Disentangling regular and chaotic motion in the standard map using complex network analysis of recurrences in phase space

In recent years, complex network theory has provided many conceptual insights based on recurrence characteristics of time series from various fields, which is referred to as recurrence network analysis. While recent applications of this novel concept have been restricted almost exclusively to dissipative dynamics (i.e., the quantitative characterization of attractors), we demonstrate in this work that some of the characteristic features of recurrence networks are useful for disentangling the complex
dynamics of low-dimensional conservative systems as well. In the standard map, a typical chaotic orbit can be temporarily trapped in the vicinity of the regular domains in phase space, resulting in a possibly rather long time necessary to homogeneously fill the chaotic domain—a phenomenon known as stickiness. The presence of sticky orbits (i.e., intermittent laminar phases of chaotic trajectories) presents an ongoing challenge to
numerically characterizing the associated phase portraits.
In this work, we demonstrate that in the standard map, the geometric organization of regular orbits as well as sticky versus filling parts of chaotic orbits in phase space can be successfully discriminated based on relatively short time series by using several recurrence network measures, including network transitivity, global clustering coefficient, and average path length. This result provides the first documented finding pointing to
the relevance of recurrence network analysis for studying conservative dynamical systems.