Periodic points of complex quadratic mappings

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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.

Definitions[edit]

Let

be the complex quadric mapping, where and are complex-valued.

Notationally, is the -fold composition of with itself—that is, the value after the k-th iteration of 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

Stability of periodic points (orbit) - multiplier[edit]

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 p 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 points[edit]

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

then .

Thus fixed points are symmetrical around

This image shows fixed points (both repelling)

Complex dynamics[edit]

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

Using Viète's formulas one can show that:

Since the derivative with respect to z is

then

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

These points are distinguished by the facts that:

  • is :
    • the landing point of external ray for angle=0 for
    • the most repelling fixed point, belongs to 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 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

Special cases[edit]

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[edit]

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 point[edit]

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

and extending polynomial such that

Then infinity is :

  • superattracting
  • a fixed point of polynomial [6]

Period-2 cycles[edit]

Bifurcation from period 1 to 2 for complex quadratic map

Period-2 cycles are two distinct points and such that and .

We write

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 2 applications (or more).

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

First method of factorization[edit]

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 factorization[edit]

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 cases[edit]

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 2[edit]

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

References[edit]

Further reading[edit]

External links[edit]