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Gamma-ray bursts (GRBs) come in two types, short and long. The distribution of logarithmic durations of long GRBs is asymmetric rather than Gaussian. Such an asymmetry, when modelled with a mixture of Gaussian distributions, requires an introduction of an additional component, often associated with another class of GRBs. However, when modelled with inherently asymmetric distributions, there is no need for such a component. The cosmological dilation was already ruled out as a source of the asymmetry, hence its origin resides in the progenitors. GRB light curves (LCs) are usually well described by a series of fast-rise-exponential-decay pulses. A statistical analysis of ensembles of simulated LCs shows that the asymmetry is a natural consequence of the pulse shape and the multi-pulse character of the LCs., Comment: 8 pages, 3+2 figures; accepted in MNRAS

Blazars, a subclass of active galactic nuclei (AGNs), are highly variable objects. Quasiperiodic oscillations (QPOs), which might originate from a binary black hole located at the AGN core,have been found in some blazar light curves. For the blazars Mrk 421 and Mrk 501, we test the possible QPO behavior using a variety of methods (generalized Lomb-Scargle periodogram,continuous-time autoregressive moving average models, wavelet scalograms, and A−T plane), studying systematic effects in detail. We use gamma-ray light curves from FACT, a ground-based imaging air Cherenkov telescope. We report herein on a 3 sigma significant QPO in Mrk 421., PoS: Proceedings of Science, 395, ISSN:1824-8039, 37th International Cosmic Ray Conference (ICRC 2021)

We present the results of the Fermi-Large Area Telescope (LAT) light curve (LC) modelling of selected blazars: six flat spectrum radio quasars (FSRQs) and five BL Lacertae (BL Lacs). All objects have densely sampled and long-term LCs, over 10 years. For each blazar we generated three LCs with 7, 10, and 14 days binning, using the latest LAT 8-year source catalog and binned analysis provided within the fermipy package. The LCs were modelled with several tools: the Fourier transform, the Lomb-Scargle periodogram (LSP), the autoregressive moving average (ARMA), the fractional autoregressive integrated moving average, the continuous-time autoregressive moving average (CARMA) processes, the Hurst exponents ($H$), the $\mathcal{A}-\mathcal{T}$ plane, and the wavelet scalogram. The power law indices $\beta$ calculated from the Fourier and LSP modelling are consistent with each other. Many objects yield $\beta\simeq 1$, with PKS 2155$-$304 even flatter, but some are significantly steeper, e.g. Mrk 501 and B2 1520+31. The power law PSD is indicative of a self-affine stochastic process characterised by $H$, underlying the observed variability. Several algorithms for the $H$ estimations are employed. For some objects we found $H>0.5$, indicating long-term memory. We confirm a quasi-periodic oscillation (QPO) in the PKS 2155$-$304 data with the period of $612\pm 42$ days at a $3\sigma$ significance level, but do not detect any QPOs in other objects. The ARMA results give in general higher orders for 7 days binned LCs and lower orders for 10 and 14 days binned LCs, implying temporal variations in the LCs are consistently captured by the fitted models. CARMA modelling leads to featureless PSDs. The recently introduced $\mathcal{A}-\mathcal{T}$ plane allows us to successfully classify the PSDs based on the LCs alone and clearly separates the FSRQ and BL Lac types of blazars., Comment: To appear in the proceedings of the High Energy Astrophysics in Southern Africa 2021 (HEASA2021) in Proceedings of Science (PoS)

Blazar variability appears to be stochastic in nature. However, a possibility of low-dimensional chaos was considered in the past, but with no unambiguous detection so far. If present, it would constrain the emission mechanism by suggesting an underlying dynamical system. We rigorously searched for signatures of chaos in Fermi-Large Area Telescope light curves of 11 blazars. The data were comprehensively investigated using the methods of nonlinear time-series analysis: phase-space reconstruction, fractal dimension, and maximal Lyapunov exponent (mLE). We tested several possible parameters affecting the outcomes, in particular the mLE, in order to verify the spuriousness of the outcomes. We found no signs of chaos in any of the analysed blazars. Blazar variability is either truly stochastic in nature or governed by high-dimensional chaos that can often resemble randomness. © 2021 The Author(s)., OO thanks the Astronomical Observatory of the Jagiellonian University for the summer internship during which this research began. MT acknowledges support by the Polish National Science Center through the OPUS grant No. 2017/25/B/ST9/01208. The work of N. Z is supported by the South African Research Chairs Initiative (grant no. 64789) of the Department of Science and Innovation and the National Research Foundation of South Africa. J.P.-G. acknowledges funding support from Spanish public funds for research under project ESP2017-87676-C5-5-R and financial support from the State Agency for Research of the Spanish MCIU through the `Center of Excellence Severo Ochoa' award to the Instituto de Astrof ' isica de Andaluc ' ia (SEV-2017-0709).

We investigated the prompt light curves (LCs) of long gamma-ray bursts (GRBs) from the Swift/BAT catalog. We aimed to characterize their power spectral densities (PSDs), search for quasiperiodic oscillations (QPOs), and conduct novel analyses directly in the time domain. We analyzed the PSDs using Lomb-Scargle periodograms, and searched for QPOs using wavelet scalograms. We also attempted to classify the GRBs using the Hurst exponent, $H$, and the $\mathcal{A}-\mathcal{T}$ plane. The PSDs fall into three categories: power law (PL; $P(f)\propto 1/f^\beta$) with index $\beta\in(0,2)$, PL with a non-negligible Poisson noise level (PLC) with $\beta\in(1,3)$, and a smoothly broken PL (SBPL; with Poisson noise level) yielding high-frequency index $\beta_2\in(2,6)$. The latter yields break time scales on the order of 1--100\,seconds. The PL and PLC models are broadly consistent with a fully developed turbulence, $\beta=5/3$. For an overwhelming majority of GRBs (93\%), $H>0.5$, implying ubiquity of the long-term memory. We find no convincing substructure in the $\mathcal{A}-\mathcal{T}$ plane. Finally, we report on 34 new QPOs: with one or more constant leading periods, as well as several chirping signals. The presence of breaks and QPOs suggests the existence of characteristic time scales that in at least some GRBs might be related to the dynamical properties of plasma trajectories in the accretion disks powering the relativistic jets., Comment: 22 pages, 9 figures, 2 tables (one online only ancillary file); accepted in ApJ

We present results of a variability study in the optical band of 44 newly identified blazar candidates behind the Magellanic Clouds. Our sample contains 27 flat spectrum radio quasars (FSRQs) and 17 BL Lacertae objects (BL Lacs). However, only nine of them are considered as secure blazar candidates, while the classification of the remaining 35 objects is still uncertain. All studied blazar candidates possess infrequently sampled optical light curves (LCs) in I filter provided by the Optical Gravitational Lensing Experiment group. The LCs were analysed with the Lomb-Scargle periodogram, the Hurst exponent $H$, and the $\mathcal{A}-\mathcal{T}$ plane, to look for blazar-like characteristic features and to study the long-term behaviour of the optical fluxes. The power law (PL) indices of the Lomb-Scargle power spectral density (PSD) of the FSRQ blazar candidates mostly lie in the range $(1,2)$. In case of the BL Lacs they are located in the range $(1,1.8)$. The PL PSD is indicative of a self-affine stochastic process characterised by $H$, underlying the observed variability. We find that the majority of analysed objects have $H\leq 0.5$, indicating short-term memory, whereas four BL Lacs and two FSRQs have $H>0.5$, implying long-term memory. 41 blazar candidates are located in the $\mathcal{A}-\mathcal{T}$ plane in the region available to PL plus Poisson noise processes. Interestingly, one FSRQ is located marginally below this region, while two FSRQs lie above the line $\mathcal{T}=2/3$, i.e. they are even more noisy than white noise. The BL Lac candidates are characterised by higher $\mathcal{A}$ values than FSRQs, i.e. $0.71\pm 0.06$ and $0.29\pm 0.05$, respectively.

In order to explain the origin of skewness in the gamma-ray burst (GRB) duration distribution, a statistical model, based on the cosmological time dilation and taking into account the GRB formation rate and instrumental effects, is examined. It is concluded that the considered effects are able to account for only a small fraction of the observed skewness. Therefore, its origin needs to be searched for in the processes governing the progenitors of GRBs., Comment: 9 pages, 2 figures, 1 table. Accepted in ApJ

We present the results of the Fermi-Large Area Telescope 10 yr long light curve (LC) modeling of selected blazars: six flat-spectrum radio quasars (FSRQs) and five BL Lacertae (BL Lacs), examined in 7, 10, and 14 day binning. The LCs and power spectral densities (PSDs) were investigated with various methods: Fourier transform, Lomb-Scargle periodogram (LSP), wavelet scalogram, autoregressive moving average (ARMA) process, continuous-time ARMA (CARMA), Hurst exponent (H), and the A-T plane. First, with extensive simulations we showed that parametric modeling returns unreliable parameters, with a high dispersion for different realizations of the same stochastic model. Hence, any such analysis should be supported with Monte Carlo simulations. For our blazar sample, we find that the power-law indices beta calculated from the Fourier and LSP modeling mostly fall in the range 1 less than or similar to beta less than or similar to 2. Using the wavelet scalograms, we confirm a quasi-periodic oscillation (QPO) in PKS 2155-304 at a 3 sigma significance level, but do not detect any QPOs in other objects. The ARMA fits reached higher orders for 7 day binned LCs and lower orders for 10 and 14 day binned LCs for the majority of blazars, suggesting there might exist a characteristic timescale for the perturbations in the jet and/or accretion disk to die out. ARMA and CARMA modeling revealed breaks in their PSDs at timescales of a few hundred days. The estimation ofHwas performed with several methods. We find that most blazars exhibit H > 0.5, indicating long-term memory. Finally, the FSRQ and BL Lac subclasses are clearly separated in the A-T plane. © 2020. The American Astronomical Society. All rights reserved., M.T. acknowledges support by the Polish National Science Center (NSC) through OPUS grant No. 2017/25/B/ST9/01208. The work of N.Z. is supported by the South African Research Chairs Initiative (grant No. 64789) of the Department of Science and Innovation and the National Research Foundation27 of South Africa. V.M. is supported by NSC grant No. 2016/22/E/ST9/00061. J.P.-G. acknowledges financial support from the State Agency for Research of the Spanish MCIU through the "Center of Excellence Severo Ochoa" award to the Instituto de Astrofisica de Andalucia (SEV-2017-0709) and from Spanish public funds for research under project ESP2017-87676-C5-5-R.

Aims. An attempt to classify gamma-ray bursts (GRBs) with a low level of supervision using the state-of-the-start approaches stemming from graph theory was undertaken. Methods. Graph-based classification methods, relying on different variants of the $k$-nearest neighbour graph, were applied to various GRB samples in the duration-hardness ratio parameter space to infer the optimal partitioning. Results. In most cases it is found that both two and three groups are feasible, with the outcome being more ambiguous with an increasing sample size. Conclusions. There is no clear indication of the presence of a third GRB class; however, such a possibility cannot be ruled out with the employed methodology. There are no hints at more than three classes though., 8 pages, 6 figures; accepted in A&A

We present an optical variability study of 44 newly identified blazar candidates behind the Magellanic Clouds, including 27 flat spectrum radio quasars (FSRQs) and 17 BL Lacertae objects (BL Lacs). All objects in the sample possess high photometric accuracy and irregularly sampled optical light curves (LCs) in I filter from the long-term monitoring conducted by the Optical Gravitational Lensing Experiment. We investigated the variability properties to look for blazar-like characteristics and to analyze the long-term behaviour. We analyzed the LCs with the Lomb-Scargle periodogram to construct power spectral densities (PSDs), found breaks for several objects, and linked them with accretion disk properties. In this way we constrained the black hole (BH) masses of 18 FSRQs to lie within the range $8.18\leq\log (M_{\rm BH}/M_\odot)\leq 10.84$, assuming a wide range of possible BH spins. By estimating the bolometric luminosities, we applied the fundamental plane of active galactic nuclei variability as an independent estimate, resulting in $8.4\leq\log (M_{\rm BH}/M_\odot)\leq 9.6$, with a mean error of 0.3. Many of the objects have very steep PSDs, with high frequency spectral index in the range $3-7$. An alternative attempt to classify the LCs was made using the Hurst exponent, $H$, and the $\mathcal{A}-\mathcal{T}$ plane. Two FSRQs and four BL Lacs yielded $H>0.5$, indicating presence of long-term memory in the underlying process governing the variability. Additionally, two FSRQs with exceptional PSDs, stand out also in the $\mathcal{A}-\mathcal{T}$ plane., Accepted for publication in ApJ

The two widely accepted classes of gamma-ray bursts (GRBs), short and long, are with confidence ascribed to mergers of compact objects and collapse of massive stars, respectively. A third, intermediate/soft class, remains putative. Its existence was claimed based on univariate and bivariate analyses of GRB observables modeled with Gaussian distributions. This, however, may not be the appropriate approach, as it has been already shown that the univariate distributions of durations are better described by mixtures of two skewed components rather than three Gaussian ones. This paper investigates whether data in the duration--hardness ratio plane is better modeled by mixtures of skewed bivariate distributions than by normal ones. The archival data set of the Compton Gamma-Ray Observatory/BATSE and Fermi/GBM data from the most recent catalogue release are examined. The preferred model is chosen based on two information criteria, Akaike ($AIC$) and Bayesian ($BIC$). It is found that the best description is given by a two-component mixture of skewed Student-$t$ distributions, which outperforms any other model considered. This implies that the distribution of the studied parameters is intrinsically skewed, introducing spurious Gaussian components, and hence the third class is unlikely to be a real phenomenon. Its existence, based on statistical inference, is therefore rejected as unnecessary to explain the observations., 15 pages; accepted in ApJ

The number of classes of gamma-ray bursts (GRBs), besides the well-established short and long ones, remains a debatable issue. It was already shown, however, that when invoking skewed distributions, the $\log T_{90}$ and $\log T_{90} - \log H_{32}$ spaces are adequately modeled with mixtures of only two such components, implying two GRB types. Herein, a comprehensive multivariate analysis of several multi-dimensional parameter spaces is conducted for the BATSE sample of GRBs, with the usage of skewed distributions. It is found that the number of extracted components varies between the examined parameter sets, and ranges from 2 to 4, with higher-dimensional spaces allowing for more classes. A Monte Carlo testing implies that these additional components are likely to be artifacts owing to the finiteness of the data and be a result of examining a particular realization of the data as a random sample, resulting in spurious identifications., 13 pages, 10 figures, 1 table; accepted in ApJ

The mechanism responsible for the prompt emission of gamma-ray bursts (GRBs) is still a debated issue. The prompt phase-related GRB correlations can allow discriminating among the most plausible theoretical models explaining this emission. We present an overview of the observational two-parameter correlations, their physical interpretations, and their use as redshift estimators and possibly as cosmological tools. The nowadays challenge is to make GRBs, the farthest stellar-scaled objects observed (up to redshift z=9.4), standard candles through well established and robust correlations. However, GRBs spanning several orders of magnitude in their energetics are far from being standard candles. We describe the advances in the prompt correlation research in the past decades, with particular focus paid to the discoveries in the last 20 years.

Gamma-ray bursts (GRBs) were confirmed to be of extragalactic origin due to their isotropic angular distribution, combined with the fact that they exhibited an intensity distribution that deviated strongly from the $-3/2$ power law. This finding was later confirmed with the first redshift, equal to at least $z=0.835$, measured for GRB970508. Despite this result, the data from $CGRO$/BATSE and $Swift$/BAT indicate that long GRBs are indeed distributed isotropically, but the distribution of short GRBs is anisotropic. $Fermi$/GBM has detected 1669 GRBs up to date, and their sky distribution is examined in this paper. A number of statistical tests is applied: nearest neighbour analysis, fractal dimension, dipole and quadrupole moments of the distribution function decomposed into spherical harmonics, binomial test, and the two point angular correlation function. Monte Carlo benchmark testing of each test is performed in order to evaluate its reliability. It is found that short GRBs are distributed anisotropically on the sky, and long ones have an isotropic distribution. The probability that these results are not a chance occurence is equal to at least 99.98\% and 30.68\% for short and long GRBs, respectively. The cosmological context of this finding and its relation to large-scale structures is discussed., 14 pages, 5 figures; accepted in MNRAS

Let us consider a classical high school exercise concerning two weights on a pulley and a string, illustrated in Fig. 1(a). A system like this is called an Atwood's machine and was invented by George Atwood in 1784 as a laboratory experiment to verify the mechanical laws of motion with constant acceleration. Nowadays, Atwood's machine is used for didactic purposes to demonstrate uniformly accelerated motion with acceleration arbitrarily smaller than the gravitational acceleration g. The simplest case is with a massless and frictionless pulley and a massless string. With little effort one can include the mass of the pulley in calculations. The mass of a string has been incorporated previously in some considerations and experiments. These include treatments focusing on friction, justifying the assumption of a massless string, incorporating variations in Earth's gravitational field, comparing the calculated value of g based on a simple experiment, taking the mass of the string into account in such a way that the resulting acceleration is constant, or in one exception solely focusing on a heavy string, but with a slightly different approach. Here we wish to provide i) a derivation of the acceleration and position dependence on the weights' masses based purely on basic dynamical reasoning similar to the conventional version of the exercise, and ii) focus on the influence of the string's linear density, or equivalently its mass, on the outcome compared to a massless string case.

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. $e\approx 0.005$., 19 pages, 9 figures; accepted for publication in Celestial Mechanics and Dynamical Astronomy

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., Comment: 12 pages, 10 figures; accepted in Astrophysics and Space Science

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., 9 pages, 7 figures, 3 tables; matches the version accepted by MNRAS. arXiv admin note: text overlap with arXiv:1602.02363

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., 6 pages, 1 figure; matches the version to published in Ap&SS

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 $��^2$ 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., 6 pages, 3 figures; matches the version to be published

The Chirikov standard map and the 2D Froeschl\'e map are investigated. A few thousand values of the Hurst exponent (HE) and the maximal Lyapunov exponent (mLE) are plotted in a mixed space of the nonlinear parameter versus the initial condition. Both characteristic exponents reveal remarkably similar structures in this space. A tight correlation between the HEs and mLEs is found, with the Spearman rank $\rho=0.83$ and $\rho=0.75$ for the Chirikov and 2D Froeschl\'e maps, respectively. Based on this relation, a machine learning (ML) procedure, using the nearest neighbor algorithm, is performed to reproduce the HE distribution based on the mLE distribution alone. A few thousand HE and mLE values from the mixed spaces were used for training, and then using $2-2.4\times 10^5$ mLEs, the HEs were retrieved. The ML procedure allowed to reproduce the structure of the mixed spaces in great detail., Comment: 20 pages, 9 figures; accepted in Physica A

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., Comment: 9 pages, 6 figures; matches the before-proof version accepted for publication in MNRAS

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 article 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.

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., Comment: 7 pages, 5 figures; matches the published version. arXiv admin note: text overlap with arXiv:1506.07801

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., Comment: 20 pages in one-column format, 7 figures; matches the version accepted for publication

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., 9 pages, 4 figures; accepted in A&A