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.

DefinitionsEdit

Let

 

be the complex quadric mapping, where   and   are complex numbers.

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

 

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

 

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

We can introduce a new function:

 

so periodic points are zeros of function  : points z satisfying

 

which is a polynomial of degree  

Number of periodic pointsEdit

The degree of the polynomial   describing periodic points is   so it has exactly   complex roots (= periodic points), counted with multiplicity.

Stability of periodic points (orbit) - multiplierEdit

 
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 a rational map   iterated   times at cyclic point   is defined as:

 

where   is the first derivative of   with respect to   at  .

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  

A periodic point is[2]

  • attracting when  
    • super-attracting when  
    • attracting but not super-attracting when  
  • indifferent when  
    • rationally indifferent or parabolic if   is a root of unity;
    • irrationally indifferent if   but multiplier is not a root of unity;
  • repelling when  

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)Edit

Finite fixed pointsEdit

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

 

which can be rewritten as

 

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

  and  

So for   we have two finite fixed points   and  .

Since

  and   where  

we have  .

Thus fixed points are symmetrical about  .

 
This image shows fixed points (both repelling)

Complex dynamicsEdit

 
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  

and

  with multiplier  

Again we have

 

Since the derivative with respect to z is

 

we have

 

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

These points are distinguished by the facts that:

  •   is:
    • the landing point of the external ray for angle=0 for  
    • 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]
  •   is:
    • the landing point of several rays
    • attracting when   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  

Special casesEdit

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 pointEdit

We have   exactly when   This equation has one solution,   in which case  . In fact   is the largest positive, purely real value for which a finite attractor exists.

Infinite fixed pointEdit

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

 

and extend   such that  

Then infinity is:

  • superattracting
  • a fixed point of  :[6]
     

Period-2 cyclesEdit

 
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   and   such that   and  , and hence

 

for  :

 

Equating this to z, we obtain

 

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   and  , computed above, since if these points are left unchanged by one application of  , then clearly they will be unchanged by more than one application of  .

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

First method of factorizationEdit

 

This expands directly as   (note the alternating signs), where

 
 
 
 

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

 

and

 

Adding these to the above, we get   and  . Matching these against the coefficients from expanding  , we get

  and  

From this, we easily get

  and  .

From here, we construct a quadratic equation with   and apply the standard solution formula to get

  and  

Closer examination shows that:

  and  

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

Second method of factorizationEdit

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

 

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

 

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

Special casesEdit

Again, let us look at  . Then

  and  

both of which are complex numbers. We have  . 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 greater than 2Edit

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

The degree of the equation   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   is equivalent to the logistic map case r = 4:   Here the equivalence is given by   One of the k-cycles of the logistic variable x (all of which cycles are repelling) is

 

ReferencesEdit

  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. Archived 2011-07-17 at the Wayback Machine
  5. ^ Periodic attractor by Evgeny Demidov Archived 2008-05-11 at the Wayback Machine
  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 Archived 2008-05-11 at the Wayback Machine
  8. ^ Gvozden Rukavina : Quadratic recurrence equations - exact explicit solution of period four fixed points functions in bifurcation diagram

Further readingEdit

External linksEdit