A boundary equation of the principal period-2 component in the degree-n bifurcation set

Abstract

The degree-n bifurcation set is a generalized Mandelbrot set for the complex polynomial P _c (z)=z ⁿ +c. The boundary of the principal period-2 component in the degree-n bifurcation set is first defined and then formulated by a parametrization of its image, which is the unit circle under the multiplier map. We investigate the boundary equation using the geometric symmetry of the degree-n bifurcation set with respect to rays of symmetry in the complex plane. In addition, an algorithm drawing the boundary curve with Mathematica codes is proposed.

Keywords:

• degree-n bifurcation set
• main component
• Mandelbrot set
• boundary equation
• branch

AMS 2000 Subject Classification :

• 00A05
• 00A69
• 00A99

Acknowledgement

This work was supported by the Korea Research Foundation Grant (MOEHRD), (KRF-037-C00010).