Graph-let based approach to evolutionary behaviors in chaotic time series

In the Graph-let based time series analysis, a time series is mapped into a series of graph-lets, representing the local states respectively. The bridges between successive graph-lets are reduced simply to a linkage with an information of occurrence. In the present work, we focus our attention on the bridge series, i.e., preserve the structures of the bridges and reduce the states into nodes. The bridge series can tell us how the system evolves. Technically, the ordinal partition algorithm is adopted to construct the graph-lets and the bridges. Results for the Logistic Map, the Hénon Map, and the Lorenz System show that the statistical properties for transition frequency network for the bridges, e.g., the number of visited bridges and the average out-entropy-degree, have the capability of characterizing chaotic processes, being equivalent with the Lyapunov exponent. What is more, the topological structure can display the details of the contributions of the transitions between the bridges to the statistical properties.