Deterministic chaos in the XRay sources
M. Grzedzielski,
P. Sukova,
and A. Janiuk
Center for Theoretical Physics, Polish Academy of Sciences, Al. Lotników 32/46 02668 Warsaw, Poland
Abstract. Hardly any of the observed black hole accretion disks in XRay binaries and active galaxies shows constant flux. When the local stochastic variations of the disk occur at specific regions where a resonant behaviour takes place, there appear the QuasiPeriodic Oscillations (QPOs). If the global structure of the flow and its nonlinear hydrodynamics affects the fluctuations, the variability is chaotic in the sense of deterministic chaos. Our aim is to solve a problem of the stochastic versus deterministic nature of the black hole binaries variability. We use both observational and analytic methods. We use the recurrence analysis and we study the occurence of long diagonal lines in the recurrence plot of observed data series and compare it to the surrogate series. We analyze here the data of two XRay binaries  XTE J1550564, and GX 3394 observed by Rossi Xray Timing Explorer. In these sources, the nonlinear variability is expected because of the global conditions (such as the mean accretion rate) leading to the possible instability of an accretion disk. The thermalviscous instability and fluctuations around the fixedpoint solution occurs at high accretion rate, when the radiation pressure gives dominant contribution to the stress tensor.
1 Introduction
Appearance of Quasi Periodic Oscillations (QPOs) may prove the evolution of the accretion disk governed by the limit cycle oscillations due to the radiation presure instability. The classic theory of accretion as proposed by Shakura, Sunyaev (1973) and Lightman, Eardley (1974) predicts that the disk is unstable and undergoes the limit cycle oscillation, if the viscous stress tensor scales with the total (gas plus radiation) pressure and the global accretion rate is large enough for the radiation pressure to dominate. The accretion disks may undergo the limitcycle oscillations around a fixed point due to the two main types of thermalviscous instabilities  radiation pressure instability and ionization instability. Both these instability types are known for over 40 years in theoretical astrophysics. In classical theory of Shakura, Sunyaev (1973), the accretion flow structure is based on description for the viscous energy dissipation. It assumes that the nonzero component of the stress tensor is proportional to the total pressure. The latter includes the radiation pressure which scales with temperature as and blows up in hot disks for large accretion rates. This in turn affects the heating and cooling balance between the energy dissipation and radiative losses. If the accretion rate is small, then most of the disk is gas pressure dominated and stable. For large enough accretion rates, there appears a zone where some of its annuli are dominated by radiation pressure and unstable. The XRay binaries, such as GRS1915 or IGR J17091, present many states of variability (Belloni et al., 2011). The QuasiPeriodic Oscillations with heartbeat patterns usually occur in diskdominated soft or softintermediate states (Nandi et al., 2012; Janiuk et al., 2015). We investigate the behaviour of time series of two other sources (GX 3394 and XTE J1550564) to find the tracks of deterministic chaos to answer the question  are the luminosity oscillations driven by deterministic or stochastic system. For that purpose we employ the recurrence analysis and we compare the results with the surrogate data. Here we pursue a similar analysis as Misra et al. (2006) to other sources, but we use a novel method which is the recurrence analysis.
2 Recurrence analysis
To answer the question about the dynamics of the system  is it stochastic or deterministic, we need a method, which reconstructs the multidimensional phase space, and investigates its behaviour. We use the recurrence analysis, tool used to study the time series (Eckmann et al., 1987). The recurrence analysis works in a broad range of application. We combine this method with the surrogate data approach developped by Theiler et al. (1992) in the following way. We first pose the “null hypothesis” about the measured time series, that the data are product of temporally independent white noise or linearly autocorrelated gaussian noise. We construct surrogates using the publicly available software package software package TISEAN (Schreiber et al., 2000; Hegger et al., 1999) and apply the recurrence analysis (Marwan et al., 2007). We define significance of the nonlinear behaviour as a weighted difference between the estimate of Rényi’s entropy for the observed data and the ensamble of its surrogates (Sukova et al., 2015).
The basic object of the analysis is the recurrence matrix, which describes the times, when the trajectory returns close to itself (closer than certain threshold ). The recurrence matrix is defined as follows:
(1) 
where are () points of the reconstructed phase trajectory and is the Heaviside step function.
The recurrence plot (RP) is a visualisation of the recurrence matrix, in which the matrix elements equalled to 1 are plotted as black dots. In such plot the long diagonal lines corresponds to the case, that the trajectory behaves similarly in two different times. Hence RP of a regular trajectory would consists of a set of diagonal lines only, contrary to the randomly scattered points in RP of stochastic system.
Rényi’s entropy is related with the cumulative histogram of diagonal lines , describing the probability of finding a line of minimal length in the RP, by the relation
where is a known function of , we can estimate the value of as the slope of the logarithm of the cumulative histogram versus for constant . Because measured data do not provide the phase space trajectory, it has to be reconstructed from the observed time series with the time delay technique. The resulting phase space vector is given as
(2) 
where is the time series, is the embedding delay and
is the embedding dimension.
We extract the lightcurve for every observation, rescale it to zero mean and zero variance.
We find appropriate guess of the embedding dimension. We produce 100 surrogates for
the different recurrence threshold . For each recurrence threshold we compute the
estimator of Renyi entropy. Later we compute the average significance in respect to
the surrogates. The Renyi entropy of the order is defined as follows:
(3) 
where is the probability that the random variable has a value of . It describes the randomness of the system. The definition of significance of chaotic process, depending on threshold is following:
(4) 
sign, is the total number of surrogates and is the number of surrogates which have enough long diagonal lines for the estimate of , so that . and are the natural logarithms of for observed and surrogate data. and is the significance computed only from the surrogates, which have enough long lines according to relation with respect to the standard deviation of , computed for set of surrogates. The standard deviation is denoted as
(5) 
3 Observations and results
3.1 Observations
We obtained some RXTE PCA observations of GX3394 and XTE J1550564 with small binning time. We extract the lightcurves using Heasoft 6.16 high energy astrophysics software package. We adjust the proper binning time to minimize the error and simultanously to not lose the information about oscillations at the scale of several seconds or several tens of seconds.
In Figure 1, we present lightcurve of GX 3394. The data were extracted using sefilter in General Event mode from channels 024 of RXTE PCA (210 keV). The binning was 0.5s.
In Figure 2, we present lightcurve of XTE J1550564. The data were extracted using saextrct in standard1 mode. The binning was 0.125s. Both Figures 1 and 2 present lightcurves with characteristic variability pattern, that resembles the socalled “heartbeat oscillations” already known for the sources IGR J170913624 and GRS 1915+105 (Altamirano et al., 2011; Belloni et al., 2011).
3.2 Results
We analyse these observations using the recurrence analysis, which can give us the important hint for existence of deterministic chaos in the time series.
In Figure 3 we show length of the longest line in the recurrence plot .
The length of the longest line on the recurrence plot is a hint for deterministic and regular character of the probed sequence (lightcurve). The most regular sequences have the longest lines spreading across all the Figure, otherwise if we have stochastic sequence like the surrogate data, there exist the values of for which the longest line is short in comparison to the longest line in the sequence with the same Power Density Spectrum, but with greater regularity.
The Figure 3 describes the times, when the time series trajectory returns to itself closer the than . The strong difference between the real data and surrogate series is seen and for a wide range of . This result indicates nonlinear features in the dynamics of the source GX 3394.
Figure 4 was made for the data of the source XTE J1550564. It shows, that the significance is dependent on the recurrence threshold , and is usually decreasing for very high . In our analysis we consider the averaged significance, which is represented in the plot by the red horizontal line. For details about the computations of the significance see Sukova et al. (2015).
Apart from , the significance also depends on the other parameters used for computing the recurrence matrix, like embedding delay and embedding dimension .
As was shown by Thiel et al. (2004) , the estimation of dynamical invariants like Renyi entropy should not depend on these parameters for a trajectory of a particular dynamical system. However, we are dealing with real observational data instead of ideal time series of certain dynamical system.
These data are contaminated both by the uncertainties emanating from the observing instrument and also by the stochastic nature of thermal radiation, which is the underlying physical process emitting the observed flux. Hence in our case, the significance depends also on these parameters. In Fig. 5 you can see the dependence of significance for the observation from Fig 1 on the used embedding dimension. The dependence shows a plateau of almost constant values in the range of , decreasing slightly for , but it has a sharp decline for m=10.
We presume, that this behaviour is due to the noise contained in the data and that the strength of the dependence is induced by the strength of the noise in the data. For comparison we show also Figure 6 computed for the observation from Figure 2. In this case, the decrease of the significance with embedding dimension is much slighter.
4 Discussion and conclusions
4.1 Discussion
The high energy radiation emitted by black hole Xray binaries originates in an accretion disk. Most of the sources undergo fast and complicated variability patterns on different timescales. The variations that are purely stochastic in their nature, are expected since the viscosity of the accretion disk is connected with its turbulent behaviour induced by magnetic instabilities. The variability of the disk that reflects its global evolution governed by the nonlinear differential equations of hydrodynamics, may not be only purely stochastic. Instead, if the global conditions in the accretion flow are such that the system finds itself in an unstable configuration, the large amplitude fluctuations around the fixed point solution will be induced. The observed behaviour of the disk will then be characterized by the deterministic chaos. The recent hydrodynamical simulations of the global accretion disk evolution confirm that the quasiperiodic flarelike events observed in couple sources, are in a good quantitative agreement with the radiation pressure instability model (Janiuk et al., 2015).
At least eight of the known BH Xray binaries should have their Eddington accretion rates large enough for the radiation pressure instability to develop.
We used the recurrence analysis method to study the nonlinear behaviour of several Xray sources mentioned in Janiuk, Czerny (2011). Our present analysis confirmed that the variability in these sources is significantly governed by the nonlinear dynamics of accretion process. We confirm that the GRS 1915 and IGR J17091, which show deterministic chaos, and are not the only sources with that feature.
XTE J1550564 is classified as a microquasar on the basis of its largescaled moving jets, detected at Xray and radio wavelengths (Corbel et al., 2002), similarly with GX 3394 (Corbel, Fender, 2002). We expect that these two objects should have similar characteristics of the disk variability, and along with the two well studied microquasars the nonlinear dynamical processes in these sources should also occur. Our current analysis does confirm these expectations.
4.2 Conclusions
We applied the recurrence analysis on observations of two black hole Xray binaries observed by RXTE satellite. We developed a method for distinguishing between stochastic, nonstochastic linear and nonlinear processes using the comparison of the quantification of recurrence plots with the surrogate data. We tested our method on the sample of observations of the microquasar IGR J170913624, which spectral states were provided by Pahari et al. (2014). Significant results for the heartbeat state were obtained. We examined several observations of the other microquasars. Aside from the wellstudied binaries GRS 1915+105 and IGR J170913624, we found significant traces of nonlinear dynamics also in three other sources: GX 3394, XTE J1550564 as discussed in this article, and GRO J165540, as presented in Sukova et al. (2015). Further details can be found in Sukova et al. (2015).
5 Acknowledgements
This work was supported in part by the grant DEC2012/05/E/ST9/03914 from the Polish National Science Center.
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