# Definition

Forward orbit[1] of a critical point[2][3] is called a critical orbit.

# Importance

Critical orbits are very important because every attracting periodic orbit[4] attracts a critical point, so studying the critical orbits helps us understand the dynamics in the Fatou set.[5][6][7]

${\displaystyle z_{0}=z_{cr}=0\,}$

${\displaystyle z_{1}=f_{c}(z_{0})=c\,}$

${\displaystyle z_{2}=f_{c}(z_{1})=c^{2}+c\,}$

${\displaystyle z_{3}=f_{c}(z_{2})=(c^{2}+c)^{2}+c\,}$

${\displaystyle ...\,}$

This orbit falls into an attracting periodic cycle.

## Shape

Relation between shape types and dynamics:

• n-th arm spiral: attracting or repelling n-periodic orbit ( cycle)
• closed curve: Siegel disc ( rotation)
• n-th arm star = period n parabolic root

The shape of critical orbit can show the type of dynamics and the period

## Level sets

Points of critical orbit ( including crirital point and fixed point = finite attractor) are on the level curves like notes on the musical staff ( dots on curves) .

# Code

```"https://github.com/conanite/rainbow/blob/master/src/arc/rainbow/spiral.arc
This software is copyright (c) Conan Dalton 2008. Permission to use it is granted under the Perl Foundations's Artistic License 2.0.
This software uses javacc which is copyright (c) its authors
"
(def plot (plt c)
(with (z 0+0i
n 0
repeats 0)
(while (and (small z) (< n 10000) (< repeats 1000))
(assign n       (+ n 1)
z       (+ c (* z z))
repeats (if (apply plt (complex-parts z))
(+ repeats 1)
0)))))
```

# References

1. Wikipedia: orbit (dynamics)
2. Wikipedia: Complex quadratic polynomial - Critical point
3. MandelOrbits - A visual real-time trace of Mandelbrot iterations by Ivan Freyman
4. Wikipedia: Periodic points of complex quadratic mappings
5. M. Romera, G. Pastor, and F. Montoya: Multifurcations in nonhyperbolic fixed points of the Mandelbrot map. Fractalia 6, No. 21, 10-12 (1997)
6. Burns A M: Plotting the Escape: An Animation of Parabolic Bifurcations in the Mandelbrot Set. Mathematics Magazine, Vol. 75, No. 2 (Apr., 2002), pp. 104-116
7. Khan Academy: Mandelbrot Spirals 2
8. Complex Power Towers (Or ‘mucking around with Mathematica’) by Mike Croucher
9. /DarkHeart by Chris King
10. Alexandre Devert blog
11. codeproject: Fractals-in-theory-and-practice