Symmetry Groups of Basins of Attraction in Equivariant Dynamical Systems

Abstract

This paper investigates the symmetry properties of basins of attraction and their boundaries in equivariant dynamical systems. While the symmetry groups of compact attractors are well understood, the corresponding analysis for non-compact basins and their boundaries has remained underdeveloped. We establish a rigorous theoretical framework demonstrating the hierarchical inclusion of a chain of symmetry groups, showing that boundary symmetries can strictly exceed those of attractors and their basins. To determine admissible symmetry groups of basin boundaries, we propose three complementary approaches: (i) thickening transfer, which connects admissibility results from compact attractors to basins; (ii) algebraic constraints, which exploit the closed nature of boundaries to impose structural restrictions; and (iii) connectivity and flow analysis, which analyzes how the system's dynamics permute states. Numerical experiments on the Thomas system confirm these theoretical results, illustrating that cyclic group actions permute basins while preserving their common boundary, whereas central inversion leaves both basins and boundaries invariant. These findings reveal that basin boundaries often exhibit higher symmetry than the attractors they separate, providing new insights into the geometry of multistable systems and suggesting broader applications to physical and biological models where basin structure determines stability and predictability.