#### Revealing deterministic chaos in the X-ray lightcurves of accreting black holes

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 will be induced. The observed behaviour of the disk will then be characterized by the deterministic chaos. Our 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 of the disc coupled with strong outflows in form of a wind. At least 8 of the known BH X-ray binaries should have their Eddington accretion rates large enough for the radiation pressure instability to develop. We aim to tackle the problem of stochastic versus deterministic nature of the BH accretion disk variability from the analytic and observational point of view.

In the two figures below, we show the lightcurves of the microquasars XTE J1550-564 and GX 339-4, as taken by the RXTE satellite. In these objects, we found the signatures of an underlying deterministic chaos type of behaviour (data reduction: Mikołaj Grzędzielski).

We use the capabilities of the recurrence analysis, which is a powerful tool for studying the time series

and is known to work in broad range of applications.

We first pose the “null hypothesis” about the measured time series, that the data are product of linearly autocorrelated gaussian noise. Then according to this null hypothesis we produce the set of surrogate time series sharing the spectrum and the value distribution with the original time series.The second order Renyi entropy, which is a measure of positive Lyapunov’s exponents and indication of deterministic chaos, is estimated for the observation and its surrogates using the program rp described by Marwan (2007) and the significance of the difference between the real and artificial data is measured. If the Renyi entropy of the observation differs significantly from the values of the surrogates, we claim, that the observation was produced by a system governed by non-linear dynamics, which is the trace of deterministic chaos in the system.

The example of the recurrence plot of the observation of XTE J1550-564 (red color) and one of its surrogates (green color) is given below. On this plot the times, when the trajectory in the phase space reconstructed from the time series returns close to itself, are plotted by color dots. Long diagonal lines in the recurrence plot correspond to the fact, that the trajectory is evolving similarly (inside ε-tube), which is typical for deterministic regular trajectories. Longer diagonal lines distributed more regularly can be seen in the plot of the real observation (red color).

Recurrence plot (author: Petra Sukova).

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