Cyclic symmetric dynamics in chaotic maps

In a recent paper (Liu et al., 2024), we reported on the microscopic mechanism underlying multistability in discrete dynamical systems, suggesting the potential for higher, even arbitrary-dimensional multistability in our conclusions. Before we can validate it, a fundamental question arises: what method can preserve the global dynamics of systems while allowing for an increase in dimensionality? This paper identifies the cyclic symmetric structure as a crucial solution and establishes two two-dimensional maps model based on it. The presence of multistability in any direction is affirmed, with this phenomenon representing either homogeneous or heterogeneous infinite expansion of the medium in multidimensional space. Furthermore, we uncover a range of dynamical characteristics, including grid-like phase trajectories, scale-free attractor clusters, fractal basin structures, symmetric attractors, and chaotic diffusion, all rooted in the system’s symmetric dynamical nature. This research not only enhances the comprehension of high-dimensional symmetric dynamics, but also offers a novel perspective for elucidating related models and phenomena.