Periodic points of complex quadratic mappings

From formulasearchengine
Jump to navigation Jump to search

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.

This theory is applied in relation with the theories of Fatou and Julia sets.



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.

is the -fold compositions of with itself = iteration of function or,

Periodic points of a complex quadratic mapping of period are points of the dynamical plane such that :

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

Stability index of periodic points along horizontal axis
boundaries of regions of parameter plane with attracting orbit of periods 1-6
Critical orbit of discrete dynamical system based on complex quadratic polynomial. It tends to weakly attracting fixed point with abs(multiplier)=0.99993612384259

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 periodic point is :[2]

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.[3] 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

then .

It means that fixed points are symmetrical around .

This image shows fixed points (both repelling)

Complex dynamics

Fixed points for c along horizontal axis
Fatou set for F(z)=z*z with marked fixed point

Here different notation is commonly used:[4]

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:

Special cases

An important case of the quadratic mapping is . In this case, we get and . In this case, 0 is a superattractive fixed point, and 1 belongs to the Julia set.

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

We can extend complex plane to Riemann sphere (extended complex plane) by adding infinity

and extend polynomial such that

Then infinity is :

Period-2 cycles

Bifurcation from period 1 to 2 for complex quadratic map

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 :

First method

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.[7]

Special cases

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

There is no general solution in radicals to polynomial equations of degree five or higher, so it must be computed using numerical methods.


  1. Alan F. Beardon, Iteration of Rational Functions, Springer 1991, ISBN 0-387-95151-2, p. 41
  2. Alan F. Beardon, Iteration of Rational Functions, Springer 1991, ISBN 0-387-95151-2, page 99
  3. Some Julia sets by Michael Becker
  4. On the regular leaf space of the cauliflower by Tomoki Kawahira Source: Kodai Math. J. Volume 26, Number 2 (2003), 167-178.
  5. Periodic attractor by Evgeny Demidov
  6. 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
  7. Period 2 orbit by Evgeny Demidov

Further reading

External links