Nonlinear time series analysis Delay embedding technique –reconstructs dynamics from single variable Correlation sum –provides an estimator of fractal dimension –tests stochastic vs chaotic nature in system Application to accreting compact objects –Different behavior seen for NS vs BHs Application to radio pulsar data –No chaotic behavior seen so far –What I might do (if I ever find the time to do it!) (talk slides available at (a simple-minded approach - DRL 11/19/2009)

Search for nonlinear and chaotic behavior in time series Reference: PRD, 50, 346 (1983) Apply to systems where only one parameter is measurable. Time series analysis is a very common “mode” of astronomy. Ideally want to be able to distinguish between a stochastic or chaotic process and make some inferences about source.

The Lorenz model (X-Y plane)

Lorenz model (X-only)

Delay-embedding scheme Define M-dimensional vectors from a time series s(t) M is the “embedding” dimension; tau is a delay time. Takens’ theorem (stated without proof!) “If the attractor has a fractal dimension D, then this mapping will reconstruct original phase space if M > 2D + 1.” Plot of s(t) against s(t+tau) to show these vector positions…

Reconstructed Lorenz dynamics

Correlation sum…. C M (R) For a given M, the average number of data points within a distance R from a given data point …then C M (R) is ~ a power law for small r with slope which closely follows the fractal dimension, D. where the Heaviside function

Search for convergent slopes

Nonlinear time series analysis (a)Random data showing D 2 ~ M and Lorenz model which saturates at D 2 ~ 2 around M~3. (b) Data from a light curve of a BH system showing evidence for saturation. Different values of tau are shown.

Astronomical applications

Accreting X-ray binaries

Compact object binaries

Sco X-1 light curve data taken over 4 yr interval McNamara et al. ApJSS (1997)

Recently submitted paper

Goals of light curve analysis Characterizing the temporal variability –Geometry/physics of emitting regions and accretion –Deterministic chaos or some stochastic process? Origin of variability currently unclear –Instabilities in the inner region of accretion disk? Flow of material may be nonlinear and turbulent –Affected by variations in external parameters? For example, accretion rate or nearby magnetic field Can it tell us about the nature of the companion? –Do black holes (BHs) have different light curves compared to neutron stars (NSs)? If so, perhaps use light curve analysis to infer the object in other systems? Plots of X-ray count rate as a function of time

Application: Sco X-1 and Cyg X-1

Application: Cyg X-2 and Cyg X-3

What about presence of noise? Generate “surrogate data” –Draw uniform deviates in time series –Modify data such that ACF and FT are comparable –Dataset has ~ characteristics of noise, but no signal Cyg X-3Cyg X-2

Summary of results BH systems appear stochastic –Is the “no-hair” theorem making this cleaner? NS systems show evidence for low-D attractor –Is the effect of the NS magnetic field causing chaos? –Describe system by ~4 equations? My “conclusions” –Delay embedding technique potentially powerful –Correlation sum can be used to search for chaos –Can also see something unusual by eye! –Authors need further analysis of other BH/NS systems –Technique shows promise, but underlying mechanism?

Quick introduction to pulsars

Previous application to pulsar data

Model pulse “Probe” is local electric field in the pulsar magnetosphere.

Evidence for chaos in model Model involves a non-linear Schrodinger-like equation which has fractal dimension of 2.3 (Russell & Ott 1981)

What I might do with all of this