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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.
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).
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
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 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.
Period-2 cycles are two distinct points and such that and .
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 factorization
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
Adding these to the above, we get and . Matching these against the coefficients from expanding , we get
From this, we easily get
From here, we construct a quadratic equation with and apply the standard solution formula to get
Closer examination shows that :
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 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.
Again, let us look at . Then
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
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.
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
^Alan F. Beardon, Iteration of Rational Functions, Springer 1991, ISBN0-387-95151-2, p. 41
^Alan F. Beardon, Iteration of Rational Functions, Springer 1991, ISBN0-387-95151-2, page 99