Mixing properties of the Rössler system and consequences for coherence and synchronization analysis

Cross-spectral and synchronization analysis of two independent, identical chaotic Rössler systems suggest a coupling although there is no interaction. This spuriously detected interaction can either be explained by the absence of mixing or by finite size effects. To decide which alternative holds the phase dynamics is studied by a model of the fluctuations derived from the system’s equations. The basic assumption of the model is a diffusive character for the system which corresponds to mixing. Comparison of theoretical properties of the model with empirical properties of the Rössler system suggests that the system is mixing but the rate of mixing appears to be rather low.

Figure 1

(a) Power spectrum of the $x$ component of the Rössler system at the vicinity of the main oscillating frequency. (b) Coherency of two independent, identical Rössler systems. The coherency at frequencies of approximately 0.17 and its multiples are lying above the 5% level of significance (dashed line).

Figure 2

Time evolution of ${\Phi }_{1,1}$ for two independent Rössler systems, left graph. The distribution of ${\Psi }_{1,1}$, right figure, is showing a sharp peak.

Figure 3

(a) Variance evolution over a sample of 1000 independent Rössler systems. (b) Same as (a) within a time window of 700–750 (solid line). The dashed line indicates the modeled variance of the phase fluctuations. The mean phase of the oscillators is shown by the grey-scale strip on the lower part of the graph, ranging from 0 (black) to $2\pi$ (white).

Figure 4

Distribution of the synchronization index $R_{1,1}$ for two independent processes of type (13). The parameters $D,\Delta t,\omega$ and the amount of data $N$ are chosen to allow a comparison of the results presented in Sec. 3.