Horizontal Visibility graphs generated by type-II intermittency

Abstract

In this contribution we study the onset of chaos via type-II intermittency within the framework of Horizontal Visibility graph theory. We construct graphs associated to time series generated by an iterated map close to a Neimark-Sacker bifurcation and study, both numerically and analytically, their main topological properties. We find well defined equivalences between the main statistical properties of intermittent series (scaling of laminar trends and Lyapunov exponent) and those of the resulting graphs, and accordingly construct a graph theoretical description of type-II intermittency. We finally recast this theory into a graph-theoretical renormalization group framework, and show that the fixed point structure of RG flow diagram separates regular, critical and chaotic dynamics.

On the degree distribution of horizontal visibility graphs associated with Markov processes and dynamical systems: diagrammatic and variational approaches

“…In this work we focus on the analytical properties of the degree distribution P (k) (and P → (k)), when the HVG (DHVG) is associated with some important classes of dynamical processes. The amount of closed analytical results on these degree distributions is also so far scarce, most of them being found for HVGs which are nontrivial graph theoretical fixed points of some renormalization group transformation, associated with critical dynamics generated at accumulation points of the canonical routes to chaos [10][11][12][13]. Almost no exact results exist for other nonperiodic dynamical processes, with the exception of uncorrelated random processes [1,14].…”

Horizontal visibility graph of a random restricted growth sequence

“…A horizontal visibility graph (HVG) [17] constitutes a paradigmatic complex network representation of sequential data, typically used to reveal order structures within the data set [8,35]. HVG-based algorithms have been employed to characterize fractal behavior of dynamical systems [21,31], study canonical routes to chaos (see [24] and references therein), discriminate between chaotic and stochastic time series [26], and test time series irreversibility [33]. There is a growing body of literature using these combinatorial data analysis techniques in applied fields such as optics [1], fluid dynamics [22], geophysics [30], physiology and neuroscience [18,27], finance [25], image processing [13], and more [8,35].…”

A Scalable Linear-Time Algorithm for Horizontal Visibility Graph Construction Over Long Sequences

“…dynamic properties of the physical system generating the time series. For example, the degrees of HVG graph vertices can be used to quantify dynamic irreversibility in nonlinear systems [10] and to estimate Lyapunov exponents and other measures of chaos [11], [12]. In stochastic physics the vertex degrees of HVGs can be used to estimate Hurst exponents to quantify long-term auto-correlations in data [13].…”