Global dynamics for symmetric planar maps

Abstract

We consider sufficient conditions to determine the global dynamics for equivariant maps of the plane with a unique fixed point which is also hyperbolic. When the map is equivariant under the action of a compact Lie group, it is possible to describe the local dynamics. In particular, if the group contains a reflection, there is a line invariant by the map. This allows us to use results based on the theory of free homeomorphisms to describe the global dynamical behaviour. We briefly discuss the case when reflections are absent, for which global dynamics may not follow from local dynamics near the unique fixed point.

Global Saddles for Planar Maps

“…If f is invertible then we define α 2 (p) in a similar way. Lemma 2.8 (Lemma 3.1 in [3]). Let f : R n → R n be a homeomorphism such that f (0) = 0.…”

“…In Alarcón et al [3], we provide a list of symmetry groups for which the corresponding equivariant maps may possess a local saddle. There it is shown that the only symmetry groups that admit a local saddle are Z 2 ( −Id ),…”

“…Note that both Z 2 ( κ ) and D 2 = Z a 2 ⊕ Z b 2 contain a reflection, and hence an f -invariant line. A description of the admissible ω-limit set of a point in some cases, when the symmetry group contains a flip is also given in [3].…”

“…That happens because the hypotheses in Theorem 2.9 are not necessary conditions. An adaptation of Proposition 3.4 in [3] shows the absence of homoclinic contacts, as follows: let r = 0, ∞ be a homoclinic contact, that is…”

“…The presence of symmetry in a dynamical system creates special features that may be used to obtain global results. Planar dynamics with symmetry, when the fixed point is either an attractor or a repellor, has been addressed by the authors in [1,2,3]. There results, as well as those without extra assumptions on symmetry, ignore the important case when the fixed point is a local saddle.…”