We analyze the time series obtained from different dynamical regimes of evolving maps and flows by constructing their equivalent time series networks, using the visibility algorithm. The regimes analyzed include periodic, chaotic, and hyperchaotic regimes, as well as intermittent regimes and regimes at the edge of chaos. We use the methods of algebraic topology, in particular, simplicial complexes, to define simplicial characterizers, which can analyze the simplicial structure of the networks at both the global and local levels. The simplicial characterizers 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.
We use the visibility algorithm to construct the time series networks obtained from the time series of different dynamical regimes of the logistic map. We define the simplicial characterisers of networks which can analyse the simplicial structure at both the global and local levels. These characterisers are used to analyse the TS networks obtained in different dynamical regimes of the logisitic map. It is seen that the simplicial characterisers are able to distinguish between distinct dynamical regimes. We also apply the simplicial characterisers to time series networks constructed from fMRI data, where the preliminary results indicate that the characterisers are able to differentiate between distinct TS networks. * Electronic address: gupte@physics.iitm.ac.in † Electronic address: nirmalthyagu@gmail.com ‡ Electronic address: malayajac@physics.iitm.ac.in arXiv:1707.00013v1 [stat.ME] 22 Jun 2017 I. INTRODUCTION The analysis of time series of evolving dynamical systems is a well established area of research. There are numerous well developed techniques for the analysis of such time series. These include Fourier transforms, power spectra, dimensions and entropies, Lyapunov exponents etc. These characterisers provide valuable insights into the dynamical behaviours of the evolving systems. In recent years, new techniques have emerged for the analysis of time series. These consist of mapping the time series to networks, using a variety of algorithms such as the visibility algorithms, recurrence times, identification of cycles or correlations, etc. See [1] for a brief review. Since networks are also a well established paradigm in the study of complex systems, there are well established metrics for their analysis. These include path lengths, clustering co-efficients, degree distributions etc. Here, we introduce a series of network characterisers which go beyond these usual characterisers, and provide new insightsinto the dynamical behaviour of systems. The characterisers are based on the methods of algebraic topology. We demonstrate the utility of these characterisers in the time series arising from the logistic map, and demonstrate that the characterisers can differentiate between different dynamical regimes. We also analyse the time series obtained from the fMRI
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.