Parabolic Bifurcations in Complex Dimension 2

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 $F_\epsilon: M\to M$givenlocallyby$$F_\epsilon(x,y) = (x + x^2 + \epsilon^2+ …, b_\epsilon y+…)$$where$|b_\epsilon|<1$,andthe\mathrm{‘}$…${}^{\mathrm{\prime }}termsinvolve$x$,$y$and$\epsilon$.