Non-autonomous Parabolic Bifurcation

Abstract

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 $\{{\epsilon }_i{\}}_{1\le i\le N}$, 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 $f_N\circ \dots f_1\to \text{Id}$ (the Lavaurs map of $f_0$). We apply our results to prove parabolic bifurcation phenomenon in two dimensions for some class of maps.

12 pages, comments welcome