We study the bifurcation of the complex Arnold family $f(z) = z + \alpha + \beta \sin z$, where $z$ is a complex variable, $\beta$ is a small real parameter and the bifurcation is studied in the complex $\alpha$-plane. One can see a structure called ``fingers'' or ``tentacles'' in the parameters space. (Also observed in Henon parameter space.) We study the qualitative and quantitative aspects of the fingers via parabolic bifurcation.

This is a work in progress joint with David Marti Pete (Kyoto University).