Abstract
The recently formulated theory of horizontal visibility graphs transforms time series into graphs and allows the possibility of studying dynamical systems through the characterization of their associated networks. This method leads to a natural graph-theoretical description of nonlinear systems with qualities in the spirit of symbolic dynamics. We support our claim via the case study of the period-doubling and band-splitting attractor cascades that characterize unimodal maps. We provide a universal analytical description of this classic scenario in terms of the horizontal visibility graphs associated with the dynamics within the attractors, that we call Feigenbaum graphs, independent of map nonlinearity or other particulars. We derive exact results for their degree distribution and related quantities, recast them in the context of the renormalization group and find that its fixed points coincide with those of network entropy optimization. Furthermore, we show that the network entropy mimics the Lyapunov exponent of the map independently of its sign, hinting at a Pesin-like relation equally valid out of chaos.
Highlights
We expose a remarkable relationship between nonlinear dynamical systems and complex networks by means of the horizontal visibility (HV) algorithm [1,2,3] that transforms time series into graphs
We offer a distinct view of the Feigenbaum scenario through the specific HV formalism, and provide a complete set of graphs, which we call Feigenbaum graphs, that encode the dynamics of all stationary trajectories of unimodal maps
We have shown that the horizontal visibility theory combines power with straightforwardness as a tool for the analytical study of nonlinear dynamics
Summary
We expose a remarkable relationship between nonlinear dynamical systems and complex networks by means of the horizontal visibility (HV) algorithm [1,2,3] that transforms time series into graphs. According to the HV algorithm, a time series generated by the Logistic map for a specific value of m (after an initial transient of approach to the attractor) is converted into a Feigenbaum graph (see figure 1).
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