Bridge successive states for a complex system with evolutionary matrix

A concept called temporal network of evolutionary matrices is proposed to identify the evolutionary laws for complex systems. The key idea is to separate the trajectory into overlapping segments. The time step between each pair of successive states is assumed to be so short that they can be bridged by a matrix. And the time duration covered by the segment is assumed to be so short that the matrices bridging the pairs of successive states are identical, called evolutionary matrix. The trajectory is them mapped to a temporal network of evolutionary matrices (“bridges”), describing the evolutionary law. Investigations on the series generated with the fractional Brownian motion, and the records for stock markets distributed over the world show that, there exist in all the evolutionary angle series long-range correlations. For the fBm increment series, the series generated with the Heston model, and stock index series, the influences of variables on themselves and the influences between adjacent variables form the backbone of the temporal networks. Non-adjacent impacts can fluctuate simultaneously. The markets in Japan as the center affects Mainland China and is unilaterally affected by the America and Mainland China. After financial crisis, there appear some abrupt and large fluctuations of the evolutionary matrices for the components of the Dow Jones stock market.