Detecting generalized synchronization by the generalized angle

Although much progress has been made in analyzing complex systems, our analysis tools are not enough from the points of convenience and efficiency. We here introduce a novel tool, i.e., the generalized angle, to analyze complex systems and show that generalized synchronization can be measured by an order parameter depending only on the generalized angle of an arbitrary high-dimensional (n>2) trajectory. For generalized synchronization, the order parameter will be close to unity. With the increase of coupling strength, the slope of the order parameter will have a drastic change at the transition point of generalized synchronization. Numerical simulations have confirmed the theoretical predictions.