Editing Mandelbrot set (section)

===Main cardioid and period bulbs===
<!--[[Douady rabbit]] links directly here.-->[[File:Mandelbrot Set – Periodicities coloured.png|right|thumb|Periods of hyperbolic components]]

The ''main [[cardioid]]'' is the period 1 continent.<ref>{{Cite book |last1=Brucks |first1=Karen M. |url=https://books.google.com/books?id=p-amwZp0R-0C |title=Topics from One-Dimensional Dynamics |last2=Bruin |first2=Henk |date=2004-06-28 |publisher=Cambridge University Press |isbn=978-0-521-54766-6 |pages=264 |language=en}}</ref> It is the region of parameters <math>c</math> for which the map <math>f_c(z) = z^2 + c</math> has an [[Periodic points of complex quadratic mappings|attracting fixed point]].<ref>{{Cite book |last=Devaney |first=Robert |url=https://books.google.com/books?id=YEIPEAAAQBAJ |title=An Introduction To Chaotic Dynamical Systems |date=2018-03-09 |publisher=CRC Press |isbn=978-0-429-97085-6 |pages=147 |language=en}}</ref> It consists of all parameters of the form <math> c(\mu) := \frac\mu2\left(1-\frac\mu2\right)</math> for some <math>\mu</math> in the [[open unit disk]].<ref name=":5">{{Cite book |last1=Ivancevic |first1=Vladimir G. |url=https://books.google.com/books?id=mbtCAAAAQBAJ |title=High-Dimensional Chaotic and Attractor Systems: A Comprehensive Introduction |last2=Ivancevic |first2=Tijana T. |date=2007-02-06 |publisher=Springer Science & Business Media |isbn=978-1-4020-5456-3 |pages=492–493 |language=en}}</ref>{{close paraphrasing inline|date=March 2025}}

To the left of the main cardioid, attached to it at the point <math>c=-3/4</math>, a circular bulb, the ''period-2 bulb'' is visible.<ref name=":5" />{{close paraphrasing inline|date=March 2025}} The bulb consists of <math>c</math> for which <math>f_c</math> has an [[Periodic points of complex quadratic mappings|attracting cycle of period 2]]. It is the filled circle of radius 1/4 centered around −1.<ref name=":5" />{{close paraphrasing inline|date=March 2025}}

[[File:Animated cycle.gif|left|thumb|Attracting cycle in 2/5-bulb plotted over [[Julia set]] (animation)]]
More generally, for every positive integer <math>q>2</math>, there are <math>\phi(q)</math> circular bulbs tangent to the main cardioid called ''period-q bulbs'' (where <math>\phi</math> denotes the [[Euler's totient function|Euler phi function]]), which consist of parameters <math>c</math> for which <math>f_c</math> has an attracting cycle of period <math>q</math>.{{Citation needed|date=March 2025}} More specifically, for each primitive <math>q</math>th root of unity <math>r=e^{2\pi i\frac{p}{q}}</math> (where <math>0<\frac{p}{q}<1</math>), there is one period-q bulb called the <math>\frac{p}{q}</math> bulb, which is tangent to the main cardioid at the parameter <math> c_{\frac{p}{q}} := c(r) = \frac{r}2\left(1-\frac{r}2\right),</math> and which contains parameters with <math>q</math>-cycles having combinatorial rotation number <math>\frac{p}{q}</math>.<ref>{{Cite book |last1=Devaney |first1=Robert L. |url=https://books.google.com/books?id=4XrHCQAAQBAJ |title=Complex Dynamical Systems: The Mathematics Behind the Mandelbrot and Julia Sets: The Mathematics Behind the Mandelbrot and Julia Sets |last2=Branner |first2=Bodil |date=1994 |publisher=American Mathematical Soc. |isbn=978-0-8218-0290-8 |pages=18–19 |language=en}}</ref> More precisely, the <math>q</math> periodic [[Classification of Fatou components|Fatou components]] containing the attracting cycle all touch at a common point (commonly called the ''<math>\alpha</math>-fixed point''). If we label these components <math>U_0,\dots,U_{q-1}</math> in counterclockwise orientation, then <math>f_c</math> maps the component <math>U_j</math> to the component <math>U_{j+p\,(\operatorname{mod} q)}</math>.<ref name=":5" />{{close paraphrasing inline|date=March 2025}}

[[File:Juliacycles1.png|right|thumb|Attracting cycles and [[Julia set]]s for parameters in the 1/2, 3/7, 2/5, 1/3, 1/4, and 1/5 bulbs]]

The change of behavior occurring at <math>c_{\frac{p}{q}}</math> is known as a [[bifurcation theory|bifurcation]]: the attracting fixed point "collides" with a repelling period-''q'' cycle. As we pass through the bifurcation parameter into the <math>\tfrac{p}{q}</math>-bulb, the attracting fixed point turns into a repelling fixed point (the <math>\alpha</math>-fixed point), and the period-''q'' cycle becomes attracting.<ref name=":5" />{{close paraphrasing inline|date=March 2025}}