Identification and prediction of bifurcation tipping points using complex networks based on quasi-isometric mapping

Many models and real systems possess tipping points at which the state of the model or real system shifts dramatically. The ability to find any early-warnings in the vicinity of tipping points is of great importance to estimate how far the system is away from the critical point. Meanwhile, among the many schemes to convert time series into complex networks, the one-dimensional recurrence method has been proved to be a quasi-isometric mapping, and therefore retains geometric information. The quasi-isometric transformation method is adopted to discover underlying changes in systems. By measuring the characteristics of the resultant networks from time series, the changes in the system are captured. Furthermore, curve fitting is applied to expose the relation between the measures of networks and the distance between the parameter of the current state and the parameter at the tipping point for a given system. According to such relation, we can predict the vicinity of critical states hidden in the observational time series. This strategy is proven to be effective over a wide range of noise levels. One real electrocardiogram data-set and two real dynamical systems are employed to demonstrate the capability and reliability of the complex network method for identification of different exercise states and bifurcation behaviors.