An epicycloidal boundary of the main component in the degree-n bifurcation set



It is known that the parametric boundary equation for the main component in the Mandelbrot set represents a cardioid. We derive an epicycloidal boundary equation of the main component in the degree-n bifurcation set by extending the parameter which describes the cardioid in the Mandelbrot set. Computational results as well as some useful properties are presented together with the programming source codes written inMathematica. Various boundaries are displayed for 2≤n≤7 and show a good agreement with the theory presented here. The known boundary equation enables us to significantly reduce the construction time for the degree-n bifurcation set.

AMS Mathematics Subject Classification

00A05 00A69 00A99 

Key words and phrases

Epicycloid bifurcation main component boundary Mandelbrot set ParametricPlot degree-n bifurcation set 


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Copyright information

© Korean Society for Computational and Applied Mathematics 2004

Authors and Affiliations

  1. 1.Department of MathematicsDankook UniversitySeoulKorea
  2. 2.Department of Applied MathematicsDankook UniversityCheonan CityKorea

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