The author investigates one-parameter analytic maps from the complex plane onto itself. The author approximates the set of parameter values for the stable periodic orbits, which arise due to subsequent bifurcation from the period-1 orbit, with the aid of normal forms. This approximated set consists of a cactus of touching circles, whose sizes obey a very simple scaling law. From this scaling law the Hausdorff dimension D of the boundary of this approximate set is computed analytically, giving D=1.2393 . . .. Numerical experiments, determining the dimension of the equivalent part of the Mandelbrot set, are consistent with this number. Moreover, this number seems to be independent of the precise form of the map, as predicted by the same analysis.