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 and the parameters are complex numbers. A periodic point of a map is a value of the variable that occurs repeatedly after intervals of a fixed length.

These periodic points play a role in the theories of Fatou and Julia sets.

## Contents

* 1 Definitions
* 2 Number of periodic points
* 3 Stability of periodic points (orbit) - multiplier
* 4 Period-1 points (fixed points)
* 4.1 Finite fixed points
* 4.1.1 Complex dynamics
* 4.1.2 Special cases
* 4.1.3 Only one fixed point
* 4.2 Infinite fixed point
* 5 Period-2 cycles
* 5.1 First method of factorization
* 5.2 Second method of factorization
* 5.2.1 Special cases
* 6 Cycles for period greater than 2
* 7 References
* 8 Further reading
* 9 External links

## Definitions

Let

fc(z)=z2+c

be the complex quadratic mapping, where z and c are complex numbers.

Notationally, fc(k)(z) is the k-fold composition of fc with itself (not to be confused with the kth derivative of fc)—that is, the value after the k-th iteration of the function fc. Thus

fc(k)(z)=fc(fc(k−1)(z)).

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

fc(p)(z)=z,

where p is the smallest positive integer for which the equation holds at that z.

We can introduce a new function:

Fp(z,f)=fc(p)(z)−z,

so periodic points are zeros of function Fp(z,f): points z satisfying

Fp(z,f)=0,

which is a polynomial of degree 2p.

## Number of periodic points

The degree of the polynomial Fp(z,f) describing periodic points is d=2p so it has exactly d=2p complex roots (= periodic points), counted with multiplicity.

## 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) m(fp,z0)=λ of a rational map f iterated p times at cyclic point z0 is defined as:

m(fp,z0)=λ={fp′(z0),if z0≠∞1fp′(z0),if z0=∞

where fp′(z0) is the first derivative of fp with respect to z at z0.

Because the multiplier is the same at all periodic points on a given orbit, it is called a multiplier of the periodic orbit.

The multiplier is:

* a complex number;
* invariant under conjugation of any rational map at its fixed point;[1]
* used to check stability of periodic (also fixed) points with stability index abs(λ).

A periodic point is[2]

* attracting when abs(λ)<1;
* super-attracting when abs(λ)=0;
* attracting but not super-attracting when 0<abs(λ)<1;
* indifferent when abs(λ)=1;
* rationally indifferent or parabolic if λ is a root of unity;
* irrationally indifferent if abs(λ)=1 but multiplier is not a root of unity;
* repelling when abs(λ)>1.

Periodic points

* that are attracting are always in the Fatou set;
* that are repelling are in the Julia set;
* that are indifferent fixed points may be in one or the other.[3] A parabolic periodic point is in the Julia set.

## Period-1 points (fixed points)

### Finite fixed points

Let us begin by finding all finite points left unchanged by one application of f. These are the points that satisfy fc(z)=z. That is, we wish to solve

z2+c=z,

which can be rewritten as

z2−z+c=0.

Since this is an ordinary quadratic equation in one unknown, we can apply the standard quadratic solution formula:

α1=1−1−4c2 and α2=1+1−4c2.

So for c∈ℂ∖{1/4} we have two finite fixed points α1 and α2.

Since

α1=12−m and α2=12+m where m=1−4c2,

we have α1+α2=1.

Thus fixed points are symmetrical about z=1/2.

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]

αc=1−1−4c2 with multiplier λαc=1−1−4c

and

βc=1+1−4c2 with multiplier λβc=1+1−4c.

Again we have

αc+βc=1.

Since the derivative with respect to z is

Pc′(z)=ddzPc(z)=2z,

we have

Pc′(αc)+Pc′(βc)=2αc+2βc=2(αc+βc)=2.

This implies that Pc can have at most one attractive fixed point.

These points are distinguished by the facts that:

* βc is:
* the landing point of the external ray for angle=0 for c∈M∖{1/4}
* the most repelling fixed point of the Julia set
* the one on the right (whenever fixed point are not symmetrical around the real axis), it is the extreme right point for connected Julia sets (except for cauliflower).[5]
* αc is:
* the landing point of several rays
* attracting when c is in the main cardioid of the Mandelbrot set, in which case it is in the interior of a filled-in Julia set, and therefore belongs to the Fatou set (strictly to the basin of attraction of finite fixed point)
* parabolic at the root point of the limb of the Mandelbrot set
* repelling for other values of c

#### Special cases

An important case of the quadratic mapping is c=0. In this case, we get α1=0 and α2=1. In this case, 0 is a superattractive fixed point, and 1 belongs to the Julia set.

#### Only one fixed point

We have α1=α2 exactly when 1−4c=0. This equation has one solution, c=1/4, in which case α1=α2=1/2. In fact c=1/4 is the largest positive, purely real value for which a finite attractor exists.

### Infinite fixed point

We can extend the complex plane ℂ to the Riemann sphere (extended complex plane) C^ by adding infinity:

C^=ℂ∪{∞}

and extend fc such that fc(∞)=∞.

Then infinity is:

* superattracting
* a fixed point of fc:[6]fc(∞)=∞=fc−1(∞).

## Period-2 cycles

Bifurcation from period 1 to 2 for complex quadratic map Bifurcation of periodic points from period 1 to 2 for fc(z)=z*z +c

Period-2 cycles are two distinct points β1 and β2 such that fc(β1)=β2 and fc(β2)=β1, and hence

fc(fc(βn))=βn

for n∈{1,2}:

fc(fc(z))=(z2+c)2+c=z4+2cz2+c2+c.

Equating this to z, we obtain

z4+2cz2−z+c2+c=0.

This equation is a polynomial of degree 4, and so has four (possibly non-distinct) solutions. However, we already know two of the solutions. They are α1 and α2, computed above, since if these points are left unchanged by one application of f, then clearly they will be unchanged by more than one application of f.

Our 4th-order polynomial can therefore be factored in 2 ways:

### First method of factorization

(z−α1)(z−α2)(z−β1)(z−β2)=0.

This expands directly as x4−Ax3+Bx2−Cx+D=0 (note the alternating signs), where

D=α1α2β1β2,

C=α1α2β1+α1α2β2+α1β1β2+α2β1β2,

B=α1α2+α1β1+α1β2+α2β1+α2β2+β1β2,

A=α1+α2+β1+β2.

We already have two solutions, and only need the other two. Hence the problem is equivalent to solving a quadratic polynomial. In particular, note that

α1+α2=1−1−4c2+1+1−4c2=1+12=1

and

α1α2=(1−1−4c)(1+1−4c)4=12−(1−4c)24=1−1+4c4=4c4=c.

Adding these to the above, we get D=cβ1β2 and A=1+β1+β2. Matching these against the coefficients from expanding f, we get

D=cβ1β2=c2+c and A=1+β1+β2=0.

From this, we easily get

β1β2=c+1 and β1+β2=−1.

From here, we construct a quadratic equation with A′=1,B=1,C=c+1 and apply the standard solution formula to get

β1=−1−−3−4c2 and β2=−1+−3−4c2.

Closer examination shows that:

fc(β1)=β2 and fc(β2)=β1,

meaning these two points are the two points on a single period-2 cycle.

### Second method of factorization

We can factor the quartic by using polynomial long division to divide out the factors (z−α1) and (z−α2), which account for the two fixed points α1 and α2 (whose values were given earlier and which still remain at the fixed point after two iterations):

(z2+c)2+c−z=(z2+c−z)(z2+z+c+1).

The roots of the first factor are the two fixed points. They are repelling outside the main cardioid.

The second factor has the two roots

−1±−3−4c2.

These two roots, which are the same as those found by the first method, form the period-2 orbit.[7]

#### Special cases

Again, let us look at c=0. Then

β1=−1−i32 and β2=−1+i32,

both of which are complex numbers. We have |β1|=|β2|=1. Thus, both these points are "hiding" in the Julia set. Another special case is c=−1, which gives β1=0 and β2=−1. This gives the well-known superattractive cycle found in the largest period-2 lobe of the quadratic Mandelbrot set.

## Cycles for period greater than 2

Periodic points of f(z) = z*z−0.75 for period =6 as intersections of 2 implicit curves

The degree of the equation f(n)(z)=z is 2n; thus for example, to find the points on a 3-cycle we would need to solve an equation of degree 8. After factoring out the factors giving the two fixed points, we would have a sixth degree equation.

There is no general solution in radicals to polynomial equations of degree five or higher, so the points on a cycle of period greater than 2 must in general be computed using numerical methods. However, in the specific case of period 4 the cyclical points have lengthy expressions in radicals.[8]

In the case c = –2, trigonometric solutions exist for the periodic points of all periods. The case zn+1=zn2−2 is equivalent to the logistic map case r = 4: xn+1=4xn(1−xn). Here the equivalence is given by z=2−4x. One of the k-cycles of the logistic variable x (all of which cycles are repelling) is

sin2(2π2k−1),sin2(2⋅2π2k−1),sin2(22⋅2π2k−1),sin2(23⋅2π2k−1),…,sin2(2k−12π2k−1).

## References

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". http://www.ijon.de/mathe/julia/some_julia_sets_1_en.html.
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
8. ↑ Gvozden Rukavina : Quadratic recurrence equations - exact explicit solution of period four fixed points functions in bifurcation diagram

## Further reading

* Geometrical properties of polynomial roots
* 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

## External links

| 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.

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