Parabolic Bifurcations in Complex Dimension 2

Abstract

Parabolic bifurcations in one complex dimension demonstrate a wide variety of interesting dynamical phenomena. In this paper we consider parabolic bifurcations of families of diffeomorphisms in two complex dimensions. Specifically we consider a two variable family of diffeomorphisms $F_\epsilon: M\to M$ given locally by $$F_\epsilon(x,y) = (x + x^2 + \epsilon^2+ ..., b_\epsilon y+...)$$ where $|b_\epsilon|<1$, and the `$...$' terms involve $x$, $y$ and $\epsilon$.