page_title: Mandelbrot set
section_title: Main cardioid and period bulbs
hierarchical_section_title: Mandelbrot set / Other properties / Main cardioid and period bulbs
caption_attribution_description: rays and cycles in Mandelbrot set bulbs
context_page_description: The Mandelbrot set is the set of complex numbers for which the function does not diverge when iterated from, i.e., for which the sequence, etc., remains bounded in absolute value.
  Its definition is credited to Adrien Douady who named it in tribute to the mathematician Benoit Mandelbrot. The set is connected to a Julia set, and related Julia sets produce similarly complex fractal shapes.
  Mandelbrot set images may be created by sampling the complex numbers and testing, for each sample point, whether the sequence goes to infinity. Treating the real and imaginary parts of as image coordinates on the complex plane, pixels may then be coloured according to how soon the sequence crosses an arbitrarily chosen threshold, with a special color used for the values of for which the sequence has not crossed the threshold after the predetermined number of iterations. If is held constant and the initial value of —denoted by —is variable instead, one obtains the corresponding Julia set for each point in the parameter space of the simple function.
context_section_description: Upon looking at a picture of the Mandelbrot set, one immediately notices the large cardioid-shaped region in the center. This main cardioid
  is the region of parameters  for which  has an attracting fixed point. It consists of all parameters of the form
  for some  in the open unit disk.
  To the left of the main cardioid, attached to it at the point , a circular-shaped bulb is visible. This bulb consists of those parameters  for which  has an attracting cycle of period 2. This set of parameters is an actual circle, namely that of radius 1/4 around −1.
  There are infinitely many other bulbs tangent to the main cardioid: for every rational number , with p and q coprime, there is such a bulb that is tangent at the parameter
  This bulb is called the -bulb of the Mandelbrot set. It consists of parameters that have an attracting cycle of period  and combinatorial rotation number . More precisely, the  periodic Fatou components containing the attracting cycle all touch at a common point (commonly called the -fixed point). If we label these components  in counterclockwise orientation, then  maps the component  to the component .
  The change of behavior occurring at  is known as a bifurcation: the attracting fixed point "collides" with a repelling period q-cycle. As we pass through the bifurcation parameter into the -bulb, the attracting fixed point turns into a repelling fixed point (the -fixed point), and the period q-cycle becomes attracting.
caption: Cycle periods and antennae
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