In these notes, we introduce a geometric characterization of the Z 2 -equivariance of the phase space of a complex generic real analytic family with an elliptic equilibrium. The equivariance allows us to understand the interaction between complex and real foliations in terms of an antiholomorphic involution or real structure. The existence of the latter provides an explanation to some rigid phenomena observed in the complex phase portrait (e.g. the appearance of complex singularities in the real phase space; the presence of conformal symmetries affecting the invariant of analytic classication; etc.) In the complex phase space the equivariance is called the real character. It plays a fundamental role in the characterization of the conformal structure of elliptic equilibria of analytic families of vector elds.
|Number of pages||18|
|Journal||REVUE ROUMAINE DE MATHEMATIQUES PURES ET APPLIQUEES|
|Publication status||Published - Jan 1 2019|
- Hopf bifurcation
- Weak focus
ASJC Scopus subject areas
- Applied Mathematics