F(z) = k cos(z) has a super-attracting fixed point at z = (-pi, 0) when k = pi = 3.14159....

If we draw the Julia set of k cos(z) for k = pi, we see nothing on the screen since all points are attracted to that fixed point. As we decrease the parameter k, the fixed point slowly increases and the derivative at the fixed point slowly increases un til we reach the value k approximately 2.97165.... where there is a saddle-node bifurcation and the Julia Set explodes. Below is a picture of what happens to the Julia Set as we vary the parameter k from 2.8006 down to 2.7992 (in increments of 0.0002) and then back to 2.8006. (If we start with a small value of the parameter k, for example at k = 1 there is an attracting fixed point at z = (0.7385...,0) and all points are attracted to that point, and increase the parameter, we find that there is a period-d oubling bifurcation at a = 1.319156...)

"The Disappearance of Julia"

>

An Indifferent fixed point

This picture shows the Julia Set of f(z) = z + z^5 which has an indifferent fixed point at z = 0.

( f(0) = 0 and f ' (0) = 1 .)

The 4 lines Re z = 0 and Im z = 0 and Re z = Im z and Re z = -Im z are invariant under iteration of f.

1. On Im z = 0: f(x) = x + x^5
2. On Re z = 0: f(ix) = ix + (ix)^5 = ix + i^5 x^5 = i(x+x^5)
3. On Re z = Im z , f(z) = r e^(i pi/4) + r^5 e^(i 5 pi/4) = e^ (i pi/4)(r - r^5)
4. On Re z = -Im z, f(z) = = r e^(i 3pi/4) + r^5 e^(i 15 pi/4) = e^ (i 3pi/4)(r - r^5)

Using one dimensional analysis it is easily shown that f(x) = x + x^5 has a repelling fixed point at x = 0 and f(x) = x - x^5 has an attracting fixed point at x = 0. Thus along the four invariant lines 0 is attracting on the first two and repelling on the second two. The points repelled from 0 are shown in shades of blue, while those attracted to 0 are shown in shades of brown. 0 has four attracting petals, which are in shades of brown. (A simply connected region C is a petal for an indifferent fixed point p if p is contained in the boundary of C and for each z in C,

F^n(z) -> p (see Devaney - 1987)

Spirals - An Iterated Function System

An iterated function system has as one of its affine transformations a rotation around the origin of pi/12 radians. This produces the "wheel" with 12 spokes. In each of 8 consecutive frames we increase the angle of rotation in equal increments until we reach the angle pi/11 which gives us the "wheel" with 11 spokes.

The Julia Set for Newton's Method Applied to finding the roots of z^7 -1 = 0

You can easily identify the seven roots of unity and their intertwined basins of attraction.

---

The Julia Set of f(z) = k sinh(z)

---

To get in touch: aburns@liu.edu

Back to Anne's home page