CHAPTER 14(a)   Complex Dynamics

14.1 Complex Analytic Dynamics

Much of the current interest in fractals is a consequence of the recent work of Benoit Mandelbrot, whose computer simulations of maps of the complex plane have resulted in extremely complicated and beautiful fractals.  We describe how fractals arise from the study of complex analytic maps.

14.2 Quadratic Maps

Complex analytic dynamics is a branch of dynamical systems where one is mainly concerned with the behavior of functions  p : C ⟶ C  under iteration (here C = complex plane).  Actually, it is more convenient to consider  p  defined on the Riemann sphere, i.e. C together with the point at infinity,  (SuperStar[C])=C ⋃ {∞}.  We can treat  ∞ in a similar manner to the way we deal with other points in the complex plane.

Suppose  p : (SuperStar[C]) ⟶ (SuperStar[C])  is a quadratic polynomial (we always assume that p(∞)=∞). It turns out that from a dynamical point of view it suffices to consider quadratic maps of the form
                                                    p(z) = (z^2) + c,
where  c  is a constant. This is because every quadratic polynomial is conjugate to a map of this form.

Example:

Let  p(z) = (z^2),  then for  |z|<1,  (p^n)(z) ⟶ 0,  as  n ⟶ ∞,  and for  |z|>1,  (p^n)(z) ⟶ ∞  as n ⟶ ∞. (p(0) = 0, p(∞) = ∞). We say that 0 and  ∞  are attracting fixed points for  p, whilst  p(1) = 1  is a repelling fixed point, i.e., |p '(1)|>1).

When  |z| = 1  (i.e. on the unit circle  (S^1)), the dynamics is more complicated (the map is chaotic). To say that p|(S^1)  is chaotic means that the following three properties hold.

(i)  sensitive to changes in initial conditions,

(ii) topologically transitive (i.e. there is a point (Subscript[z, 0])∈(S^1), which has a dense orbit under  p),

(iii) periodic points  (i.e. points with  (p^n)(z) = z,  (for some natural number  n),  are dense in  K.

(S^1)  is called the Julia set for  p(z) = (z^2). This is the set where all the interesting dynamics takes place. In general, for a polynomial  p : (SuperStar[C]) ⟶ (SuperStar[C]), the Julia set  J(p)  is defined to be the closure of the repelling periodic points of  p  ((Subscript[z, 0])  is a repelling periodic point of  p  if  (p^n)((Subscript[z, 0])) = (Subscript[z, 0]), for some natural number  n  and  |((p^n))'((Subscript[z, 0]))|>1).

Although the map  p(z)=(z^2) exhibits highly chaotic behavior on its Julia set  (S^1), the study of  p may lead one to believe the Julia set is generally a nice smooth set. Nothing could be farther from the truth, as a variation of the parameter c gives rise to quadratic maps whose Julia sets are fractals. In fact D. Ruelle has shown that for quadratic maps  (Subscript[p, c])(z)=(z^2)+c  with c  small, the fractal dimension is approximately

(Subscript[d, c])=1+|c((|)^2)/4ln(2)

and so is indeed a fractal in these cases. The question of what value the fractal dimension can take for other values of c is still an open and difficult question. In 1980, Misuirewicz solved an outstanding open question of Fatou by showing that for the exponential function f(z)=(e^z),  J(f)=C. This result implies that (e^z) is topologically transitive on C. 
    Activate the following code to sketch the Julia sets of different quadratic maps. Experiment with the following values for c before trying some randomly chosen values of your own.
(i)  c = -0.12256117 + 0.74486177I
(ii) c =  0.32 + 0.043I
(iii)c =  0.3950541 - 0.586788I

Coordinates={Complex[x_,y_]->{x,y}};

JuliaSet[c_,z_,Npoints_]:=Module[{Nc=N[c],Nz=N[z]},
NestList[(1-2 Random[Integer]) Sqrt[#-Nc] &,Nz, Npoints]/. Coordinates];

JuliaData=Drop[Evaluate[JuliaSet[-1+0 I,0+5 I, 2^18]],100];

ListPlot[JuliaData, Axes->False, AspectRatio->Automatic,PlotStyle->PointSize[0.001]]

Exercises 14.2

1. Find a 3-cycle for the map f(z)=(z^3)-(1/2+i/2)((z^2)+z)+i, and classify it as attracting, repelling or neutral.

14.3 Julia Sets.

In fact, Mathematica has a built in command for graphing Julia sets: JuliaSetPlot[f[z],z]:

f[z_]:=z^2+.3-.3I

JuliaSetPlot[f[z],z]

g[z_]:=z^2+-1+.1I

JuliaSetPlot[g[z],z]

h[z_]:=z^2+-.4-.3I

JuliaSetPlot[h[z],z]

k[z_]:=z^2+-.1+.2I

JuliaSetPlot[k[z],z]

l[z_]:=z^2+1/4

JuliaSetPlot[l[z],z]

Exercises 14.3

1. Use the Mathematica command: JuliaSetPlot[f[z],z], to plot Julia sets for f(z)=(z^3), f(z)=(z^3)+i, f(z)=(z^5)-z+i and f(z)=(z^3)+1/2.

14.4 The Mandelbrot Set.

Quadratic Julia sets and the Mandelbrot set arise simply from the sequence of complex numbers defined inductively by  
(Subscript[z, n+1])=((Subscript[z, n])^2)+c, where c is a constant.

    Julia sets are defined by fixing  c  and letting (Subscript[z, 0]) vary over  C, whilst the Mandelbrot set is obtained from fixing  (Subscript[z, 0])=0 and letting the parameter c vary.

Let c be fixed, then if (Subscript[z, 0]) (or (Subscript[z, n])) has large absolute value (for some n) then 
(Subscript[z, n])->∞ as n->∞. But there are some starting values of (Subscript[z, 0]) for which (Subscript[z, n]) remains bounded in absolute value for all n. The set of all such (Subscript[z, 0]) is called the filled in Julia set, K(c) of p(z)=(z^2)+c. (The Julia set being the boundary of K(c)).

By varying  c  the types of Julia sets arising vary considerably, some being in one piece (connected), others consisting of many disjoint (disconnected) sets, these are referred to as fractal dust.

The Mandelbrot set is defined to be the set of all c∈C for which  K(c)  is a connected set.

Although the Mandelbrot set does not have the same type of linear self similarity as the Koch snowflake, its boundary is a fractal whose fractal dimension has been shown to be equal to 2 (Shishikura, 1994). It is a set with many incredible properties. It is a connected set (Douady and Hubbard, 1982), which contains an infinite number of small copies of itself. If c lies in the interior of the main body of M, the corresponding Julia set J(c) is a fractally deformed circle surrounding one attractive fixed point. If  c  lies in the interior of one of the buds, the Julia set consists of infinitely many fractally deformed circles connected to each other, each surrounding a periodic attractor. Other possibilities arise by taking  c  on the boundary.  In particular, as  c  passes through the boundary to the outside of the Mandelbrot set there is a most dramatic change in the corresponding Julia sets. They decompose into a cloud of infinitely many points (the fractal dust). 
     Here are two different modules for generating the Mandelbrot set:

DensityPlot[Length[FixedPointList[#^2+(x+I*y)&,x+I*y,20,SameTest->(Abs[#2]>2.0&)]],{x,-2.0,1.0},{y,-1.0,1.0},PlotPoints->175,Mesh->False,Frame->False,AspectRatio->Automatic,ColorFunction->Hue]

xmin=-2.2;xmax=0.6;ymin=-1.2;ymax=1.2;

Itermax=50;stepsize=0.005;

Counter=Compile[{x,y,{Itermax,_Integer}},Module[{w,z,count=0},w=x+I y;z=w;
While[(Abs[z]<4.0) && (count≤Itermax),++count;
z=z^2+w;];Return[count]]];

MandelbrotData=Table[Counter[x,y,Itermax],{y,ymin,ymax,stepsize},{x,xmin,xmax,stepsize}];

Rasterize[ListDensityPlot[MandelbrotData,Frame->False,Mesh->False,MeshRange->{{xmin,xmax},{ymin,ymax}},AspectRatio->Automatic,ColorFunction->(Hue[Log[1+#]]&)],ImageResolution->300]

14.5 The Main Bulb of the Mandelbrot Set.

The main bulb of the Mandelbrot set is determined by the parametric equations  x(t)=1/2cos(t)-1/4cos(2t); y(t)=1/2sin(t)-1/4sin(2t). The secondary bulb is given by x(t)=-1+1/4cos(t), y(t)=1/4sin(t).

Clear[x,y]
x[t_] := 1/2Cos[t]-1/4Cos[2t]; y[t_]:=1/2Sin[t]-1/4Sin[2t]

Clear[a,b]
a[t_] := -1+1/4Cos[t]; b[t_]:=1/4Sin[t]

ParametricPlot[{{a[t],b[t]},{x[t],y[t]}}, {t,0,2Pi},AspectRatio -> Automatic]

Exercises 14.5

1. Find the Cartesian equation of the secondary bulb of the Mandelbrot set, and show that it is a circle.