Abstract
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.
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References
Lars V. Ahlfors,Complex Analysis, 3rd ed., Mcgraw-Hill Inc, 1979.
Michael F. Barnsley,Fractals Everywhere, 2nd ed., Academic Press Professional, 1993.
Lennart Carleson and Theodore W. Gamelin,Complex Dynamics, Springer-Verlag, 1995.
Robert L. Devaney,An Introduction to Chaotic Dynamical Systems, The Benjamin/Cummings publishing Company, Inc, 1986.
Robert L. Devaney,Chaos, Fractals, and Dynamics, Addison-Wesley Inc, 1990.
Robert L. Devaney,The Complex Dynamics of Quadratic Polynomials, Proceedings of symposia in Applied Mathematics, Vol. 49, 1994.
Robert L. Devaney, The Mandelbrot Set, the Farey Tree, and the Fibonacci Sequence,The American mathematical monthly: the official journal of the Mathematical Association of America, Vol. 106, No. 4, 1999, pp. 289–302.
Young H. Geum & Young I. Kim, A Study on Computation of Component Centers in the Degree-n Bifurcation Set,Inter. J. Computer Math., Vol. 80, No. 2, 2003, pp. 223–232.
J. D. Lawrence,A Catalog of Special Plane Curves, Dover Publications Inc., New York, 1972, pp. 160–164 and 169.
J. S. Madachy,Madachy's Mathematical Recreations, Dover Publications Inc., New York, 1979, pp. 219–225.
James R. Munkres,Topology, 3rd ed., Prentice-Hall Inc, 1975.
H. O. Peitgen and P. H. Richter,The Beauty of Fractals, Springer-Verlag, Berlin-Heidelberg, 1986.
Mitsuhiro Shishikura, The Boundary of the Mandelbrot Set has Hausdorff Dimension Two,Astérisque, Vol. 222, 1994, pp. 389–405.
Murray R. Spigel,Theory and Problems of Mathematical Handbook of Formulas and Tables, Schaum's Outline Series in Mathematics, McGraw-Hill Inc., 1968.
Dirk J. Struik,Lectures on Classical Differential Geometry, 2nd ed., Dover Publications Inc., New York, 1988.
S. Wagon,Mathematica in Action, W. H. Freeman Inc., 1991, pp. 50–52.
Stephen Wolfram,The Mathematica Book, 4th ed., Cambridge University Press, 1999.
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Young Hee Geum is a lecturer in the Department of Mathematics, Dankook University. Department of Mathematics, Dankook University, Hannamdong, Seoul, 140-714, Korea.
Young Ik Kim received his BS from Seoul National University, KOREA and MS, MA, Ph. D. from Arizona State University, USA. He is currently a professor of mathematics at Dankook University, Korea. His research interest is in numerical analysis on the problems arising in many applied sciences including chaos and fractals in applied mathematics. He is also interested in developing visual mathematics application software using c++, Mathematica and Maple and HTML languages.
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Geum, Y.H., Kim, Y.I. An epicycloidal boundary of the main component in the degree-n bifurcation set. JAMC 16, 221–229 (2004). https://doi.org/10.1007/BF02936163
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DOI: https://doi.org/10.1007/BF02936163