|San José State University|
& Tornado Alley
The Mandelbrot set is surely one of the most visually fascinating structures in
mathematics. The Mandelbrot set is the set of complex numbers c such
that the iteration scheme
zn+1 = zn² + c
is bounded when starting from the point z0=0. A significant subset of the Mandelbrot 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 z0=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 z0=0 will approach them.
Consider the deviations of the iteration values from the corresponding limit point; i.e.,
For values of zn close to z* this reduces to 2|z*|<1. The boundary between the stable and unstable limit points is given by |z*|=1/2. Such limit points are given by the equation
The question is what are the values of c which give those limit points. Those values of c are simply
This equation is a parametric equation for the set of c values. It shows how the points on the circle of radius 1/2 in the z* space map into the c space. For example, z*=½ maps into c=¼, z*=−½ maps into c=−3/4, z*=i½ maps into c=½+i¼ and z*=−i½ maps into c=−½+i¼.
The plot below shows the full set of c values.
And there is the familiar cardioid shape that bounds the main body of the mandelbrot set, as seen below.
The iteration may approach a limit cycle rather than a limit point. For a two-period cycle of z1* and z2* the values would have to satisfy the equations
From these equations defining z1* and z2* it follows that
The deviations from the limit values satisfy the equations
Therefore if |zn+2-z2*| is to be less than |zn-z2*| 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.,
It was previously determined that z2* = −(1 + z1*), therefore the condition for the boundary of stability reduces to:
The allowable values of z1 are solutions to the quadratic equation
There are two solutions to this quadratic equation; one corresponding to z1* and the other to z2*.
It is not the solutions for z1* or z2* that are of most interest; it is the set of values of c corresponding to these solutions.
The defining equation
It is known that
This is the locus of a circle of radius ¼, the center of which is shifted one unit to the left on the real axis. This is shown below in a diagram with the values of c corresponding to limit points.
The analysis for 3-cycles is more difficult but by same procedures as in the previous it can be established that the boundaries between the stable and unstable cycles are circles of radius 1/8. In general the boundaries between stable and unstable m-cycles are circles of radius 1/2m. The diagram below shows the placement of circles having radii of 1/8, 1/16 and 1/32.
There are also easily discernible circles in the Mandelbrot set of radius 1/64.
The limit points z1*, z2* and z3* satisfy the equations
The substitution of the equation for z1* into the equation for z2*² and the substitution of that equation into the equation for z3* produces a polynomial equation of degree 8.
There will be eight solutions to this equation and two pairs of triples which satisfy the conditions for limiting 3-cycles for each given value of c. The product of the eight solutions will be equal to the constant term divided by the coefficient of the highest power of z*. This means that the product of the solutions is equal to (c² + c)² + c.
In order to establish the conditions for stability of the 3-cycles consider the deviations between zn+3 and z3*, zn+2 and z2*, zn+1 and z1*, and zn and z3*. These deviations satisfy the equations
Successive substitutions establish that if |zn+3 − z3*| is to be less than |zn − z3*| it is necessary that
For values very close to the limiting 3-cycles this reduces to
and for the boundary between the stable and unstable 3-cycles
From the defining equations for the limit points it follows that
(To be continued.)
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