In these notes I will use mathematical notation similar to the syntax of Mathematica.

13. Fractals associated with Newton's method.

We have discussed Newton's method before. It is a mthod for finding the roots of polynomials or other functions. Given a function f[x] with a root near some x0, we can get a better approximation to the root by replacing the estimate x0 by x1 = x0 - f[x0]/f'[x0]. By repeating this procedure a few times we can usually get a a sufficiently precise estimate of the root in question, but not always.

Given a function f[x], define its associated Newton function to be

g[x_] := x - f[x]/f'[x]

If f[x] is a polynomial, then g[x_] is a rational function, i.e. a fraction in which the numerator and denominator are polynomials. It is not defined when f'[x] = 0, i.e. at critical points of f. Applying Newton's method for the function f[x] is the same thing as iterating the function Nf[x].

If x is a root of f, then g[x] = x, i.e., x is a fixed point of g. The derivative of g is

g'[x] = f[x]*f''[x]/(f'[x])^2.

It follows that if a root x is not a critical point (meaning that f'[x] is not zero) then it is a superstable fixed point of g. It also follows that if x is a point of inflection of f (meaning that f''[x]=0) and not a critical point of f, then it is a critical point of g.

The dynamics of the function g[x] are easy to analyze when f is the quadratic function

f[x_] := (x - r1)(x - r2).

Then g[x] has superstable fixed points at the roots x = r1 and x = r2. The orbit of any point that is not equidistant from from the two roots will converge to the nearer one.

Newton fractals for cubic polynomials.

The situation can be much more complicated when f is a cubic. Suppose we have f[x_] := x^3 -1, so the roots are 1, (-1 + I*Sqrt[3])/2, and (-1 -I*Sqrt[3])/2. The following picture, hioch was produced by Fractint using newtbasin, colors each pixel according to which of the three roots its orbit converges to. Each colored region is called a basin of attraction for the root in question. Note that wherever two colors meet, the third color can be found in between them.

Figure 13.1. Newton basins of attraction for the function f[z] = z^3 - 1 with magninfication .666 and .05.

• Explain why almost all points on the real axis are colored blue, meaning that their orbits converge to 1. Find the real values in the picture for which this does not happen.
• Explain why this picture shows large scale self-similarity with a scaling factor of 1.5.
• Fractint allows one to make similar pictures for f[x_] := x^d-1 for other exponents d. What is the corresponding scaling factor?

Here is another example. Suppose we have f[x_] := x^3 - x, so the roots are 0, 1 and -1. Then the Newton function is

g[x_] := 2x^3/(3x^2 - 1)

We see that g[1/2] = -1, even though -1 is the furthest root from 1/2, and g[-0.45541] = .5, so g[g[-0.45541]] = -1 even though the nearest root is 0. Thus the orbit of x need not converge to the nearest root.

What is worse, the orbit might not converge to any root. For example

g[Sqrt[1/5]] = - Sqrt[1/5]    and

g[-Sqrt[1/5]] = Sqrt[1/5],

so the orbit of Sqrt[1/5] as period 2. Hence this point is fixed by the iterate g2[x_] := g[g[x]]. Is it an attracting fixed point?

What can we say about the Julia set of g? There are two things about it that make the question quite different from that of the Julia set associated with polynomial function.

1. The function g[x] is undefined when 3x^2 - 1 = 0, i.e., when x = 1/Sqrt[3] or x = -1/Sqrt[3]. This means that if any iterate of g sends x to one of these two numbers, then the orbit of x is undefined.
2. Since Limit[g[x]/x, x -> Infinity] = 2/3, if x is large, then g[x] is approximately 2x/3, which is smaller than x. Thus there are no unbounded orbits.

Thus the Julia set as defined previously consists of all points except those with undefined orbits.

A more interesting question is which points have orbits that do not converge to a root of f, i.e. for which points does Newton's method fail to produce a root? For this we use the following Fractint code, taken from dcr95.frm.

CubicNewtonJulia2{; Douglas C. Ravenel January 26, 1995

; Use floating point option for this program.

; This is similar to the Newton-Julia set shown in the

; Mandelbrot-Lorenz video

z = (10^p2)*pixel, c = p1:

fz = z*z*z + (c-1)*z -c,

f1z= 3*z*z + c - 1,

z = z - fz/f1z,

|fz| > .0000000001 }

Here we are iterating the Newton function for f[z_] := z^3 + (c-1)z - c, and the iteration stops when f[z] gets sufficiently small, i.e., when the orbit gets close to a root. The parameter p2 is included to allow arbitrary zoom-outs, which are not possible otherwise.

Figure 13.2. CubicNewtonJulia2 for c = 0, with 10-fold magnification near z = 1/Sqrt[5].

• Does this image have large scale self-similarity, and if so why and with what scaling factor?
• Is there small scale self-similarity near z = 1/Sqrt[5] or other such points?

By varying c, we can get an analog of the Mandelbrot set. The Newton function for z^3 + (c-1)z - c is

g[z_] := (c + 2*z^3)/(-1 + c + 3*z^2).

When c is not 1, g has a critical point at z =0 in addition to ones at the roots of f. The orbits of the roots are obvious; they are fixed by g. The other critical orbit, the one for z = 0, may or may not converge to a root. The following Fractint code (in dcr95.frm) tests this.

CubicNewton2{; Douglas C. Ravenel January 26, 1995

; Use floating point option for this program.

; This is similar to the Newton-Mandelbrot set shown in the

; Mandelbrot-Lorenz video

c = (10^p1)*pixel, z = 0:

fz = z*z*z + (c-1)*z - c,

f1z= 3*z*z + c - 1,

z = z - fz/f1z,

|fz| > .0000000001 }

As before we iterate the Newton function, this time on the critical point z = 0, until we get sufficiently close to a root of f. The magnification parameter is p1.

Figure 13.3. The Newton-Mandelbrot set for f[z] = z^3 + (c-1)z - c with detail near c = .15 + 1.7*I.

The surprising thing about this image is the appearance of copies of the classical Mandelbrot set within it. This was first reported by Curry-Garnett-Sullivan in 1983. Here are some possible topics for a group project.

• Find some miniature Mandelbrot sets on the real axis in this picture. Plot the cubic curve for the corresponding value of c and see what happens with Newton's method for z = 0.
• Find the periods associated with some miniature Mandelbrot sets in this picture.
• Try to find an equation for the main cardioid on such a miniature Mandelbrot set.
• Study the Newton-Julia sets corresponding to some points in some of these miniature Mandelbrot sets. see if there might be an analog of Lei's theorem (discussed in Lecture 8) for them.

Newton fractals for some quartics.

Suppose we have a quartic of the form f[z_] := z^4 + 6*a^2*z^2 + b. Then f''[z] = 12(z^2 - a^2), so there are points of inflection at z = a, and z = -a. Hence these are the two orbits that have to be tested to construct the Newton-Mandelbrot set. However, since f is an even function, these two orbits behave in the same way and only one of them has to tested. This is convnenient because Fractint seems to have trouble when testing two orbits. As in the case of cubic polynomials, this family of functions is apramtrized by two complex variable a and b, so it leads to an unviewable 4-dimensional fractal. We can only look at 2-dimensional slices of it. Here are some examples in dcr95.frm.

• QuarticNewton examines the family f[z_] := (z^2 - 1)(z^2 + c) for varying c. The image is concentrated near the positive real axis. It appears to be a mixtures of miniature Ms and diamond shaped regions.

Figure 13.4. QuarticNewton zoomed out by a factor of 10 and magnified 30 times at c = 7.96.

• Quartic2Newton examines the family f[z_] := f(z) = (z^2 - c)*(z^2 - 6 + c), for which f''(z) = 12*(z^2 - 1), for varying c. This image display large scale self-similarity and 6-fold rotational symmetry centered at c = 3. The only solid regions appear to be minature Ms.

Figure 13.5. QuarticNewton zoomed out by a factor of 20 with unmagnified detail at c = -1.25.

Now here are some study questions.

• Why are these two images so different?
• Do the diamond shaped regions in Figure 13.5 have periodic orbits associated with them?
• Find the periods of the largest miniature Ms in both pictures.
• Look for preperiodic points in both pictures.
• Look at Julia sets (using QuarticNewtonJulia and Quartic2NewtJulia) associated with such points.
• Look for a main body equation in both pictures.
• Look for other interesting 2-dimensional slices of this 4-dimensional Newton-Mandelbrot set.