The Geometry of Chaotic Dynamics -- A Complex Network Perspective

Abstract

Recently, several complex network approaches to time series analysis have been developed and applied to study a wide range of model systems as well as real-world data, e.g., geophysical or financial time series. Among these techniques, recurrence-based concepts and prominently $\epsilon$-recurrence networks, most faithfully represent the geometrical fine structure of the attractors underlying chaotic (and less interestingly non-chaotic) time series. In this paper we demonstrate that the well known graph theoretical properties local clustering coefficient and global (network) transitivity can meaningfully be exploited to define two new local and two new global measures of dimension in phase space: local upper and lower clustering dimension as well as global upper and lower transitivity dimension. Rigorous analytical as well as numerical results for self-similar sets and simple chaotic model systems suggest that these measures are well-behaved in most non-pathological situations and that they can be estimated reasonably well using $\epsilon$-recurrence networks constructed from relatively short time series. Moreover, we study the relationship between clustering and transitivity dimensions on the one hand, and traditional measures like pointwise dimension or local Lyapunov dimension on the other hand. We also provide further evidence that the local clustering coefficients, or equivalently the local clustering dimensions, are useful for identifying unstable periodic orbits and other dynamically invariant objects from time series. Our results demonstrate that $\epsilon$-recurrence networks exhibit an important link between dynamical systems and graph theory.

21 pages, 12 figures

References in corpus (10)

• Recurrence Plots for the Analysis of Complex Systems
• Synchronization in complex networks
• From time series to complex networks: the visibility graph
• Complex networks in climate dynamics - Comparing linear and nonlinear network construction methods
• Complex Network Approach for Recurrence Analysis of Time Series
• The backbone of the climate network
• Historical Review of Recurrence Plots
• Ambiguities in recurrence-based complex network representations of time series
• Degree distribution of the visibility graphs mapped from fractional Brownian motions and multifractal random walks
• Identifying complex periodic windows in continuous-time dynamical systems using recurrence-based methods

Cited by in corpus (41)

• Complex network approaches to nonlinear time series analysis
• Dynamic Analysis of Digital Chaotic Maps via State-Mapping Networks
• Network structure of multivariate time series
• Unified functional network and nonlinear time series analysis for complex systems science: The pyunicorn package
• Analytical framework for recurrence-network analysis of time series
• Geometric detection of coupling directions by means of inter-system recurrence networks
• Disentangling different types of El Niño episodes by evolving climate network analysis
• Recurrence threshold selection for obtaining robust recurrence characteristics in different embedding dimensions
• Complex network based techniques to identify extreme events and (sudden) transitions in spatio-temporal systems
• Evaluation of selected recurrence measures in discriminating pre-ictal and inter-ictal periods from epileptic EEG data
• Multiplex Recurrence Networks
• Analysing spatially extended high-dimensional dynamics by recurrence plots
• Sequential visibility-graph motifs
• Multiscale characterization of recurrence-based phase space networks constructed from time series
• A unified and automated approach to attractor reconstruction
• Horizontal Visibility graphs generated by type-I intermittency
• Time Series Analysis via Network Science: Concepts and Algorithms
• Uniform framework for the recurrence-network analysis of chaotic time series
• Sequential motif profile of natural visibility graphs
• Topological properties and fractal analysis of recurrence network constructed from fractional Brownian motions
• Fractal and complex network analyses of protein molecular dynamics
• Cross and joint ordinal partition transition networks for multivariate time series analysis
• Recurrence based quantification of dynamical complexity in the Earth's magnetosphere at geospace storm timescales
• Phase space reconstruction for non-uniformly sampled noisy time series
• Geometric signature of complex synchronisation scenarios
• Quasiperiodic graphs: structural design, scaling and entropic properties
• Disentangling regular and chaotic motion in the standard map using complex network analysis of recurrences in phase space
• Analyzing long-term correlated stochastic processes by means of recurrence networks: Potentials and pitfalls
• Recurrence network measures for hypothesis testing using surrogate data: application to black hole light curves
• Fluctuation of similarity (FLUS) to detect transitions between distinct dynamical regimes in short time series
• Geometric and dynamic perspectives on phase-coherent and noncoherent chaos
• Can recurrence networks show small world property?
• Classification of Close Binary Stars Using Recurrence Networks
• Recurrence Network Analysis of Exoplanetary Observables
• Regime change detection in irregularly sampled time series
• Network analysis of chaotic dynamics in fixed-precision digital domain
• Recurrence network analysis in a model tripartite quantum system
• Quantifying information loss on chaotic attractors through recurrence networks
• Quasiperiodic graphs at the onset of chaos
• Time-series and network analysis in quantum dynamics: Comparison with classical dynamics
• Stability of exoplanetary systems retrieved from scalar time series