Periodic points of complex quadratic mappings
This article describes periodic points of some complex quadratic maps. A map is a formula for computing a value of a variable based on its own previous value or values; a quadratic map is one that involves the previous value raised to the powers one and two; and a complex map is one in which the variable is a complex number. A periodic point of a map is a value of the variable that occurs repeatedly after intervals of a fixed length.
- 1 Definitions
- 2 Stability of periodic points (orbit) - multiplier
- 3 Period-1 points (fixed points)
- 4 Period-2 cycles
- 5 Cycles for period>2
- 6 References
- 7 Further reading
- 8 External links
where and are complex-valued. (This is the complex quadratic mapping mentioned in the title.) This article explores the periodic points of this mapping - that is, the points that form a periodic cycle when is repeatedly applied to them.
where is the smallest positive integer.
We can introduce a new function:
so periodic points are zeros of function :
which is a polynomial of degree
Stability of periodic points (orbit) - multiplier
The multiplier ( or eigenvalue, derivative ) of rational map at fixed point is defined as :
where is first derivative of with respect to at .
Because the multiplier is the same at all periodic points, it can be called a multiplier of the periodic orbit.
The multiplier is:
- a complex number,
- invariant under conjugation of any rational map at its fixed point
- used to check stability of periodic (also fixed) points with stability index :
A periodic point is :
- attracting when
- super-attracting when
- attracting but not super-attracting when
- indifferent when
- repelling when
Where do periodic points belong?
- attracting is always in Fatou set
- repelling is in the Julia set
- Indifferent fixed points may be in one or the other. Parabolic periodic point is in Julia set.
Period-1 points (fixed points)
Finite fixed points
Let us begin by finding all finite points left unchanged by 1 application of . These are the points that satisfy . That is, we wish to solve
which can be rewritten
Since this is an ordinary quadratic equation in 1 unknown, we can apply the standard quadratic solution formula. Look in any standard mathematics textbook, and you will find that there are two solutions of are given by
In our case, we have , so we will write
So for we have two finite fixed points and .
- and where
It means that fixed points are symmetrical around .
Here different notation is commonly used:
- with multiplier
- with multiplier
Using Viète's formulas one can show that:
Since derivative with respect to z is :
It implies that can have at most one attractive fixed point.
This points are distinguished by the facts that:
- is :
- landing point of several rays
- is :
- attracting when c is in main cardioid of Mandelbrot set, then it is in interior of Filled-in Julia set, it means belongs to Fatou set ( strictly to basin of attraction of finite fixed point )
- parabolic at the root point of the limb of Mandelbrot set
- repelling for other c values
Only one fixed point
We might wonder what value should have to cause . The answer is that this will happen exactly when . This equation has 1 solution: (in which case, ). This is interesting, since is the largest positive, purely real value for which a finite attractor exists.
Infinite fixed point
and extend polynomial such that
Then infinity is :
Suppose next that we wish to look at period-2 cycles. That is, we want to find two points and such that , and .
Let us start by writing , and see where trying to solve this leads.
Thus, the equation we wish to solve is actually .
This equation is a polynomial of degree 4, and so has 4 (possibly non-distinct) solutions. However, actually, we already know 2 of the solutions. They are and , computed above. It is simple to see why this is; if these points are left unchanged by 1 application of , then clearly they will be unchanged by 2 applications (or more).
Our 4th-order polynomial can therefore be factored in 2 ways :
This expands directly as (note the alternating signs), where
We already have 2 solutions, and only need the other 2. This is as difficult as solving a quadratic polynomial. In particular, note that
Adding these to the above, we get and . Matching these against the coefficients from expanding , we get
From this, we easily get : and .
From here, we construct a quadratic equation with and apply the standard solution formula to get
Closer examination shows (the formulas are a tad messy) that :
meaning these two points are the two halves of a single period-2 cycle.
Second method of factorization
The roots of the first factor are the two fixed points . They are repelling outside the main cardioid.
The second factor has two roots
These two roots form period-2 orbit.
Again, let us look at . Then
both of which are complex numbers. By doing a little algebra, we find . Thus, both these points are "hiding" in the Julia set. Another special case is , which gives and . This gives the well-known superattractive cycle found in the largest period-2 lobe of the quadratic Mandelbrot set.
Cycles for period>2
- Alan F. Beardon, Iteration of Rational Functions, Springer 1991, ISBN 0-387-95151-2, p. 41
- Alan F. Beardon, Iteration of Rational Functions, Springer 1991, ISBN 0-387-95151-2, page 99
- Some Julia sets by Michael Becker
- On the regular leaf space of the cauliflower by Tomoki Kawahira Source: Kodai Math. J. Volume 26, Number 2 (2003), 167-178.
- Periodic attractor by Evgeny Demidov
- R L Devaney, L Keen (Editor): Chaos and Fractals: The Mathematics Behind the Computer Graphics. Publisher: Amer Mathematical Society July 1989, ISBN 0-8218-0137-6 , ISBN 978-0-8218-0137-6
- Period 2 orbit by Evgeny Demidov
- Alan F. Beardon, Iteration of Rational Functions, Springer 1991, ISBN 0-387-95151-2
- Michael F. Barnsley (Author), Stephen G. Demko (Editor), Chaotic Dynamics and Fractals (Notes and Reports in Mathematics in Science and Engineering Series) Academic Pr (April 1986), ISBN 0-12-079060-2
- Wolf Jung : Homeomorphisms on Edges of the Mandelbrot Set. Ph.D. thesis of 2002
- The permutations of periodic points in quadratic polynominials by J Leahy
|Wikibooks has a book on the topic of: Fractals|
- Algebraic solution of Mandelbrot orbital boundaries by Donald D. Cross
- Brown Method by Robert P. Munafo
- arXiv:hep-th/0501235v2 V.Dolotin, A.Morozov: Algebraic Geometry of Discrete Dynamics. The case of one variable.
- Gvozden Rukavina : Quadratic recurrence equations - exact explicit solution of period four fixed points functions in bifurcation diagram