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On periodic and chaotic regions in the Mandelbrot set

Author

Listed:
  • Pastor, G.
  • Romera, M.
  • Álvarez, G.
  • Arroyo, D.
  • Montoya, F.

Abstract

We show here in a graphic and simple way the relation between the periodic and chaotic regions in the Mandelbrot set. Since the relation between the periodic and chaotic regions in a one-dimensional (1D) quadratic set is already well known, we shall base on it to extend the results to the Mandelbrot set. We shall see that in the same way as the hyperbolic components of the period-doubling cascade determines the chaotic bands structure in 1D quadratic sets, the periodic region determines the chaotic region in the Mandelbrot set.

Suggested Citation

  • Pastor, G. & Romera, M. & Álvarez, G. & Arroyo, D. & Montoya, F., 2007. "On periodic and chaotic regions in the Mandelbrot set," Chaos, Solitons & Fractals, Elsevier, vol. 32(1), pages 15-25.
    • Handle: RePEc:eee:chsofr:v:32:y:2007:i:1:p:15-25
    DOI: 10.1016/j.chaos.2005.10.099
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    References listed on IDEAS

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    1. Romera, M. & Pastor, G. & Montoya, F., 1996. "Misiurewicz points in one-dimensional quadratic maps," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 232(1), pages 517-535.
    2. Pastor, G. & Romera, M. & Alvarez, G. & Montoya, F., 2005. "External arguments for the chaotic bands calculation in the Mandelbrot set," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 353(C), pages 145-158.
    3. Pastor, G. & Romera, M. & Alvarez, G. & Montoya, F., 2001. "Misiurewicz point patterns generation in one-dimensional quadratic maps," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 292(1), pages 207-230.
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    Cited by:

    1. San Martín, Jesús & Moscoso, Ma José & González Gómez, A., 2009. "The universal cardinal ordering of fixed points," Chaos, Solitons & Fractals, Elsevier, vol. 42(4), pages 1996-2007.
    2. Yu, Dakuan & Ta, Wurui & Zhou, Youhe, 2021. "Fractal diffusion patterns of periodic points in the Mandelbrot set," Chaos, Solitons & Fractals, Elsevier, vol. 153(P1).
    3. Adhikari, Nabaraj & Sintunavarat, Wutiphol, 2024. "The Julia and Mandelbrot sets for the function zp−qz2+rz+sincw exhibit Mann and Picard–Mann orbits along with s-convexity," Chaos, Solitons & Fractals, Elsevier, vol. 181(C).
    4. Geum, Young Hee & Hare, Kevin G., 2009. "Groebner basis, resultants and the generalized Mandelbrot set," Chaos, Solitons & Fractals, Elsevier, vol. 42(2), pages 1016-1023.

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    1. Pastor, G. & Romera, M. & Alvarez, G. & Montoya, F., 2001. "Misiurewicz point patterns generation in one-dimensional quadratic maps," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 292(1), pages 207-230.

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