  We analyze the time series obtained from different dynamical regimes of the
logistic map by constructing their equivalent time series (TS) networks, using
the visibility algorithm. The regimes analyzed include both periodic and
chaotic regimes, as well as intermittent regimes and the Feigenbaum attractor
at the edge of chaos. We use the methods of algebraic topology to define the
simplicial characterizers, which can analyse the simplicial structure of the
networks at both the global and local levels. The simplicial characterisers
bring out the hierarchical levels of complexity at various topological levels.
These hierarchical levels of complexity find the skeleton of the local dynamics
embedded in the network which influence the global dynamical properties of the
system, and also permit the identification of dominant motifs. We also analyze
the same networks using conventional network characterizers such as average
path lengths and clustering coefficients. We see that the simplicial
characterizers are capable of distinguishing between different dynamical
regimes and can pick up subtle differences in dynamical behavior, whereas the
usual characterizers provide a coarser characterization. However, the two taken
in conjunction, can provide information about the dynamical behavior of the
time series, as well as the correlations in the evolving system. Our methods
can therefore provide powerful tools for the analysis of dynamical systems.
