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The Mandelbrot sets are the sets of complex numbers c such that the iteration scheme
is bounded when starting from the point z_{0}=0. For example the above image is for m=4. A significant subset of this set consists of those values of c such that the iteration scheme approaches limits for which
Such a limit point z* satisfies the equation
For any c there is a limit point z*; i.e., such that if z_{0}=z* the iteration will remain at z* forever.
The crucial question is what are the limit points that are stable so that the iteration starting from z_{0}=0 will approach them.
Consider the deviations of the iteration values from the corresponding limit point; i.e.,
Thus
For values of z_{n} close to z* this reduces to mz*^{m1}<1. The boundary between the stable and unstable limit points is given by
Such limit points are given by the equation
It is noteworthy that the stability of points is independent of the value of the iteration parameter c. The values of c which give those limit points are simply
This equation is a parametric equation for the set of c values. It shows how the points on the circle of radius (1/m)^{1/(m1)} in the z* space map into the c space.
The plots below shows the full set of c values for m=2, 3, 4, 5, 6, 7 and 8.
The main structure of the Mandelbrot set for m has (m1) lobes. It has the symmetry of the (m1) cyclic group; i.e., a rotation through an angle of 2π/(m1) leaves the figure unchanged. It also has a symmetry with respect to the real axis. Thus it has the symmetry of the (m1) dihedral group. The complete Mandelbrot sets have this same symmetry.
The iteration may approach a limit cycle rather than a limit point. For a twoperiod cycle of z_{1}* and z_{2}* the values would have to satisfy the equations
From these equations defining z_{1}* and z_{2}* it follows that
This is a preliminary result which will be used later. For convenience let
Thus the above result can be expressed as
The deviations from the twocycle satisfy the equations
Therefore if z_{n+2}z_{2}* is to be less than z_{n}z_{2}* it must be that
For values very close to a cycle pair this reduces to:
For the boundary of the stable set equality prevails; i.e.,
This means that z_{2}*z_{1}* is on a circle in the complex plane of radius (1/m²)^{1/(m1)} . This means that
Now consider the equation which determines the values for a twocycle. It was previously established that
Multiplying this equation by z_{1}*^{m1} and grouping together terms of the form (z_{1}*z_{2}*)^{k} gives
where re^{iφ}=z_{1}*z_{2}*, r being equal to (1/m²)^{1/(m1)}. This is the polynomial equation whose solutions give z_{1}. Because of symmetry of the previous equation with respect to z_{1} and z_{2} the equation for z_{2} is exactly the same. The order of the polynomial is 2(m1) so the solutions are grouped into (m1) pairs.
There are roots of a polynomial which are of multiplicity two. These correspond to the case in which an infinitesimal change in coefficients of the polynomial would result in two nearby roots rather than one root. The condition for a root of multiplicity two is it makes both the polynomial and its derivative equal to zero.
Before examining the case of the boundary between sequences converging to a fixed point (if effect onecycles) and sequences converging to a two cycle, it worthwhile to look at the case of two of the fixed point solutions converging. This not the same as the elements for a twocycle converging into a onecycle.
A complex number z' is of multiplicity two if
Therefore, from the second equation
In order for these values to be roots of the polynomial it is necessary that
These values of c are just discrete points on the boundary curve of the main body of the Mandelbrot set. They are separated by an angle of 2π/(m1). The diagram below shows these points in green for the case of m=4.
Now let us consider the bifurcation points. Since the condition for stability is
if z_{2}* → z_{1}* then the condition to be satisfied by z_{1}* is
The condition on z_{1}*² means that
The plot of the values of c corresponding to these values of z_{1}* are shown below.
The points shown in blue are the ones which correspond to the bifurcation points. Those in green at the cusps of the curve correspond to multiplicitytwo fixed point solutions.
It is shown elsewhere that the entire Mandelbrot set for exponent m has this type of symmetry for rotations through an angle of 2π/(m1).
If w is defined as z_{1}/(re^{iφ})^{1/2} the polynomial can be put into the form
In principle for a given value of m the polynomial can be solved for each value of φ and the value of c determined from
where z_{2}* = re^{iφ}/z_{1}*.
Multiplying the above equation through by (w²1) introduces an additional two roots of w=±1 but puts the polynomial into a more amenable form of
This above form can be used for obtaining the numerical solutions. There are (m1) different components. These can be obtained by starting with the (m1) bifurcation points; i.e.,
where θ_{0} = 2π/(m1) and r=(1/m)^{1/(m1)}.
These values for w are entered into the polynomial and the values of φ to which they correspond are determined, say {φ_{k}}. The polynomial is then solved for values of φ in the range [φ_{k},φ_{k}+2π]. The solutions for w are converted into values for z*_{1} and then into values for c.
(To be continued.)
For more on the structure of other mandelbrot sets Click here.
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