Non-autonomous Parabolic Bifurcation

Let $f(z)=z+z^2+O(z^3)$ and $f_{ϵ}(z)=f(z)+{ϵ}^2$. A classical result in parabolic bifurcation in one complex variable is the following: if $N-\frac{\pi }{ϵ}\to 0$ we obtain $(f_\epsilon)^{N} \to $\mathcal{L}$f$,where$$\mathcal{L}$f$istheLavaursmapof$f$. In this paper we study a $\mathit{\text{non-autonomous}}$ parabolic bifurcation. We focus on the case of $f_0(z)=\frac{z}{1-z}$. Given a sequence $\{{ϵ}_i{\}}_{1\le i\le N}$, we denote $f_n(z)=f_0(z)+{ϵ}_n^2$. We give sufficient and necessary conditions on the sequence $\{{ϵ}_i\}$ that imply that $f{N}\circ\ldots f{1} \to $\text{Id}$$(theLavaursmapof$f_0$). We apply our results to prove parabolic bifurcation phenomenon in two dimensions for some class of maps.