Complex network analysis of recurrences

We present a complex network-based approach to characterizing the geometric properties of chaos by exploiting the pattern of recurrences in phase space. For this purpose, we utilize the basic definition of a recurrence as the mutual proximity of two state vectors in phase space (disregarding time information) and re-interpret the recurrence plot as a graphical representation of the adjacency matrix of a random geometric graph governed by the system’s invariant density. The resulting recurrence networks contain exclusively geometric information about the system under study, which can be exploited for inferring quantitative information on the geometric properties of the system’s attractor without explicitly studying scaling characteristics as in the case of “classical” fractal dimension estimates.

Similar as the established recurrence quantification analysis, recurrence networks can be utilized for studying dynamical transitions in non-stationary systems, as well as for automatically discriminating between chaos and order without the necessity of extensive computations typically necessary when inferring this distinction based on the systems’ maximum Lyapunov exponents. Moreover, we provide a thorough re-interpretation of two bi- and multivariate generalizations of recurrence plots in terms of complex networks, which allow tracing geometric signatures of asymmetric coupling and complex synchronization processes between two or more chaotic oscillators.