Fast Variables Research Articles

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.

VFFVA: dynamic load balancing enables large-scale flux variability analysis

BackgroundGenome-scale metabolic models are increasingly employed to predict the phenotype of various biological systems pertaining to healthcare and bioengineering. To characterize the full metabolic spectrum of such systems, Fast Flux Variability Analysis (FFVA) is commonly used in parallel with static load balancing. This approach assigns to each core an equal number of biochemical reactions without consideration of their solution complexity.ResultsHere, we present Very Fast Flux Variability Analysis (VFFVA) as a parallel implementation that dynamically balances the computation load between the cores in runtime which guarantees equal convergence time between them. VFFVA allowed to gain a threefold speedup factor with coupled models and up to 100 with ill-conditioned models along with a 14-fold decrease in memory usage.ConclusionsVFFVA exploits the parallel capabilities of modern machines to enable biological insights through optimizing systems biology modeling. VFFVA is available in C, MATLAB, and Python at https://github.com/marouenbg/VFFVA.

Θ-SEIHRD mathematical model of Covid19-stability analysis using fast-slow decomposition.

In general, a mathematical model that contains many linear/nonlinear differential equations, describing a phenomenon, does not have an explicit hierarchy of system variables. That is, the identification of the fast variables and the slow variables of the system is not explicitly clear. The decomposition of a system into fast and slow subsystems is usually based on intuitive ideas and knowledge of the mathematical model being investigated. In this study, we apply the singular perturbed vector field (SPVF) method to the COVID-19 mathematical model of to expose the hierarchy of the model. This decomposition enables us to rewrite the model in new coordinates in the form of fast and slow subsystems and, hence, to investigate only the fast subsystem with different asymptotic methods. In addition, this decomposition enables us to investigate the stability analysis of the model, which is important in case of COVID-19. We found the stable equilibrium points of the mathematical model and compared the results of the model with those reported by the Chinese authorities and found a fit of approximately 96 percent.

Program and Results of Investigations Rapid Variability of the BL Lac Object 3C 371 in Radio and Optical Ranges

BL Lac object 3C 371 is variable in optical and radio range, according to long-term monitoring data, for example AAVSO (American Association of Variable Star Observers) and OVRO (Owens Valley Radio Observatory). In addition, some authors note intra-night variability. However, in terms of access, just a few works are devoted to this topic, and questions remain about intra-day variability in radio range. The purpose of the work is to search for fast variability in radio (5, 6.1, 6.7 GHz) and optical bands (V, R, I) using international cooperation in 2019 and 2020 observation sessions. The 16-m radio telescope VIRAC, in Latvia, as well as optical telescopes AZT-3 (Mayaki, Ukraine), VNT (Vihorlat, Slovakia), and Schmidt camera (Baldone, Latvia) were used. To analyze variability, the STFT method of spectrograms and Lomb–Scargle periodograms for non-uniform time series were used. As result of the work, there is no correlation between optical and radio observations, and no significant quasi-harmonic variability was detected in radio range, but there is irregular low amplitude variability. In the optical range, there is variability with a characteristic time of about seven days and possibly less. Cyclical variability of 3C 371 was found in the optical range, and intra-day variability in radio range is most likely absent, as there are irregular variations and noise. It is planned to continue joint radio-optical observations 3C 371 to accumulate the necessary statistics.

NICER observations reveal that the X-ray transient MAXI J1348−630 is a black hole X-ray binary

ABSTRACT We studied the outburst evolution and timing properties of the recently discovered X-ray transient MAXI J1348−630 as observed with NICER. We produced the fundamental diagrams commonly used to trace the spectral evolution, and power density spectra to study the fast X-ray variability. The main outburst evolution of MAXI J1348−630 is similar to that commonly observed in black hole transients. The source evolved from the hard state (HS), through hard- and soft-intermediate states, into the soft state in the outburst rise, and back to the HS in reverse during the outburst decay. At the end of the outburst, MAXI J1348−630 underwent two reflares with peak fluxes approximately one and two orders of magnitude fainter than the main outburst, respectively. During the reflares, the source remained in the HS only, without undergoing any state transitions, which is similar to the so-called ‘failed outbursts’. Different types of quasi-periodic oscillations (QPOs) are observed at different phases of the outburst. Based on our spectral-timing results, we conclude that MAXI J1348−630 is a black hole candidate.

Application of the asymptotic homogenization in a parametric space to the modeling of structurally heterogeneous materials

An asymptotic averaging of differential equations with fast oscillating quasi-periodic coefficients so-called “the asymptotic averaging in parametric space” is developed. The system of equations corresponding to structurally heterogeneous thermoelastic media with smoothly varying microstructures is considered. Such materials are usually treated as functionally graded ones. Their description is satisfied by introducing two types of variables into the coefficients of the system of thermoelastic equations: “fast” and “slow”. The fast variables describe the geometry of the inhomogeneities provided their periodic arrangement. The additional variables allow one to describe a smooth spatial variation of a periodic structure of inclusions, i.e. to describe within the proposed approach various materials with functionally graded properties. In addition, the coefficients may depend on the temperature, which also has a given regular distribution in the space. The allocation of this dependence in the separate parameter makes sense, since it is important in practice. It is shown that the asymptotic averaging solves the smooth aperiodic dependencies for the coefficients of the considered equations system parametrically through the functions of rapid variables. A two-level model system for such structurally heterogeneous materials is formulated as a result of averaging, and an algorithm of accurate definition of thermomechanical properties including the functionally graded and thermally depending ones could be found. The combined numerical-analytical block method based on the Papkovich–Neuber​ representation is proposed for low-level problems with arbitrary structures (e.g. for the problem defined on a cell for functions of fast variables; such a problem determines the effective properties of a material). The structures with spherical and cylindrical inclusions and interphase layers are especially considered and the complete function systems allowing one an efficient solution approximation on a cell with exactly satisfied contact conditions on interlayers is constructed.

Electroencephalographic prediction of global and domain specific cognitive performance of clinically active Australian Nurses

Objective: To investigate the relationship between EEG activity and the global and domain specific cognitive performance of healthy nurses, and determine the predictive capabilities of these relationships. Approach: Sixty-four nurses were recruited for the present study, and data from 61 were utilised in the present analysis. Global and domain specific cognitive performance of each participant was assessed psychometrically using the Mini-mental state exam and the Cognistat, and a 32-lead monopolar EEG was recorded during a resting baseline phase and an active phase in which participants completed the Stroop test. Main results: Global cognitive performance was successfully predicted (81%–85% of variance) by a combination of fast wave activity variables in the alpha, beta and theta frequency bands. Interestingly, predicting domain specific performance had varying degrees of success (42%–99% of the variance predicted) and relied on combinations of both slow and fast wave activity, with delta and gamma activity predicting attention performance; delta, theta, and gamma activity predicting memory performance; and delta and beta variables predicting judgement performance. Significance: Global and domain specific cognitive performance of Australian nurses may be predicted with varying degrees of success by a unique combination of EEG variables. These proposed models image transitory cognitive declines and as such may prove useful in the prediction of early cognitive impairment, and may enable better diagnosis, and management of cognitive impairment.

Local Adiabatic Invariants Near a Homoclinic Set of a Slow–Fast Hamiltonian System

In slow–fast systems, fast variables change at a rate of the order of one, and slow variables, at a rate of the order of $$\varepsilon\ll 1$$ . The system obtained for $$\varepsilon=0$$ is said to be frozen. If the frozen (fast) system has one degree of freedom, then in the region where the level curves of the frozen Hamiltonian are closed there exists an adiabatic invariant. A. Neishtadt showed that near a separatrix of the frozen system the adiabatic invariant exhibits quasirandom jumps of order $$\varepsilon$$ . In this paper we partially extend Neishtadt’s result to the multidimensional case. We show that if the frozen system has a hyperbolic critical point possessing several transverse homoclinics, then for small $$\varepsilon$$ there exist trajectories shadowing homoclinic chains. The slow variables evolve in a quasirandom way, shadowing trajectories of systems with Hamiltonians similar to adiabatic invariants. This paper extends the work of V. Gelfreich and D. Turaev, who considered similar phenomena away from critical points of the frozen Hamiltonian.

Hierarchical sparse observation models and informative prior for Bayesian inference of spatially varying parameters

We develop a new computational approach for the speedy and accurate recovery of unknown spatially varying parameters of (stochastic) PDE models based on the sequentially arriving noisy observations of the evolving system variables. Two essential ingredients in our strategy for the identification of the target spatial fields are given by the Fourier diagonalization (FD) method and the hierarchical Bayesian model.We first apply the FD framework to perform an effectively parallelized computation in quantifying the propagating uncertainties of the dynamic variables in the Fourier space, and to facilitate a drastic saving of computational resources required for the Bayesian estimation of the associated parameters via sequential data assimilation. Yet our case study reveals that a very poor performance of the FD scheme occurs when the fast and slow variables coexist in the system evolution. The key observation is that, estimating the parameters in association with some slow variables, the FD-based Bayesian solver significantly underperforms compared to the remaining cases of faster variables. Due to this highly non-uniform discrepancy in the accuracy across distinct Fourier modes, the approximation of the parameter field cannot be so desired if the consequence is represented in the physical space.As one effort to circumvent this problem, we provide a systematic approach for a radical improvement of the naive FD technique. To do this, we make use of the hierarchical Bayesian model; it refers to the process of gradually enriching the knowledge of the unknown spatial fields from coarse to medium and to fine resolutions by using the Bayesian inference obtained at coarser levels to provide prior information for the estimation at finer levels. One major contribution of this paper is the development of a variant of the classical application of the multi-resolution approach through the close integration with the FD method, leading to the emergence of a new Bayesian paradigm for the data-driven parametric identification. Numerical experiments are performed to corroborate our demonstration concerning the efficacy and effectiveness of the proposed algorithm in obtaining a good degree of accuracy together with a significantly reduced computational cost.

Asymptotics of the Solution to a Singularly Perturbed Time-Optimal Control Problem with Two Small Parameters

The paper continues the authors’ previous studies. We consider a time-optimal control problem for a singularly perturbed linear autonomous system with two independent small parameters and smooth geometric constraints on the control in the form of a ball \(\left\{\begin{array}[]{llll}\phantom{\varepsilon^{3}}\dot{x}=y,&x,\,y\in \mathbb{R}^{2},\quad u\in\mathbb{R}^{2},\\ \varepsilon^{3}\dot{y}=Jy+u,&\,\|u\|\leq 1,\quad 0<\varepsilon,\mu\ll 1,\\ x(0)=x_{0}(\varepsilon,\mu)=(x_{0,1},\varepsilon^{3}\mu\xi)^{*},\quad y(0)=y_{ 0},\\ x(T_{\varepsilon,\mu})=0,\quad y(T_{\varepsilon,\mu})=0,\quad T_{\varepsilon, \mu}\to\min,&\end{array}\right.\) where \(J=\left(\begin{array}[]{rr}0&1\\ 0&0\end{array}\right).\) The main difference of this case from the systems with fast and slow variables studied earlier is that here the matrix \(J\) at the fast variables is the second-order Jordan block with zero eigenvalue and, thus, does not satisfy the standard asymptotic stability condition. Continuing the research, we consider initial conditions depending on the second small parameter \(\mu\). We derive and justify a complete asymptotic expansion in the sense of Erdelyi of the optimal time and optimal control with respect to the asymptotic sequence \(\varepsilon^{\gamma}(\varepsilon^{k}+\mu^{k})\), \(0<\gamma<1\).

The Origin of Matter at the Base of Relativistic Jets in Active Galactic Nuclei

The generation of relativistic jets in active sources such as blazars is a complex problem with many aspects, most of them still not fully understood. Relativistic jets are likely produced by the accretion of matter and magnetic fields onto spinning black holes. Ergospheric dragging effects launch a Poynting-dominated outflow in the polar directions of these systems. Observations with very high resolution of the jet in the nearby radio galaxy M87 and evidence of extremely fast variability in the non-thermal radiation of several other objects indicate that charged particles produce synchrotron emission and gamma rays very close to the base of the jet. How these particles are injected into the magnetically shielded outflow is a mystery. Here we explore the effects of various processes in the hot accretion inflow close to the black hole that might result in the copious production of neutral particles which, through annihilation and decay in the jet’s funnel, might load the outflow with mass and charged particles on scales of a few Schwarzschild radii.

Active Management of Photovoltaic System Variability With Power Electronics

Grid-connected photovoltaic (PV) energy systems present a variable power source. They are often treated as “negative load,” especially for small systems, and require grid reserve margins and management resources to address their stochastic operation. However, inverters linked to PV energy systems probably represent the fastest resource on the grid, able to adjust operation on a cycle-by-cycle basis. This work reviews and quantifies PV variability, with emphasis on fast time scales. Operating needs if batteries are used to manage variability are presented. A reference case establishes the rates of change and storage needs in megawatt-scale installations. Alternative power electronics operating strategies are explored to mitigate variability directly at the source. Variability measures based on a time-window standard deviation lead to fast adaptive strategies. Controls that trade off energy production against variability are presented. Measured data are used to determine and compare performance expectations among several different control methods. Results, validated with scaled experiments, show that fast variability can decrease by about an order of magnitude given the right control strategy, implying battery capacity reduction of about <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$5\times $ </tex-math></inline-formula> compared to systems in which batteries manage variation. System-level economics strongly favor large variability reduction linked to modest energy reduction.

Characterization of variability in blazar light curves

AbstractBlazars represent the dominant population of the extragalactic γ‐ray sources in the universe. These sources exhibit some characteristic properties such as strong and nonthermal continuum emission over the entire electromagnetic spectrum, from radio to Tera electron Volts (TeV) γ‐rays, with rapid variability on all timescales. The emission at radio and optical wavelengths is highly polarized with significant variation. The fastest variability in the blazar emission is observed during the flaring activity, which is an important observational property of blazars. In this paper, we describe various methods to characterize the temporal variability in the multiwavelength light curves of blazars. We also provide a detailed description of the set of statistical parameters that are used to quantify the level of variability present in the time series. Implications of the information derived from the variability study to probe the physics of blazars using multiwavelength observations are also discussed.

Time lags of the type-B quasi-periodic oscillation in MAXI J1348−630

ABSTRACT The fast variability observed in the X-ray emission from black hole binaries has a very complex phenomenology, but offers the possibility to investigate directly the properties of the inner accretion flow. In particular, type-B oscillations in the 2–8 Hz range, observed in the soft-intermediate state, have been associated with the emission from a relativistic jet. We present the results of the timing and spectral analysis of a set of observations of the bright transient MAXI J1348−630 made with the NICER (Neutron Star Interior Composition Explorer) telescope. The observations are in the brightest part of the outburst and all feature a strong type-B quasi-periodic oscillation (QPO) at ∼4.5 Hz. We compute the energy dependence of the fractional rms and the phase lags at the QPO frequency, obtaining high signal-to-noise data and sampling for the first time at energies below 2 keV. The fractional rms decreases from more than 10 per cent at 9 keV to 0.6 per cent at 1.5 keV, and is constant below that energy. Taking the 2–3 keV band as reference, photons at all energies show a hard lag, increasing with the distance from the reference band. The behaviour below 2 keV has never been observed before, due to the higher energy bandpass of previous timing instruments. The energy spectrum can be fitted with a standard model for this state, consisting of a thin disc component and a harder power law, plus an emission line between 6 and 7 keV. We discuss the results, concentrating on the phase lags, and show that they can be interpreted within a Comptonization model.

A Rapidly Varying Red Supergiant X-Ray Binary in the Galactic Center

Abstract We analyzed multiwavelength observations of the previously identified Galactic center X-ray binary CXO 174528.79–290942.8 (XID 6592) and determine that the near-infrared counterpart is a red supergiant based on its spectrum and luminosity. Scutum X-1 is the only previously known X-ray binary with a red supergiant donor star and closely resembles XID 6592 in terms of X-ray luminosity (L X), absolute magnitude, and IR variability (L IR,var), supporting the conclusion that XID 6592 contains a red supergiant donor star. The XID 6592 infrared counterpart shows variability of ∼0.5 mag in the Wide-field Infrared Survey Explorer-1 band (3.4 μm) on timescales of a few hours. Other infrared data sets also show large-amplitude variability from this source at earlier epochs but do not show significant variability in recent data. We do not expect red supergiants to vary by ∼50% in luminosity over these short timescales, indicating that the variability should be powered by the compact object. However, the X-ray luminosity of this system is typically ∼1000× less than the variable luminosity in the infrared and falls below the Chandra detection limit. While X-ray reprocessing can produce large-amplitude fast infrared variability, it typically requires L IR,var to do so, indicating that another process must be at work. We suggest that this system may be a supergiant fast X-ray transient (SFXT), and that a large (∼1038 ergs s−1), fast (102-4 s) X-ray flare could explain the rapid IR variability and lack of a long-lasting X-ray outburst detection. SFXTs are normally associated with blue supergiant companions, so if confirmed, XID 6592 would be the first red supergiant SFXT, as well as the second X-ray red supergiant binary.

Reduced-Order Algorithm for Eigenvalue Assignment of Singularly Perturbed Linear Systems

In this paper, we present an algorithm for eigenvalue assignment of linear singularly perturbed systems in terms of reduced-order slow and fast subproblem matrices. No similar algorithm exists in the literature. First, we present an algorithm for the recursive solution of the singularly perturbed algebraic Sylvester equation used for eigenvalue assignment. Due to the presence of a small singular perturbation parameter that indicates separation of the system variables into slow and fast, the corresponding algebraic Sylvester equation is numerically ill-conditioned. The proposed method for the recursive reduced-order solution of the algebraic Sylvester equations removes ill-conditioning and iteratively obtains the solution in terms of four reduced-order numerically well-conditioned algebraic Sylvester equations corresponding to slow and fast variables. The convergence rate of the proposed algorithm is Oε, where ε is a small positive singular perturbation parameter.

Nonlinear waves propagation and stability analysis for planar waves at far field using quintic B-spline collocation method

A study of the dynamics of nonlinear waves at far field has been presented for real gases using quintic B-spline collocation method. To examine the dynamics at far field, “fast variable” in the governing system of equations has been introduced. Using asymptotic expansion of the flow variables in new fast variables, an evolution equation has been obtained for the behavior of waves at far from source. The evolution equation is modified Burger’s equation, which is known to have exact solution for planar flow. However, the solution for the corresponding cylindrical and spherical symmetric flow for the equation is unknown. The obtained numerical results have been compared with the exact solution for the planar flow. The error norms of numerically computed values are found very small when compared to corresponding exact solution. A Von Neumann stability analysis of the scheme is presented. The scheme is found unconditionally stable. It shows that the method is quite efficient to capture the dynamics of flow at far field. Results corresponding to other two geometries have been presented at different time. Further, a study on the effect of the real medium on the flow is presented. Schematic shows that with the increase in non-ideal property of the medium, the formation of shock becomes faster.

Spike-Adding Canard Explosion in a Class of Square-Wave Bursters

This paper examines a spike-adding bifurcation phenomenon whereby small-amplitude canard cycles transition into large-amplitude bursting oscillations along a single continuous branch in parameter space. We consider a class of three-dimensional singularly perturbed ODEs with two fast variables and one slow variable and singular perturbation parameter $$\varepsilon \ll 1 $$ under general assumptions which guarantee such a transition occurs. The primary ingredients include a cubic critical manifold and a saddle homoclinic bifurcation within the associated layer problem. The continuous transition from canard cycles to N-spike bursting oscillations up to $$N\sim {\mathcal {O}}(1/\varepsilon )$$ spikes occurs upon varying a single bifurcation parameter on an exponentially thin interval. We construct this transition rigorously using geometric singular perturbation theory; critical to understanding this transition are the existence of canard orbits and slow passage through the saddle homoclinic bifurcation, which are analyzed in detail.

Asymptotic expansion of a solution of a singularly perturbed optimal control problem with a convex integral quality index, whose terminal part additively depends on slow and fast variables

The paper deals with the problem of optimal control with a Boltz-type quality index over a finite time interval for a linear steady-state control system in the class of piecewise continuous controls with smooth control constraints. In particular, we study the problem of controlling the motion of a system of small mass points under the action of a bounded force. The terminal part of the convex integral quality index additively depends on slow and fast variables, and the integral term is a strictly convex function of control variable. If the system is completely controllable, then the Pontryagin maximum principle is a necessary and sufficient condition for optimality. The main difference between this study and previous works is that the equation contains the zero matrix of fast variables and, thus, the results of A.B. Vasilieva on the asymptotic of the fundamental matrix of a control system are not valid. However, the linear steady-state system satisfies the condition of complete controllability. The article shows that problems of optimal control with a convex integral quality index are more regular than time-optimal problems.

A hybrid neuro-fuzzy inference system-based algorithm for time series forecasting applied to energy consumption prediction

The accuracy of the prediction of buildings’ energy consumption is being tackled using existing artificial intelligence techniques. However, there is a lack of effort on the development of new techniques for solving that problem and, therefore, achieving higher performance, which is important for the efficient management of energy in many levels. This study addresses this gap by proposing a new hybrid machine learning algorithm that incorporates the adaptive neuro-fuzzy inference system model with a new version of the firefly algorithm denominated as the gender-difference firefly algorithm. We expanded the search space diversification to increase the accuracy on the prediction and adopted the autoregressive process in order to approximate the chaotic behavior of the consumption time series. A new layer, denominated as non-working time adaptation was also integrated so as to decrease the fast variability of the predictions during non-working periods of time. We have applied our algorithm for the consumption prediction on 1 h, 2 h and 3 h ahead horizons. We have obtained improvements on the MAPE and R coefficient when compared with state-of-the-art publications in both a private dataset from the Faculty of Chemistry, located in the city of Murcia, Spain and a public dataset of the consumption of a Retail building located in California, United States. We also show our method’s performance in five more buildings. Our results demonstrate the robustness and the accuracy of our proposal when compared to the traditional adaptive neuro-fuzzy inference system models and also to the different predictive techniques implemented in several pieces of literature.