Astrocytes: Arnol'd Tongues Generalization in Dynamical Systems' Parameter Plane

TL;DR

This paper introduces a new structural concept called astrocytes in the parameter plane of dynamical systems, revealing complex, self-similar patterns and bifurcations associated with regular and chaotic behaviors.

Contribution

It generalizes the Arnol'd tongues concept by identifying astrocytes, detailed morphological structures that characterize regions of regular behavior in dynamical systems' parameter space.

Findings

Astrocytes exhibit self-similarity and hierarchical organization.

Multiple periodicities coexist within astrocyte structures.

Bifurcation cascades lead to higher periodicities and chaos.

Abstract

We discovered generalized structures, named astrocytes due to their shape, that constitute a defined region characterizing regular behavior within the parameter plane (PP) of dynamical systems (DSs). Morphologically, they are characterized by a branch and a soma with several vertices (arms) and sometimes with multiple periodicities. A bunch of infinite astrocytes emerge through their branches from a region, in general, of low periodicity. Astrocytes are embedded in a quasiperiodic-chaotic scenario. The soma complexity (number of vertices) determines a kind of hierarchy of the astrocytes; moreover, bunches of subsequent structures from the astrocyte have been emphasized, revealing a self-similarity property. We conducted a detailed analysis in a Zeeman laser model, but we also observed astrocytes in many other DSs. The multiperiodicity exhibited by the astrocytes in their soma gives rise to harlequin dress-like patterns and tri-, quad-, and quint-critical points, which indicate the coexistence of different higher-order periodicities. In the concave borders of the soma, a doubling cascade of quint-points emerges as a bifurcation in the PP, defining regions of ordered sequences of higher periodicity in the route to chaos.