*k*th diagonal parallel to the LOI), e.g. RR(

*k*). The measure RR(

*k*) or RR(

*-k*) means the recurrence density in the diagonal above or below the LOI (with distance

*k*from the LOI). The variable

*k*hat the meaning of a time lag.

Then calculate a symmetry

Q(k) = ( RR(

*k*) + RR(

*-k*) ) / 2

and an asymmetry measure

q(k) = ( RR(

*k*) - RR(

*-k*) ) / 2.

The graph of both measures over the time lag

*k*shows whether both systems are coupled/synchronised and which system leads the other. These measures can also be calculated from the other RQA measures.

This plot shows these proposed symmetry and asymmetry measures derived from RR(

*k*) for two coupled Roessler oscillators. The maxima of the (A) symmetry measure at lag 0.3 reveals the lag-coupling, and (B) the negative value of the asymmetry measure athe the same lag reveals that the second system leads the first one.

I got this idea by reading

Quian Quiroga et al, Event synchronization: A simple and fast method to measure synchronicity and time delay patterns, PRE 66, 041904, 2002

Perhaps someone find it useful.

Regards,

Norbert