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Mariusz Tarnopolski

Publications and source records attributed to Mariusz Tarnopolski.

32 records · Page 2Linked to original sources

Rotation of an oblate satellite: Chaos control

Aims. This paper investigates the chaotic rotation of an oblate satellite in the context of chaos control. Methods. A model of planar oscillations, described with the Beletskii equation, was investigated. The Hamiltonian formalism was utilized to employ a control method for suppressing chaos. Results. An additive control term, which is an order of magnitude smaller than the potential, is constructed. This allows not only for significantly diminished diffusion of the trajectory in the phase space, but turns the purely chaotic motion into strictly periodic motion.

astro-ph.EP↗

Modeling the price of Bitcoin with geometric fractional Brownian motion: a Monte Carlo approach

The long-term dependence of Bitcoin (BTC), manifesting itself through a Hurst exponent $H>0.5$, is exploited in order to predict future BTC/USD price. A Monte Carlo simulation with $10^4$ geometric fractional Brownian motion realisations is performed as extensions of historical data. The accuracy of statistical inferences is 10\%. The most probable Bitcoin price at the beginning of 2018 is 6358 USD.

q-fin.CP↗

Bifurcation Diagrams and Generalized Bifurcation Diagrams for a rotational model of an oblate satellite

This paper presents bifurcation and generalized bifurcation diagrams for a rotational model of an oblate satellite. Special attention is paid to parameter values describing one of Saturn's moons, Hyperion. For various oblateness the largest Lyapunov Characteristic Exponent (LCE) is plotted. The largest LCE in the initial condition as well as in the mixed parameter-initial condition space exhibits a fractal structure, for which the fractal dimension was calculated. It results from the bifurcation diagrams of which most of the parameter values for preselected initial conditions lead to chaotic rotation. The First Recurrence Time (FRT) diagram provides an explanation of the birth of chaos and the existence of quasi-periodic windows occuring in the bifurcation diagrams.

nlin.CD↗

Influence of a second satellite on the rotational dynamics of an oblate moon

The gravitational influence of a second satellite on the rotation of an oblate moon is numerically examined. A simplified model, assuming the axis of rotation perpendicular to the (Keplerian) orbit plane, is derived. The differences between the two models, i.e. in the absence and presence of the second satellite, are investigated via bifurcation diagrams and by evolving compact sets of initial conditions in the phase space. It turns out that the presence of another satellite causes some trajectories, that were regular in its absence, to become chaotic. Moreover, the highly structured picture revealed by the bifurcation diagrams in dependence on the eccentricity of the oblate body's orbit is destroyed when the gravitational influence is included, and the periodicities and critical curves are destroyed as well. For demonstrative purposes, focus is laid on parameters of the Saturn-Titan-Hyperion system, and on oblate satellites on low-eccentric orbits, i.e. $e\approx 0.005$.

astro-ph.EP↗

On the relationship between the Hurst exponent, the ratio of the mean square successive difference to the variance, and the number of turning points

The long range dependence of the fractional Brownian motion (fBm), fractional Gaussian noise (fGn), and differentiated fGn (DfGn) is described by the Hurst exponent $H$. Considering the realisations of these three processes as time series, they might be described by their statistical features, such as half of the ratio of the mean square successive difference to the variance, $\mathcal{A}$, and the number of turning points, $T$. This paper investigates the relationships between $\mathcal{A}$ and $H$, and between $T$ and $H$. It is found numerically that the formulae $\mathcal{A}(H)=a{\rm e}^{bH}$ in case of fBm, and $\mathcal{A}(H)=a+bH^c$ for fGn and DfGn, describe well the $\mathcal{A}(H)$ relationship. When $T(H)$ is considered, no simple formula is found, and it is empirically found that among polynomials, the fourth and second order description applies best. The most relevant finding is that when plotted in the space of $(\mathcal{A},T)$, the three process types form separate branches. Hence, it is examined whether $\mathcal{A}$ and $T$ may serve as Hurst exponent indicators. Some real world data (stock market indices, sunspot numbers, chaotic time series) are analyzed for this purpose, and it is found that the $H$'s estimated using the $H(\mathcal{A})$ relations (expressed as inverted $\mathcal{A}(H)$ functions) are consistent with the $H$'s extracted with the well known wavelet approach. This allows to efficiently estimate the Hurst exponent based on fast and easy to compute $\mathcal{A}$ and $T$, given that the process type: fBm, fGn or DfGn, is correctly classified beforehand. Finally, it is suggested that the $\mathcal{A}(H)$ relation for fGn and DfGn might be an exact (shifted) $3/2$ power-law.

physics.data-an↗

Analysis of gamma-ray burst duration distribution using mixtures of skewed distributions

Two classes of GRBs have been confidently identified thus far and are prescribed to different physical scenarios -- NS-NS or NS-BH mergers, and collapse of massive stars, for short and long GRBs, respectively. A third, intermediate in duration class, was suggested to be present in previous catalogs, such as BATSE and $Swift$, based on statistical tests regarding a mixture of two or three log-normal distributions of $T_{90}$. However, this might possibly not be an adequate model. This paper investigates whether the distributions of $\log T_{90}$ from BATSE, $Swift$, and $Fermi$ are described better by a mixture of skewed distributions rather than standard Gaussians. Mixtures of standard normal, skew-normal, sinh-arcsinh and alpha-skew-normal distributions are fitted using a maximum likelihood method. The preferred model is chosen based on the Akaike information criterion. It is found that mixtures of two skew-normal or two sinh-arcsinh distributions are more likely to describe the observed duration distribution of $Fermi$ than a mixture of three standard Gaussians, and that mixtures of two sinh-arcsinh or two skew-normal distributions are models competing with the conventional three-Gaussian in the case of BATSE and $Swift$. Based on statistical reasoning, existence of a third (intermediate) class of GRBs in $Fermi$ data is rejected, and it is shown that other phenomenological models may describe the observed $Fermi$, BATSE, and $Swift$ duration distributions at least as well as a mixture of standard normal distributions.

astro-ph.HE↗

Analysis of the observed and intrinsic durations of gamma-ray bursts with known redshift

The duration distribution of 408 GRBs with measured both duration $T_{90}$ and redshift $z$ is examined. Mixtures of a number of distributions (standard normal, skew-normal, sinh-arcsinh, and alpha-skew-normal) are fitted to the observed and intrinsic durations using the maximum log-likelihood method. The best fit is chosen via the Akaike information critetion. The aim of this work is to assess the presence of the presumed intermediate GRB class, and to provide a phenomenological model more appropriate than the common mixture of standard Gaussians. While $\log T^{obs}_{90}$ are well described by a truly trimodal fit, after moving to the rest frame the statistically most significant fit is unimodal. To trace the source of this discrepancy, 334 GRBs observed only by $Swift$/BAT are examined in the same way. In the observer frame, this results in a number of statistically plausible descriptions, being uni- and bimodal, and with the number of components ranging from one to three. After moving to the rest frame, no unambiguous conclusions may be put forward. It is concluded that the size of the sample is not big enough to infer reliably GRB properties based on a univariate statistical reasoning only.

astro-ph.HE↗

Analysis of the observed and intrinsic durations of $Swift$/BAT gamma-ray bursts

The duration distribution of 947 GRBs observed by $Swift$/BAT, as well as its subsample of 347 events with measured redshift, allowing to examine the durations in both the observer and rest frames, are examined. Using a maximum log-likelihood method, mixtures of two and three standard Gaussians are fitted to each sample, and the adequate model is chosen based on the value of the difference in the log-likelihoods, Akaike information criterion and Bayesian information criterion. It is found that a two-Gaussian is a better description than a three-Gaussian, and that the presumed intermediate-duration class is unlikely to be present in the $Swift$ duration data.

astro-ph.HE↗

Distinguishing short and long $Fermi$ gamma-ray bursts

Two classes of gamma-ray bursts (GRBs), short and long, have been determined without any doubts, and are usually ascribed to different progenitors, yet these classes overlap for a variety of descriptive parameters. A subsample of 46 long and 22 short $Fermi$ GRBs with estimated Hurst Exponents (HEs), complemented by minimum variability time-scales (MVTS) and durations ($T_{90}$) is used to perform a supervised Machine Learning (ML) and Monte Carlo (MC) simulation using a Support Vector Machine (SVM) algorithm. It is found that while $T_{90}$ itself performs very well in distinguishing short and long GRBs, the overall success ratio is higher when the training set is complemented by MVTS and HE. These results may allow to introduce a new (non-linear) parameter that might provide less ambiguous classification of GRBs.

astro-ph.HE↗

On the limit between short and long GRBs

Two classes of GRBs have been identified thus far without doubt and are prescribed to different physical scenarios -- NS-NS or NS-BH mergers, and collapse of massive stars, for short and long GRBs, respectively. The existence of two distinct populations was inferred through a bimodal distribution of the observed durations $T_{90}$, and the commonly applied $2\,{\rm s}$ limit between short and long GRBs was obtained by fitting a parabola between the two peaks in binned data from BATSE 1B. Herein, by means of a maximum likelihood (ML) method a mixture of two Gaussians is fitted to the datasets from BATSE, $Swift$, $BeppoSAX$, and $Fermi$ in search for a local minimum that might serve as a new, more proper, limit for the two GRB classes. It is found that $Swift$ and $BeppoSAX$ distributions are unimodal, hence no local minimum is present, $Fermi$ is consistent with the conventional limit, whereas BATSE gives the limit significantly longer (equal to $3.38\pm 0.27\,{\rm s}$) than $2\,{\rm s}$. These new values change the fractions of short and long GRBs in the samples examined, and imply that the observed $T_{90}$ durations are detector dependent, hence no universal limiting value may be applied to all satellites due to their different instrument specifications. Because of this, and due to the strong overlap of the two-Gaussian components, the straightforward association of short GRBs to mergers and long ones to collapsars is ambiguous.

astro-ph.HE↗

Analysis of Fermi gamma-ray burst duration distribution

Two classes of GRBs, short and long, have been determined without any doubts, and are usually prescribed to different physical scenarios. A third class, intermediate in $T_{90}$ durations, has been reported to be present in the datasets of BATSE, Swift, RHESSI and possibly BeppoSAX. The latest release of $>1500$ GRBs observed by Fermi gives an opportunity to further investigate the duration distribution. The aim of this paper is to investigate whether a third class is present in the $\log T_{90}$ distribution, or is it described by a bimodal distribution. A standard $χ^2$ fitting of a mixture of Gaussians is applied to 25 histograms with different binnings. Different binnings give various values of the fitting parameters, as well as the shape of the fitted curve. Among five statistically significant fits none is trimodal. Locations of the Gaussian components are in agreement with previous works. However, a trimodal distribution, understood in the sense of having three separated peaks, is not found for any binning. It is concluded that the duration distribution in Fermi data is well described by a mixture of three log-normal distributions, but it is intrinsically bimodal, hence no third class is present in the $T_{90}$ data of Fermi. It is suggested that the log-normal fit may not be an adequate model.

astro-ph.HE↗

Nonlinear time-series analysis of Hyperion's lightcurves

Hyperion is a satellite of Saturn that was predicted to remain in a chaotic rotational state. This was confirmed to some extent by Voyager 2 and Cassini series of images and some ground-based photometric observations. The aim of this aticle is to explore conditions for potential observations to meet in order to estimate a maximal Lyapunov Exponent (mLE), which being positive is an indicator of chaos and allows to characterise it quantitatively. Lightcurves existing in literature as well as numerical simulations are examined using standard tools of theory of chaos. It is found that existing datasets are too short and undersampled to detect a positive mLE, although its presence is not rejected. Analysis of simulated lightcurves leads to an assertion that observations from one site should be performed over a year-long period to detect a positive mLE, if present, in a reliable way. Another approach would be to use 2---3 telescopes spread over the world to have observations distributed more uniformly. This may be achieved without disrupting other observational projects being conducted. The necessity of time-series to be stationary is highly stressed.

astro-ph.EP↗

Nonlinear time series analysis of Hyperion's rotation: photometric observations and numerical simulations

The case of Hyperion has been studied excesively due to the fact it is the largest known celestial body of a highly aspherical shape. It also has a low mass density and remains in a 4:3 orbital resonance with Titan. Its lightcurve, obtained through photometric observations by (Klavetter 1989a,b), was initialy used to show that Hyperion's rotation exhibits no periodicity. Herein it is analyzed in the means of time series analysis. The Hurst Exponent was estimated to be H=0.87, indicating a persistent behaviour. The largest Lyapunov Exponent $λ_{max}$ unfortunately could not be given a reliable estimate because of the shortness of the dataset, consisting 38 observational points. These results are compared with numerical simulations, which gave a value H=0.88 for the chaotic zone of the phase space. The Lyapunov time $T_{Lyap}=1/λ_{max}$ is about 30 days, which is roughly 1.5 times greater than the orbital period. By conducting observations over a longer period an insight in the dynamical features of the present rotational state is possible.

nlin.CD↗

On the fractal dimension of the Duffing attractor

The box counting dimension $d_C$ and the correlation dimension $d_G$ change with the number of numerically generated points forming the attractor. At a sufficiently large number of points the fractal dimension tends to a finite value. The obtained values are $d_C\approx 1.43$ and $d_G\approx 1.38$.

nlin.CD↗