Estimation of the direction of the coupling by conditional probabilities of recurrence

We introduce a method to detect and quantify the asymmetry of the coupling between two interacting systems based on their recurrence properties. This method can detect the direction of the coupling in weakly as well as strongly coupled systems. It even allows detecting the asymmetry of the coupling in the more challenging case of structurally different systems and it is very robust against noise. We also address the problem of detecting the asymmetry of the coupling in passive experiments, i.e., when the strength of the coupling cannot be systematically changed, which is of great relevance for the analysis of experimental time series.

Figure 1

(a) Rössler driving system, (b) the RP of the Rössler system $(m=3,\tau =5)$, (c) the driven Lorenz system, (d) the RP of the Lorenz system $(m=7,\tau =5)$, (e) representation of the “joint” system, and (f) the joint recurrence plot of both systems. The threshold for the computation of the RPs has been chosen so that the recurrence rate (number of recurrence points divided by $N^2$) is equal for both systems. In this case the recurrence rate was 0.005. The equations were integrated using fourth order Runge-Kutta and the sampling time was 0.2.

Figure 2

Mean conditional probabilities of recurrence $M_{CR}(X\mid Y)$ (solid) and $M_{CR}(Y\mid X)$ (dashed) for two unidirectionally coupled identical (a) and nonidentical (c) Hénon maps. The system X is in both cases the driver, and hence, $M_{CR}(X\mid Y)>M_{CR}(Y\mid X)$. For each value of the coupling strength $\mu$, the mean value over 100 trajectories for uniformly distributed initial conditions has been computed. In (b) and (d) we have plotted the mean value of the difference $\Delta M_{CR}=M_{CR}(X\mid Y)-M_{CR}(Y\mid X)$ over 100 trajectories and the corresponding standard deviation for the identical and nonidentical Hénon systems, respectively. The zero line is also plotted for orientation (dotted-dashed-line).

Figure 3

Mean conditional probabilities of recurrence $M_{CR}(X\mid Y)$ (solid) and $M_{CR}(Y\mid X)$ (dashed) for (a) two weakly unidirectionally coupled nonidentical Lorenz systems. For each value of the coupling strength $\mu$, the mean value over 100 trajectories for uniformly distributed initial conditions has been computed. The system X is the driver, and hence, $M_{CR}(X\mid Y)>M_{CR}(Y\mid X)$. (c) Two weakly bidirectionally coupled stochastic Van der Pol oscillators. The coupling strength from $X$ to $Y$ is fixed and equal to 0.03. In (b) and (d) we have plotted the mean value of $\Delta M_{CR}=M_{CR}(X\mid Y)-M_{CR}(Y\mid X)$ and the corresponding standard deviation over 100 trajectories for each system, respectively. The zero line is also plotted for orientation (dotted-dashed-line).

Figure 4

(a) Mean conditional probabilities of recurrence $M_{CR}(X\mid Y)$ (solid) and $M_{CR}(Y\mid X)$ (dashed) for the chaotic Rössler system driven by the stochastic Van der Pol system. For each value of the coupling strength $\mu$, the mean value over 100 trajectories for uniformly distributed initial conditions has been computed. The system X is the driver, and hence, we find $M_{CR}(X\mid Y)>M_{CR}(Y\mid X)$. (b) Mean value of the difference $\Delta M_{CR}=M_{CR}(X\mid Y)-M_{CR}(Y\mid X)$ and corresponding standard deviation over 100 trajectories. The zero line is also plotted for orientation (dotted-dashed line).

Figure 5

Mean conditional probabilities of recurrence $M_{CR}(X\mid Y)$ (solid) and $M_{CR}(Y\mid X)$ (dashed) for two identical unidirectionally coupled Hénon maps for different choices of the embedding parameters: (a) $m=3$, $\tau =1$, (c) $m=2$, $\tau =3$, (e) $m=5$, $\tau =1$. For each value of the coupling strength $\mu$, the mean value over 10 trajectories for uniformly distributed initial conditions has been computed. In (b), (d), and (f) the mean value and standard deviation over 10 trajectories of the corresponding $\Delta M_{CR}$ are represented. The zero line is also plotted for orientation (dotted-dashed line).

Figure 6

$\Delta M_{CR}=M_{CR}(X\mid Y)-M_{CR}(Y\mid X)$ for two identical unidirectionally coupled Hénon maps in dependence on the coupling strength $\mu$ and the mean probability of recurrence or recurrence rate. For each value of the coupling strength $\mu$, the mean value over 10 trajectories for uniformly distributed initial conditions has been computed.

Figure 7

$M_{CR}(X\mid Y)$ and $M_{CR}(Y\mid X)$ for two identical unidirectionally coupled Hénon maps contaminated by uniformly distributed noise in dependence on the coupling strength $\mu$. (a) $\gamma =0.2$, (c) $\gamma =0.4$, (e) $\gamma =0.6$. For each value of the coupling strength $\mu$, the mean value over 10 trajectories for uniformly distributed initial conditions has been computed. In (b), (d), and (f) the corresponding mean value of the difference $\Delta M_{CR}$ and the standard deviation over 10 trajectories are plotted. The zero line is also plotted for orientation (dotted-dashed line).

Figure 8

(a) Mean conditional probability of recurrence $M_{CR}(X\mid Y)$ (solid) and 1% significance level (dotted) for two identical Hénon maps unidirectionally coupled. (b) $M_{CR}(Y\mid X)$ (dashed) and 1% significance level (dotted) for the same system. 100 twin surrogates have been generated to estimate the significance level. The data have been normalized beforehand to have zero mean and standard deviation equal to one and ${\epsilon }_X={\epsilon }_Y=0.1$.

Figure 9

Transfer entropy for two identical unidirectionally coupled Hénon maps in dependence on the coupling strength $\mu$ and the threshold. For each value of the coupling strength $\mu$, the mean value over 10 trajectories for uniformly distributed initial conditions has been computed. Solid surface: transfer entropy from $X$ to $Y$; dashed surface: transfer entropy from $Y$ to $X$.