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

Journal of Singularities
volume 6 (2012), 1-14
Proceedings of the Workshop on Singularities in Geometry and Applications, Będlewo, 5 – 21 May 2011

Received: 19 December 2011. Received in revised form: 8 May 2012.

DOI: 10.5427/jsing.2012.6a

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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 Z_n having a local attractor that is not a global attractor. The construction starts from an example with symmetry group Z_4. We show that although this example has codimension 3 as a Z_4-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.

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