Analysis of the Chaotic Itinerancy Phenomenon using Entropy and Clustering

TL;DR

This work addresses the detection and quantification of chaotic itinerancy by coupling local entropy measures with density-based clustering to identify attractor ruins and chaotic transition states, followed by residence-time analysis and randomness testing. It introduces local Shannon entropy $H_{\text{local}}$ and local permutation entropy $PE_{\text{local}}$ as biomarkers of order versus chaos, and employs HDBSCAN to uncover attractor ruins in the globally coupled logistic maps (GCM) model. Through automated parameter sweeps and corroborative PCA and statistical tests, the method distinguishes coherent from intermittent regimes and characterizes the dimensionality of attractor ruins, aligning with Kaneko’s classic observations while providing a rigorous, algorithmic pipeline. The approach is dimension-agnostic and practical for high-dimensional systems, with code availability enabling application across a wide range of dynamical systems beyond the GCM example.

Abstract

We introduce a new methodology for the analysis of the phenomenon of chaotic itinerancy in a dynamical system using the notion of entropy and a clustering algorithm. We determine systems likely to experience chaotic itinerancy by means of local Shannon entropy and local permutation entropy. In such systems, we find quasi-stable states (attractor ruins) and chaotic transition states using a density-based clustering algorithm. Our approach then focuses on examining the chaotic itinerancy dynamics through the characterization of residence times within these states and chaotic transitions between them with the help of some statistical tests. We demonstrate the effectiveness of these methods on the system of globally coupled logistic maps (GCM), a well-known model exhibiting chaotic itinerancy. In particular, we conduct comprehensive computations for a large number of parameters in the GCM system and algorithmically identify itinerant dynamics observed previously by Kaneko in numerical simulations as coherent and intermittent phases.

Figures (14)

• Figure 1: Overview of the proposed method to determine the presence of chaotic itinerancy.
• Figure 2: Time series and local Shannon entropy computed for a segment of a sample trajectory for the GCM model with $N=5$, $a=2$ and $\varepsilon=0.234$.
• Figure 3: Time series and local Shannon entropy computed for a segment of a sample trajectory for the GCM model with $N=5$, $a=2$ and $\varepsilon =0.1$.
• Figure 4: Sum of variances of local Shannon entropy for all the coordinates of points on a segment of a sample trajectory as a function of $\varepsilon$, computed for the GCM model with $N=3$ and $a=2$.
• Figure 5: Projection onto the first two coordinates of a sample trajectory in the GCM model with $N=3$, $a=2$ and (a) $\varepsilon=0.15$, (b) $\varepsilon=0.2574$ and (c) $\varepsilon=0.27$.