A local but not global attractor for a Zn-symmetric map

TL;DR

This paper constructs planar maps with Zn symmetry that have local attractors which are not global, demonstrating that local stability does not necessarily imply global stability in symmetric dynamical systems.

Contribution

It provides explicit examples of symmetric maps with local but not global attractors, highlighting limitations of local stability analysis for global dynamics.

Findings

Existence of maps with local but not global attractors under Zn symmetry

Construction of examples starting from Z4-symmetric maps

Maps exhibit rational rotation numbers due to symmetry constraints

Abstract

There are many tools for studying local dynamics. An important problem is how this information can be used to obtain global information. We present examples for which local stability does not carry on globally. To this purpose we construct, for any natural n>1, planar maps whose symmetry group is Zn having a local attractor that is not a global attractor. The construction starts from an example with symmetry group Z4. We show that although this example has codimension 3 as a Z4-symmetric map-germ, its relevant dynamic properties are shared by two 1-parameter families in its universal unfolding. The same construction can be applied to obtain examples that are also dissipative. The symmetry of these maps forces them to have rational rotation numbers.