Granger causality and cross recurrence plots in rheochaos

Our stress relaxation measurements on wormlike micelles using a Rheo-SALS (rheology + small angle light scattering) apparatus allow simultaneous measurements of the stress and the scattered depolarized intensity. The latter is sensitive to orientational ordering of the micelles. To determine the presence of causal influences between the stress and the depolarized intensity time series, we have used the technique of linear and nonlinear Granger causality. We find there exists a feedback mechanism between the two time series and that the orientational order has a stronger causal effect on the stress than vice versa. We have also studied the phase space dynamics of the stress and the depolarized intensity time series using the recently developed technique of cross recurrence plots (CRPs). The presence of diagonal line structures in the CRPs unambiguously proves that the two time series share similar phase space dynamics.

Figure 1

Bivariate time series. (a) Stress time series. (b) Depolarized intensity (VH) time series obtained at a shear rate of $25{\mathrm{s}}^{-1}$.

Figure 2

(a) Ratio of the variance of errors for $Y$ driving $X$ and (b) ratio of the variance of errors of $X$ driving $Y$ plotted against the regression model order $m$. Here $X$ stand for the stress and $Y$ for the VH intensity. The inset to (a) shows the Akaike information criteria (AIC) plotted against the regression model order.

Figure 3

(Color online) (a) Predicted time series of stress (continuous lines) obtained without using information from VH intensity overlayed on raw time series (dashed lines). (b) Predicted time series of stress (continuous lines) obtained with using VH intensity information overlayed on raw time series (dashed lines). Regression model order $m=33$.

Figure 4

(Color online) (a) Predicted time series of VH intensity (continuous lines) obtained without using information from stress overlayed on raw time series (dashed lines). (b) Predicted time series of VH intensity (continuous lines) obtained with using stress information overlayed on raw time series (dashed lines). Regression model order $m=33$.

Figure 5

(Color online) EGCI plotted against the box size $ϵ$. $Y$ driving $X$ (stars). $X$ driving $Y$ (circles). $X$ stands for the stress and $Y$ for the VH intensity.

Figure 6

CRP of stress and VH intensity in the laminar region.

Figure 7

CRP of stress and VH intensity in the chaotic region.