Non-autonomous Parabolic Bifurcation

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