable attractors. Numerical simulations complement the study, highlighting a possible competition between stable attractors and allowing us to investigate their basins of attraction. In each considered situation, a good agreement has been observed between theoretical results and numerical simulations, which validates the proposed asymptotic analysis.

Fast photometric variability of very low mass stars in IC 348: detection of superflare in an M dwarf

ABSTRACT We present here optical I-band photometric variability study down to ≃19 mag of a young (∼2–3 Myr) star-forming region IC 348 in the Perseus molecular cloud. We aim to explore the fast rotation (in the time-scales of hours) in very low-mass stars including brown dwarfs (BDs). From a sample of 177 light curves using our new I-band observations, we detect new photometric variability in 22 young M dwarfs including 6 BDs, which are bonafide members in IC 348 and well characterized in the spectral type of M dwarfs. Out of 22 variables, 11 M dwarfs including one BD show hour-scale periodic variability in the period range 3.5–11 h and rest are aperiodic in nature. Interestingly, an optical flare is detected in a young M2.75 dwarf in one night data on 2016 December 20. From the flare light curve, we estimate the emitted flared energy of 1.48 × 1035 erg. The observed flared energy with an uncertainty of tens of per cent is close to the superflare range (∼1034 erg), which is rarely observed in active M dwarfs.

The dual nature of blazar fast variability: Space and ground observations of S5 0716+714

ABSTRACT Blazar S5 0716+714 is well-known for its short-term variability, down to intraday time-scales. We here present the 2-min cadence optical light curve obtained by the TESS space telescope in 2019 December–2020 January and analyse the object fast variability with unprecedented sampling. Supporting observations by the Whole Earth Blazar Telescope Collaboration in B, V, R, and I bands allow us to investigate the spectral variability during the TESS pointing. The spectral analysis is further extended in frequency to the UV and X-ray bands with data from the Neil Gehrels Swift Observatory. We develop a new method to unveil the shortest optical variability time-scales. This is based on progressive de-trending of the TESS light curve by means of cubic spline interpolations through the binned fluxes, with decreasing time bins. The de-trended light curves are then analysed with classical tools for time-series analysis (periodogram, autocorrelation, and structure functions). The results show that below 3 d there are significant characteristic variability time-scales of about 1.7, 0.5, and 0.2 d. Variability on time-scales $\lesssim 0.2$ d is strongly chromatic and must be ascribed to intrinsic energetic processes involving emitting regions, likely jet substructures, with dimension less than about 10−3 pc. In contrast, flux changes on time-scales $\gtrsim 0.5$ d are quasi-achromatic and are probably due to Doppler factor changes of geometric origin.

Nonlinear dynamics of inertial particles in the ocean: from drifters and floats to marine debris and Sargassum

Buoyant, finite-size or inertial particle motion is fundamentally unlike neutrally buoyant, infinitesimally small or Lagrangian particle motion. The de-jure fluid mechanics framework for the description of inertial particle dynamics is provided by the Maxey-Riley equation. Derived from first principles - a result of over a century of research since the pioneering work by Sir George Stokes - the Maxey-Riley equation is a Newton-type-law with several forces including (mainly) flow, added mass, shear-induced lift, and drag forces. In this paper we present an overview of recent efforts to port the Maxey-Riley framework to oceanography. These involved: 1) including the Coriolis force, which was found to explain behavior of submerged floats near mesoscale eddies; 2) accounting for the combined effects of ocean current and wind drag on inertial particles floating at the air-sea interface, which helped understand the formation of great garbage patches and the role of anticyclonic eddies as plastic debris traps; and 3) incorporating elastic forces, which are needed to simulate the drift of pelagic Sargassum. Insight on the nonlinear dynamics of inertial particles in every case was possible to be achieved by investigating long-time asymptotic behavior in the various Maxey-Riley equation forms, which represent singular perturbation problems involving slow and fast variables.

Chlorophyll fluorescence and water content parameters are good biomarkers for selecting drought tolerant eucalyptus clones

Fields of Eucalyptus throughout the world have faced ever more frequent events of water restriction, whilst the burgeoning demand has promoted the planting of crops in even drier sites. This study evaluated the ability of fast measurement variables (gas exchange, chlorophyll fluorescence, and water content) to determine the functional diversity of eucalyptus clones with differential tolerance to drought. For this, ten commercial eucalyptus clones were submitted to two treatments: control (90% of field capacity) and water deficit (50% of field capacity). After 40 days, the following parameters were measured: stem growth parameters, predawn water potential, relative water content, leaf succulence, and photosynthetic parameters. Based on the obtained data, three groups of plants were identified: (I) high stem growth and high tolerance clones; (II) high stem growth and low tolerance; and (III) low stem growth and low tolerance. The higher tolerance and stem growth of clones from group I involved the maintenance of cell water status and central physiological process (e.g. photosynthesis). Such clones, notwithstanding, did not present the highest carbon fixation rates. Even though the gas exchange parameters are easily and quickly obtained, they do not constitute a suitable tool for the selection of eucalyptus clones tolerant to drought. Conversely, parameters, such as chlorophyll a fluorescence, F0/Fm ratio, and relative water content, represented strong predictors to describe the behavior of the clones subjected to differential water regimes. These tools will facilitate early plant selection, which is important in tree improvement programs and sustainable eucalyptus forest management.

Variability of Multisite Decadal Running Means Arising from Year-To-Year Fluctuations

AbstractDecadal mean variables are frequently used to characterize decadal climate variabilities. Decadal means are often calculated using yearly data, which can represent variability at time scales from annual to centennial. Residuals from interannual fluctuations may contribute to the variability in decadal time series. Such variability is more difficult to be predicted at the long range. Removing it from the decadal variability means that the remaining variability is more likely to arise from slowly varying multidecadal or longer time scale external forcing and internal climate dynamics, which are more likely to be predicted. Here, a new approach is proposed to understand the uncertainty, potential predictability, and drivers of decadal mean variables. The covariance matrix of multivariate decadal running means is decomposed into unpredictable fast decadal variability and the potentially predictable slow decadal variability. EOF analysis is then applied to the decomposed matrices to find the dominant modes, which may be related to the drivers of the two types of variabilities in the multivariate decadal means. The methodology has been applied to 140-yr datasets of North Pacific sea surface temperature and the Northern Hemisphere 1000-hPa geopotential height. For sea surface temperature, the Pacific decadal oscillation is the major driver of the fast decadal variability, while the radiative forcing and the Atlantic multidecadal oscillation are major drivers of the slow decadal variability. For the 1000-hPa geopotential height, fast decadal variability is associated with the northern annular mode, the east Atlantic mode, and the Pacific decadal oscillation. Slow decadal variability is associated with the northern annular mode and the Atlantic multidecadal oscillation.

Problematic Issues of Intellectualizing the Control System of Autonomous Underwater Vehicles

The paper considers ways of intellectualizing the control processes of autonomous underwater vehicles (AUVs) by the example of solving three problems, on which the successful use of AUVs largely depends. The first problem is to create an AUV control system (CS), which ensures the achievement of the mission’s objective in the event of emergencies caused by both external and internal reasons, as well as deliberate and unintentional counteraction. It is shown that for the construction of the AUV’s CS the decentralized multi-agent structure is most suitable, in which each AUV system is an independent intellectual agent with its own control system. The control system must be equipped with a set of adaptive algorithms that ensure: control of AUV in the event of emergency situations, taking into account restrictions on the supply of electric energy, speed, accuracy of autonomous underwater navigation, range of hydro-acoustic communication; rational distribution of energy resources by AUV systems in accordance with the current situation; maintaining the functional stability of the AUV with a partial malfunction of technical means. The second problem is to create an underwater navigation system that ensures the accomplishment of AUV missions at great distances from the base point. Since AUV navigation using only on-board means (inertial navigation system and lag) does not provide the necessary accuracy, a prerequisite for AUV navigation over long distances is to conduct an observation using external sources, the choice of which in the circumstances is a non-trivial task. The third problem is to create a network underwater communication system (NUCS), which provides for the group application of AUVs. The ground analogue of NUCS is network radio communication. But if the latter is fairly well developed, then the former only takes the initial steps. This is due to both the later practical relevance of NUCS, and many fundamental physical factors that impede the development of NUCS, which include: a substantially limited frequency band that can be used in practice for signal transmission; a large propagation time of the hydroacoustic signal compared to the radio signal; the formation of extended shadow zones and fading of the connected signal due to its multipath propagation; significant Doppler distortions, fast variability of the characteristics of the hydroacoustic medium.

Обґрунтування методу усереднення для нелокальної m-частотної задачі із лінійно перетвореними аргументами

The system of differential equations with delay on a finite interval with slow and fast variables is investigated. The delay in the system is characterized by linearly transformed arguments in slow and fast variables. Integral conditions are given for slow and fast variables. A characteristic feature of such systems is the appearance of resonances in the process of evolution. The condition for the resonance in the system contains a dependence on the delays in fast variables. An effective method for the research of multi-frequency systems is the method of averaging, the substantiation of which for systems without delay of the argument is obtained in the works of V. I. Arnold, E. O. Grebenikov, M. M. Khapaiev, А. М. Samoilenko, R. I. Petryshyn. This paper uses the method proposed by A. M. Samoilenko which is based on the estimation of oscillating integrals. In this paper, the procedure of averaging over fast variables is carried out both in the system of equations and in integral conditions. In the average problem, the variables are separated and the problem for slow variables is solved independently of the fast variables. Finding fast variables is reduced to the problem of integration. The existence of a unique solution for the problem in the class of continuously differentiated functions is proved. The accuracy estimation of the averaging method is obtained, which clearly depends on the small parameter and the number of fast variables and linearly transformed arguments in them. An estimate of the value of the small parameter was also found. The condition of passing resonant zones is reduced to checking the difference from zero of the Vronsky determinant, constructed by the frequency system taking into account the number of linearly transformed arguments. An example of a single-frequency system with integral conditions is constructed, on which the obtained result is illustrated; accuracy estimation and values of the small parameter are obtained.

Gait plasticity impairment as an early frailty biomarker.

To assess whether gait plasticity and gait reserve, valid measures of gait adaptation to environmental stressors, are associated with frailty. Cross-sectional sub-analysis of the FISTAC study (Identification of the Physical Attributes of the Fear of Falling Syndrome). Community-dwelling women from the Falls Unit of a Geriatrics Department. One hundred and twenty-nine women with an age ≥ 70 years old and presence of at least one previous fall in the last year. Age, comorbidity, nutritional status, cognitive status, depression, medications, disability, fear of falling, physical function, hand grip strength, 1RM leg-press strength, maximum and mean leg-press power were determined. Frailty was assessed using the frailty phenotype criteria. Gait plasticity parameters were measured by walking at normal pace, fast pace, and slow pace, and mean (left and right) stride velocity and stride variability (SD) for the three walks were determined independently and for the sum of the three walks. Gait reserve was calculated as the difference in stride velocity from normal to fast pace. ROC curves were constructed to determine the best association between gait plasticity parameters and frailty. The mean age of the participants was 79 years (SD 8.0). The median of normal, fast, slow and three-walks pace stride velocity were 68.9 cm/s (interquartile range [IQR 33.8]), 96.1 cm/s (IQR 38.3), 51.6 cm/s (IQR 19.8), and 72.7 cm/s (IQR 20.7) respectively. The median of normal, fast, slow and three-walks pace stride variability were 4.5 cm/s (IQR 3.3), 5.4 cm/s (IQR 3.8), 3.6 cm/s (IQR 2.3) and 15.9 cm/s (IQR 16.5) respectively. The median of gait reserve was 23 cm/s (IQR 46). Gait reserve and fast pace stride velocity were associated not only with frailty, but also with a lower age, disability, depression, physical function, muscle strength and power, and fear of falling, more than gait velocity. Areas under the curve (95% CI) for gait parameters with stronger association with frailty were fast pace stride velocity 0.801 (0.723-0.880), three-walk mean stride velocity 0.761 (0.678-0.845), three-walks stride variability 0.724 (0.635-0.81) and gait reserve 0.727 (0.635-0.818). Lower gait reserve and lower gait plasticity have a stronger association with frailty than gait speed in older women. Our results may support the use of these gait parameters to early identify frailty in community-dwelling older women.

Intrinsic Determination of the Criticality of a Slow–Fast Hopf Bifurcation

The presence of slow-fast Hopf (or singular Hopf) points in slow-fast systems in the plane is often deduced from the shape of a vector field brought into normal form. It can however be quite cumbersome to put a system in normal form. In the monograph "Canards from birth to transition", an intrinsic presentation of slow-fast vector fields is initiated, showing hands-on formulas to check for the presence of such singular contact points. We generalize the results in the sense that the criticality of the Hopf bifurcation can be checked with a single formula. We demonstrate the result on a slow-fast system given in non-standard form where slow and fast variables are not separated from each other. The formula is convenient since it does not require any parameterization of the critical curve.