Deterministic chaos in the X-Ray sources

M. Grzedzielski1∗,

P. Sukova1,

and A. Janiuk1

1 Center for Theoretical Physics, Polish Academy of Sciences, Al. Lotników
32/46 02-668 Warsaw, Poland

Abstract. Hardly any of the observed black hole accretion disks in X-Ray
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 Quasi-Periodic Oscillations (QPOs). If the
global structure of the flow and its non-linear 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 X-Ray binaries - XTE
J1550-564, and GX 339-4 observed by Rossi X-ray Timing Explorer. In these
sources, the non-linear variability is expected because of the global
conditions (such as the mean accretion rate) leading to the possible
instability of an accretion disk. The thermal-viscous instability and
fluctuations around the fixed-point 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 limit-cycle oscillations around a fixed point
due to the two main types of thermal-viscous 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 $\alpha$
description for the viscous energy dissipation. It assumes that the non-zero
component $T_{r\phi}$ of the stress tensor is proportional to the total
pressure. The latter includes the radiation pressure which scales with
temperature as $T^{4}$ 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 X-Ray binaries, such as GRS1915 or IGR
J17091, present many states of variability (Belloni et al., 2011). The Quasi-
Periodic Oscillations with heartbeat patterns usually occur in disk-dominated
soft or soft-intermediate states (Nandi et al., 2012; Janiuk et al., 2015). We
investigate the behaviour of time series of two other sources (GX 339-4 and
XTE J1550-564) 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 $\bar{\mathcal{S}}$ of the non-linear behaviour as a
weighted difference between the estimate of Rényi’s entropy for the observed
data $K_{2}^{\rm obs}$ 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 $\epsilon$). The recurrence matrix is defined as follows:

$\mathbf{R}_{i,j}(\epsilon)=\Theta(\epsilon-\parallel\vec{y}_{i}-\vec{y}_{j}\parallel),\qquad
i,j=1,...,N,$ (1)

where $\vec{y}_{i}=\vec{y}(t_{i})$ are ($N$) points of the reconstructed phase
trajectory and $\Theta$ 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 $K_{2}$ is related with the cumulative histogram of diagonal
lines $p_{c}(\epsilon,l)$, describing the probability of finding a line of
minimal length $l$ in the RP, by the relation

$p_{c}(\epsilon,l)\sim f(\epsilon)e^{-l\Delta tK_{2}},$

where $f(\epsilon)$ is a known function of $\epsilon$, we can estimate the
value of $K_{2}$ as the slope of the logarithm of the cumulative histogram
versus $l$ for constant $\epsilon$. 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

$\vec{y}(t)=\\{x(t),x(t+\Delta t),x(t+2\Delta t),\dots,x(t+(m-1)\Delta t)\\},$
(2)

where $x(t)$ is the time series, $\Delta t$ is the embedding delay and $m$ 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 $\epsilon$. 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 $\alpha$ is defined as follows:

$K_{\alpha}=\frac{1}{1-\alpha}\ln\sum\limits_{i=1}^{i=N}p_{i}^{\alpha},$ (3)

where $p_{i}$ is the probability that the random variable has a value of $i$.
It describes the randomness of the system. The definition of significance of
chaotic process, depending on threshold $\epsilon$ is following:

$\mathcal{S}(\epsilon)=\frac{N_{sl}}{N_{\rm
surr}}\mathcal{S}_{sl}-s\frac{N_{\mathcal{S}_{K}}}{N^{\rm
surr}}\mathcal{S}_{K_{2}}(\epsilon).$ (4)

$s=$sign$(\mathcal{Q}^{obs}(\epsilon)-\bar{\mathcal{Q}}^{\rm
surr}(\epsilon))$, $N^{\rm surr}$ is the total number of surrogates and
$N_{\mathcal{S}_{K}}$ is the number of surrogates which have enough long
diagonal lines for the estimate of $K_{2}$, so that $N_{\rm
sl}+N_{\mathcal{S}_{K}}=N^{\rm surr}$. $\mathcal{Q}^{obs}$ and
$\bar{\mathcal{Q}}^{\rm surr}$ are the natural logarithms of $K_{2}$ for
observed and surrogate data. $\mathcal{S}^{sl}=3$ and $\mathcal{S}_{K_{2}}$ is
the significance computed only from the surrogates, which have enough long
lines according to relation with respect to the standard deviation of
$\bar{\mathcal{Q}}^{\rm surr}(\epsilon)$, computed for set of surrogates. The
standard deviation is denoted as $\sigma_{\bar{\mathcal{Q}}^{\rm
surr}(\epsilon)}$

$\mathcal{S}_{K_{2}}(\epsilon)=\frac{|\mathcal{Q}^{obs}(\epsilon)-\bar{\mathcal{Q}}^{\rm
surr}(\epsilon)|}{\sigma_{\bar{\mathcal{Q}}^{\rm surr}(\epsilon)}}.$ (5)

##  3 Observations and results

###  3.1 Observations

We obtained some RXTE PCA observations of GX339-4 and XTE J1550-564 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 339-4. The data were extracted using
sefilter in General Event mode from channels 0-24 of RXTE PCA (2-10 keV). The
binning was 0.5s.

In Figure 2, we present lightcurve of XTE J1550-564. 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 so-called “heartbeat oscillations” already known for the sources IGR
J17091-3624 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
$L_{max}$.

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 $\epsilon$ 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 $\epsilon$. The strong difference between the real data
and surrogate series is seen and $L^{max}_{obs}>L^{max}_{\rm surr}$ for a wide
range of $\epsilon$. This result indicates non-linear features in the dynamics
of the source GX 339-4.

Figure 4 was made for the data of the source XTE J1550-564. It shows, that the
significance is dependent on the recurrence threshold $\epsilon$, and is
usually decreasing for very high $\epsilon$. 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 $\epsilon$, the significance also depends on the other parameters
used for computing the recurrence matrix, like embedding delay $\Delta t$ and
embedding dimension $m$.

As was shown by Thiel et al. (2004) , the estimation of dynamical invariants
like Renyi entropy $K_{2}$ 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 $m\in(3,6)$, decreasing slightly for $m\in(7,9)$, 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.

Figure 1: Lightcurve of GX 339-4 on 29.04.2010 in Soft-Intermediate State
(Nandi et al., 2012), ObsID 95409-01-16-05, extracted from PCU2 with time bin
0.5s for the energy range 2-10 keV (channels 0-24). Figure 2: Lightcurve of
XTE J1550-564 on 08.09.1998, ObsID 30188-06-03-00, extracted with time bin
0.125s from Standard 1 data mode (2-60 keV). The same observation was studied
by Sobczak et al. (2000). Figure 3:  Dependence of $L_{\rm max}$ on $\epsilon$
for the observation shown on Fig. 1 by thick red line and the ensemble of 100
surrogates by grey thin lines. The length of the longest line is expressed in
the number of points on the line. For some values of $\epsilon$ \- for example
for the interval $\epsilon\in[2.2,3]$ it clearly shows, that the length of the
longest line for the observation is significantly different from all ensemble
of surrogates. It means that there exist the tracks of nonlinear dynamics for
that observation of the microquasar GX 339-4.  Figure 4:  Dependence of
significance of the non-linear nature of the observation of XTE J1550-564 on
28.09.1998, ObsID 30191-01-14-00 on the recurrence threshold $\epsilon$ and
the mean (red horizontal line) and standard deviation (black horizontal lines)
of the values of significance. Figure 5:  Dependence of significance of the
non-linear nature of the observation from Figure 1 on embedding dimension $m$.
Figure 6:  ependence of significance of the non-linear nature of the
observation from Figure 2 on embedding dimension $m$.

##  4 Discussion and conclusions

###  4.1 Discussion

The high energy radiation emitted by black hole X-ray 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 quasi-periodic flare-like 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 X-ray 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 non-linear behaviour of
several X-ray 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 J1550-564 is classified as a microquasar on the basis of its large-scaled
moving jets, detected at X-ray and radio wavelengths (Corbel et al., 2002),
similarly with GX 339-4 (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 non-linear 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 X-ray
binaries observed by RXTE satellite. We developed a method for distinguishing
between stochastic, non-stochastic linear and non-linear 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
J17091-3624, 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 well-studied binaries
GRS 1915+105 and IGR J17091-3624, we found significant traces of non-linear
dynamics also in three other sources: GX 339-4, XTE J1550-564 as discussed in
this article, and GRO J1655-40, 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 DEC-2012/05/E/ST9/03914 from the
Polish National Science Center.

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