Fast Variables Research Articles

Anomalous stochastic dynamics induced by the slip–stick friction and leading to phantom attractors

Abstract A new phenomenon of phantom attractors appearing as a result of slip–stick friction force under the influence of stochastic forcing is revealed on the basis of simple nonlinear dynamical model. The abrupt stepwise transitions between the stable equilibria or cycles and floating phantom attractors are found at different “noise” - “no noise” shifts. Some unusual features of stochastic dynamics are connected with the fact that the system with small noise exhibits large amplitude oscillations while the noise of greater intensities induces the lesser stochastic fluctuations near the phantom attractor. The theory based on the separation of slow and fast dynamical variables is developed to predict the arrangement of phantom attractors.

Slow manifolds for dynamical systems with non-Gaussian stable Lévy noise

This work is concerned with the dynamics of a class of slow–fast stochastic dynamical systems driven by non-Gaussian stable Lévy noise with a scale parameter. Slow manifolds with exponentially tracking property are constructed, and then we eliminate the fast variables to reduce the dimensions of these stochastic dynamical systems. It is shown that as the scale parameter tends to zero, the slow manifolds converge to critical manifolds in distribution, which helps to investigate long time dynamics. The approximations of slow manifolds with error estimate in distribution are also established. Furthermore, we corroborate these results by three examples from biological sciences.

Energy Efficient and Intermittently Variable Ammonia Synthesis over Mesoporous Carbon-Supported Cs-Ru Nanocatalysts

The Cs-promoted Ru nanocatalysts supported on mesoporous carbon materials (denoted as Cs-Ru/MPC) and microporous activated carbon materials (denoted as Cs-Ru/AC) were prepared for the sustainable synthesis of ammonia under mild reaction conditions (<500 °C, 1 MPa). Both Ru and Cs species were homogeneously impregnated into the mesostructures of three commercial available mesoporous carbon materials annealed at 1500, 1800 and 2100 °C (termed MPC-15, MPC-18 and MPC-21, respectively), resulting in a series of Cs-Ru/MPC catalysts with Ru loadings of 2.5–10 wt % and a fixed Cs loading of 33 wt %, corresponding to Cs/Ru molar ratios of 2.5–10. However, the Ru and Cs species are larger than the pore mouths of microporous activated carbon (shortly termed AC) and, as a consequence, were mostly aggregated on the outer surface of the Cs-Ru/AC catalysts. The Cs-Ru/MPC catalysts are superior to the Cs-Ru/AC catalyst in catalysing mild ammonia synthesis, especially for the 2.5Cs-10Ru/MPC-18 catalyst with a Ru loading of 10 wt % and a Cs/Ru ratio of 2.5, which exhibited the highest activity across a wide SV range. It also showed an excellent response and stability during cycling tests over a severe temperature jump in a short time, presumably due to the open mesoporous carbon framework and suitable surface concentrations of CsOH and metallic Ru species at the catalytically active sites. This 2.5Cs-10Ru/MPC-18 catalyst with high activity, fast responsibility and good stability has potential application in intermittently variable ammonia synthesis using CO2-free hydrogen derived from electrolysis of water using renewable energy with fast variability.

Sub‐second variability in black‐hole X‐ray binary jets

In the last 10 years, multiwavelength fast variability studies of low‐mass X‐ray binaries have shown a dramatic development. A key discovery was the detection of O‐IR sub‐second fluctuations in two black‐hole transients, lagging the X‐rays by ≈0.1 s. This demonstrated how the fluctuations observed in the inflow could be transferred to the jet, allowing, therefore, also to study in a completely new way the physical processes that take place at the base of the jet. In this article, we review the latest developments of the study of jets with this new approach, focusing on the results obtained with cross‐spectral analysis techniques.

Leap-frog patterns in systems of two coupled FitzHugh-Nagumo units.

We study a system of two identical FitzHugh-Nagumo units with a mutual linear coupling in the fast variables. While an attractive coupling always leads to synchronous behavior, a repulsive coupling can give rise to dynamical regimes with alternating spiking order, called leap-frogging. We analyze various types of periodic and chaotic leap-frogging regimes, using numerical path-following methods to investigate their emergence and stability, as well as to obtain the complex bifurcation scenario which organizes their appearance in parameter space. In particular, we show that the stability region of the simplest periodic leap-frog pattern has the shape of a locking cone pointing to the canard transition of the uncoupled system. We also discuss the role of the timescale separation in the coupled FitzHugh-Nagumo system and the relation of the leap-frog solutions to the theory of mixed-mode oscillations in multiple timescale systems.

Reverberation reveals the truncated disc in the hard state of GX 339-4

The nature and geometry of the hard state in black hole binaries is controversial. The broadband continuum spectrum and fast variability properties can be explained in a model where the inner disc evaporates into a geometrically thick, hot flow. However these models are challenged by the persistent detection of an extremely broad iron line, which requires that the disc extends down to the last stable orbit of a high spin black hole. This line width can be considerably reduced if the Comptonisation continuum is multi-component rather than single temperature, but such models are highly degenerate. Here we show a specific model of a radially stratified continuum coupled to a model of propagating fluctuations, fit to some of the best hard state data from GX 339-4. This full spectral-timing model can fit the time averaged spectrum, the power spectra in different energy bands, and the frequency dependent lags between these bands. For the first time we also include disc reverberation and show that this same spectral-timing model successfully \textit{predicts} the lag-energy spectra on all timescales. This gives a more robust method to determine the inner radius of the disc, which is of order $20~R_g$, i.e. significantly truncated. This opens up the way to use the fast variability spectral-timing data to trace the source geometry of black hole binaries in all states.

Evolution of space tethered system’s orbit during space debris towing taking into account the atmosphere influence

Space debris removal by a low thrust tethered spacecraft is considered in this paper. The objective of the work is the development of a simplified mathematical model describing the perturbed motion of the space tethered system under the influence of the aerodynamic drag forces and low thrust of the spacecraft’s engines and study the evolution of the space tethered system’s center of mass orbital parameters on large time intervals. The mathematical model describing the plane motion of a space tethered system with inextensible massless tether is constructed. Linearization and averaging of the mathematical model over the angle of the tether deflection are carried out. Separation of fast and slow variables is performed using Van der Pol approach. The obtained system is averaged over a fast variable. It is used to study the evolution of the center of mass orbit during space debris removal. The effect of the space tethered system attitude motion on the center of mass motion is analyzed. It is concluded that the greatest influence of the relative motion is observed in the case when the tether oscillates near the local vertical so that the space tug is located above the space debris.

IGR J17503–2636: a candidate supergiant fast X-ray transient

The object IGR J17503–2636 is a hard X-ray transient discovered by INTEGRAL on 2018 August 11. This was the first ever reported X-ray emission from this source. Following the discovery, follow-up observations were carried out with Swift, Chandra, NICER, and NuSTAR. Here we report on the analysis of all of these X-ray data and the results obtained. Based on the fast variability in the X-ray domain, the spectral energy distribution in the 0.5–80 keV energy range, and the reported association with a highly reddened OB supergiant at ∼10 kpc, we conclude that IGR J17503–2636 is most likely a relatively faint new member of the supergiant fast X-ray transients. Spectral analysis of the NuSTAR data revealed a broad feature in addition to the typical power-law with exponential roll-over at high energy. This can be modeled either in emission or as a cyclotron scattering feature in absorption. If confirmed by future observations, this feature would indicate that IGR J17503–2636 hosts a strongly magnetized neutron star with B ∼ 2 × 1012 G.

D'Alembert‐type solution of the Cauchy problem for the Boussinesq‐Klein‐Gordon equation

AbstractIn this paper, we construct a weakly‐nonlinear d'Alembert‐type solution of the Cauchy problem for the Boussinesq‐Klein‐Gordon (BKG) equation. Similarly to our earlier work based on the use of spatial Fourier series, we consider the problem in the class of periodic functions on an interval of finite length (including the case of localized solutions on a large interval), and work with the nonlinear partial differential equation with variable coefficients describing the deviation from the oscillating mean value. Unlike our earlier paper, here we develop a novel multiple‐scales procedure involving fast characteristic variables and two slow time scales and averaging with respect to the spatial variable at a constant value of one or another characteristic variable, which allows us to construct an explicit and compact d'Alembert‐type solution of the nonlinear problem in terms of solutions of two Ostrovsky equations emerging at the leading order and describing the right‐ and left‐propagating waves. Validity of the constructed solution in the case when only the first initial condition for the BKG equation may have nonzero mean value follows from our earlier results, and is illustrated numerically for a number of instructive examples, both for periodic solutions on a finite interval, and localized solutions on a large interval. We also outline an extension of the procedure to the general case, when both initial conditions may have nonzero mean values. Importantly, in all cases, the initial conditions for the leading‐order Ostrovsky equations by construction have zero mean, while initial conditions for the BKG equation may have nonzero mean values.

MODELING THE MOTION OF A SOLID BODY UNDER THE ACTION OF THE MOMENT OF LIGHT PRESSURE IN THE MEDIUM WITH RESISTANCE

The paper describes the simulation of fast rotational motionof a dynamically asymmetric satellite relative to the centre of mass under the influence of the joint effect of the moment of forces of light pressure and resistance. It is assumed that the surface of the spacecraft is a surface of revolution. The medium creates a weak resistance proportional to the angular velocity of the rigid body's own rotation relative to the centre of mass. Orbital motions with an arbitrary eccentricity are considered given. The mathematical model of satellite motion in this formulation is described by a rigid system of differential equations: the fast variables are Euler angles, and the slow variables are the modulus of the angular momentum vector, the kinetic energy, and the angles of ori-entation of the angular momentum vector in space. Averaging is performed over the Euler-Poinsot motion. The averaged system of equations of body motion allows numerical simulation of the satellite's motion relative to the centre of mass. The study is carried out in a dimensionless form for a multiparameter system of equations. For numerical calculation, an implicit third-order Adams method is used to integrate systems of differential equations. A personal computational package was developed for the constructed mathe-matical model of the satellite, as well as a library for calculating the complete elliptic integrals of the first and second kinds. Numer-ical calculation allows one to obtain the functions of modulating the modulus of the satellite kinetic moment vector, its orientation angles to the orbit, as well as the satellite kinetic energy values. The analysis of the influence of the parameters of the problem on the nature of the motion of the satellite relative to the centre of mass is carried out. A qualitative picture was obtained of the influence of the initial values of the angles of orientation of the kinetic moment vector, the geometry of the masses, the eccentricity of the orbit, the characteristic numbers of disturbing moments on the hodograph character of the kinetic moment vector. The hodograph of the kinetic moment vector in three-dimensional space is simulated for various values of the system parameters. To construct three-dimensional objects on the scene, according to the carried out numerical calculations, we developed our own software using DirectX technology in C# language, simulating a virtual laboratory of a numerical experiment.

Multiwavelength spectroscopy of the black hole candidate MAXI J1813-095 during its discovery outburst

Abstract MAXI J1813-095 is an X-ray transient discovered during an outburst in 2018. We report on X-ray and optical observations obtained during this event, which indicate that the source is a new low-mass X-ray binary. The outburst lasted ∼70 d and peaked at LX(0.5–10 keV) ∼ 7.6 × 1036 erg s−1, assuming a distance of 8 kpc. Swift/XRT follow-up covering the whole activity period shows that the X-ray emission was always dominated by a hard power-law component with a photon index in the range of 1.4–1.7. These values are consistent with MAXI J1813-095 being in the hard state, in agreement with the ∼30 per cent fractional root-mean-square amplitude of the fast variability (0.1–50 Hz) inferred from the only XMM–Newton observation available. The X-ray spectra are well described by a Comptonization emission component plus a soft, thermal component (kT ∼ 0.2 keV), which barely contributes to the total flux (≲8 per cent). The Comptonization y-parameter (∼1.5), together with the low temperature and small contribution of the soft component supports a black hole accretor. We also performed optical spectroscopy using the Very Large Telescope and Gran Telescopio Canarias telescopes during outburst and quiescence, respectively. In both cases, the spectrum lacks emission lines typical of X-ray binaries in outburst. Instead, we detect the Ca ii triplet and H α in absorption. The absence of velocity shifts between the two epochs, as well as the evolution of the H α equivalent width, strongly suggest that the optical emission is dominated by an interloper, likely a G–K star. This favours a distance ≳3 kpc for the X-ray transient.

Reduction of a kinetic model for Na^{+} channel activation, and fast and slow inactivation within a neural or cardiac membrane.

A 15-state kinetic model for Na^{+} channel gating that describes the coupling between three activation sensors, a two-stage fast inactivation process, and slow inactivated states may be reduced to equations for a 6-state system by application of the method of multiple scales. By expressing the occupation probabilities for closed states and the open state in terms of activation and fast inactivation variables, and assuming that activation has a faster relaxation than inactivation and that the activation sensors are mutually independent, the kinetic equations may be further reduced to rate equations for activation, and coupled fast and slow inactivation that describe spike frequency adaptation, a repetitive bursting oscillation in the neural membrane, and a cardiac action potential with a plateau oscillation. The fast inactivation rate function is, in general, dependent on the activation variable m(t) but may be approximated by a voltage-dependent function, and the rate function for entry into the slow inactivated state is dependent on the fast inactivation variable.

On Asymptotic Stability of Homogeneous Singular Systems with Switching

This paper considers the stability of singular switched systems with homogeneous functions in the right-hand sides of the corresponding equations. Stability conditions of the complete system with an arbitrary switching mode are established through its decomposition into the isolated subsystems of fast and slow motions. A stabilizing feedback control design method based on measurements of the fast variables only is suggested.

A fast, very-high-energy γ-ray flare from BL Lacertae during a period of multi-wavelength activity in June 2015

The mechanisms producing fast variability of the γ-ray emission in active galactic nuclei (AGNs) are under debate. The MAGIC telescopes detected a fast, very-high-energy (VHE, E > 100 GeV) γ-ray flare from BL Lacertae on 2015 June 15. The flare had a maximum flux of (1.5 ± 0.3) × 10−10 photons cm−2 s−1 and halving time of 26 ± 8 min. The MAGIC observations were triggered by a high state in the optical and high-energy (HE, E > 100 MeV) γ-ray bands. In this paper we present the MAGIC VHE γ-ray data together with multi-wavelength data from radio, optical, X-rays, and HE γ rays from 2015 May 1 to July 31. Well-sampled multi-wavelength data allow us to study the variability in detail and compare it to the other epochs when fast, VHE γ-ray flares have been detected from this source. Interestingly, we find that the behaviour in radio, optical, X-rays, and HE γ-rays is very similar to two other observed VHE γ-ray flares. In particular, also during this flare there was an indication of rotation of the optical polarization angle and of activity at the 43 GHz core. These repeating patterns indicate a connection between the three events. We also test modelling of the spectral energy distribution based on constraints from the light curves and VLBA observations, with two different geometrical setups of two-zone inverse Compton models. In addition we model the γ-ray data with the star-jet interaction model. We find that all of the tested emission models are compatible with the fast VHE γ-ray flare, but all have some tension with the multi-wavelength observations.

Multi-frequency Variability Study of Ton 599 during the High Activity of 2017

Abstract In this work, I have presented a multi-frequency variability and correlation study of the blazar Ton 599, which was observed for the first time in the flaring state at the end of 2017. Data from Fermi-LAT, Swift-XRT/UVOT, Steward observatory, and Owens Valley Radio Observatory(OVRO) (15 GHz) are used and it is found that the source is more variable in γ-rays and optical/UV than in X-rays and radio. Large variations in degree of polarization and position angle are observed during the flaring period. Maximum flux during the γ-ray flare is found to be 12.63 × 10−7 at MJD 58057.5 from the 1 day bin light curve, which is the highest flux ever achieved by this source. It is further found that all the peaks of the flare are very symmetric, which suggests the cooling time of electrons is much smaller than the light-crossing time. Using 1 day as a fast variability time, the size of the γ-ray emission region is estimated as 1.88 × 1016 cm. Two 42 GeV of photons are detected during the flare, which puts a constraint on the location of the emission region, and it is found that the γ-ray emitting blob is located at the outer edge or outside the broad-line region. The trend of increasing fractional variability toward higher energies is also seen. Strong correlations were seen between γ-ray, optical/UV, X-ray, and radio (15 GHz) emission. A small time lag between γ-rays and the optical/UV suggests their emission to be co-spatial, while the lag of 27 days between γ-rays and OVRO (15 GHz) suggests two different emission zones separated by a distance of ∼5 pc.

Optimal perturbation guidance with constraints on terminal flight-path angle and angle of attack

This paper presents the design of a singular-perturbation-based optimal guidance with constraints on terminal flight-path angle and angle of attack. By modeling the flight-control system dynamics as a first-order system, the angle of attack is introduced into the performance index as a state variable. To solve the resulting high-order optimal guidance problem analytically, the posed optimal guidance problem is divided into two sub-problems by utilizing the singular perturbation method according to two time scales: range, altitude, and flight-path angle are the slow time-scale variables while the angle of attack is the fast time-scale variable. The outer solutions are the optimal control of the slow-scale subsystem. Thereafter, by applying the stretching transformation, the fast-scale subsystem establishes the relationships between the outer solutions and acceleration command. Then, the optimal command can be obtained by solving the fast-scale subsystem also using the optimal control theory. The proposed guidance can achieve a near-zero terminal acceleration as well as a small miss distance. The superior performance of the guidance is demonstrated by adequate trajectory simulations.

A New Finite Time Control Solution for Robotic Manipulators Based on Nonsingular Fast Terminal Sliding Variables and the Adaptive Super-Twisting Scheme

In this study, a new finite time control method is suggested for robotic manipulators based on nonsingular fast terminal sliding variables and the adaptive super-twisting method. First, to avoid the singularity drawback and achieve the finite time convergence of positional errors with a fast transient response rate, nonsingular fast terminal sliding variables are constructed in the position errors' state space. Next, adaptive tuning laws based on the super-twisting scheme are presented for the switching control law of terminal sliding mode control (TSMC) so that a continuous control law is extended to reject the effects of chattering behavior. Finally, a new finite time control method ensures that sliding motion will take place, regardless of the effects of the perturbations and uncertainties on the robot system. Accordingly, the stabilization and robustness of the suggested control system can be guaranteed with high-precision performance. The robustness issue and the finite time convergence of the suggested system are totally confirmed by the Lyapunov stability principle. In simulation studies, the experimental results exhibit the effectiveness and viability of our proposed scheme for joint position tracking control of a 3DOF PUMA560 robot.

Deterministic-Like Model Reduction for a Class of Multiscale Stochastic Differential Equations With Application to Biomolecular Systems

We consider a class of singularly perturbed stochastic differential equations with linear drift terms, and present a reduced-order model that approximates both the slow and fast variable dynamics when the time-scale separation is large. We show that, on a finite time interval, moments of all orders of the slow variables for the original system become closer to those of the reduced-order model as time-scale separation is increased. A similar result holds for the first and second moments of the fast variable approximation. Biomolecular systems with linear propensity functions, modeled by the chemical Langevin equation fit the class of systems considered in this paper. Thus, as an application example, we analyze the tradeoffs between noise and information transmission in a typical gene regulatory network motif, for which both the slow and fast variables are required. 1

Асимптотическое разложение решения сингулярно возмущенной задачи оптимального управления с интегральным выпуклым критерием качества, терминальная часть которого зависит от медленных и быстрых переменных

Асимптотическое разложение решения сингулярно возмущенной задачи оптимального управления с интегральным выпуклым критерием качества, терминальная часть которого зависит от медленных и быстрых переменных

Регуляризация и построение асимптотических решений в методе пограничных функций

In his brief article published in 1977, S.A. Lomov noted that the asymptotics of Vasil'eva—Butuzov’s type boundary layer for a singularly perturbed problem can be obtained by the method of transition to a space of higher dimension or regularization of the original differential operator using a fast independent variable. Unfortunately, a detailed algorithm for constructing such asymptotics is not given in the article, and it remains to be answered whether the asymptotics obtained in it will actually coincide with a boundary-layer type asymptotics. In the present paper, this question is solved positively. However, the regularization itself is divided into two stages: partial regularization and complete regularization. In the course of partial regularization, it is not possible to obtain an "extended" problem the iterative systems of which admit solutions in the form of the sum of two functions with separated independent variables (one of which is an ordinary variable, and the other is a regularizing variable) The complete regularization has at its heart the procedure for constructing an "extended" system, which was applied by the authors in their papers devoted to regularization of integral operators, the essence of which consists in regularizing the original problem in the class of formal asymptotic series in powers of a small parameter with coefficients in the form of the sum of functions with separated variables. The asymptotics obtained using such procedure coincides completely with the boundary-layer type asymptotics, a result that confirms S.A. Lomov’s hypothesis.