CHAPTER 2(a), Periodic Points

2.1 Periodic Points of Quadratic Maps and the Logistic Maps.

We now examine the iterations of one-dimensional maps, with emphasis on the logistic maps and their points of period greater than one. Recall that these are maps of the form: (Subscript[L, r])(x)=r x(1-x), 0≤x≤ 1,  where r is a parameter, 0≤ r≤ 4. It is convenient to show the iterations of such maps using the webPlot module used in Chapter 1. We start with the parameter  r= 3.45, graphing the second iterate of (Subscript[L, r])(x). We try to see if (Subscript[L, r])(x) has any period two points, i.e., points (SuperStar[x]) with the property that ((Subscript[L, r])^2)((SuperStar[x]))=(SuperStar[x]), but don’t satisfy (Subscript[L, r])((SuperStar[x]))=(SuperStar[x]).  The easiest way to find such points is to graph ((Subscript[L, r])^2)(x), and see where it intersects the line y=x.

f[x_]:=3.45x(1-x)

Plot[{f[f[x]],x},{x,0,1},AspectRatio->Automatic]

We see there are two period two points, what we call a 2-cycle (and also two fixed points).

webPlot[g_,{x_,xmin_,xmax_},{a1_,n_}, opts_]:=Module[{seq, r, pts, web, graph}, r[t_]:=N[g/.x->t];seq=NestList[r,a1,n]; pts=Flatten[Table[{{seq[[i]],seq[[i+1]]},{seq[[i+1]],seq[[i+1]]}}, {i,1,n}],1]; web=Graphics[{Hue[0], Line[PrependTo[pts, {seq[[1]], 0}]]}];graph=Plot[{x,r[x]}, {x, xmin, xmax}, DisplayFunction->Identity];Show[web, graph, opts, Frame->True, PlotRange->{{xmin, xmax}, {xmin, xmax}}]];

Choosing a point at random, we now see that its orbit approaches the 2-cycle arbitrarily closely.

webPlot[f[x], {x,0,1.2}, {.2, 16}, AspectRatio->Automatic]

Now we increase the value of r (in (Subscript[L, r])(x)=r x(1-x)), to see if we can find a 3-cycle.

f[x_]:=3.8x(1-x)

Plot[{f[x],x},{x,0,1},AspectRatio->Automatic]

Plot[{f[f[f[x]]],x},{x,0,1},AspectRatio->Automatic]

We don't quite have a period three point, but if we increase r a little more, we obtain two 3-cycles:

f[x_]:=3.84x(1-x)

Plot[{f[f[f[x]]],x},{x,0,1},AspectRatio->Automatic]

In fact, if we set r=(8^(1/2)) we get exactly one 3-cycle:

f[x_]:=(1+Sqrt[8])x(1-x)

Plot[{f[f[f[x]]],x},{x,0,1},AspectRatio->Automatic]

This is called the birth of period three. Where the graph is tangential to the line y=x, we obtain a 3-cycle (a point (SuperStar[x]) whose orbit is {(SuperStar[x]), f((SuperStar[x])), (f^2)((SuperStar[x]))}, all distinct with (f^3)((SuperStar[x]))=(SuperStar[x])). A 3-cycle is created when r=(8^(1/2)). This type of parameter value where the nature of the map changes, is called a bifurcation point. Note that we also see two fixed points in the graph.  Taking a slightly larger value of the coefficient gives two 3-cycles:

f[x_]:=(1.05+Sqrt[8])x(1-x)

Plot[{f[f[f[x]]],x},{x,0,1},AspectRatio->Automatic]

We now see two  3-cycles. Let us look at the web plots in this type of example:

webPlot[(1+Sqrt[8])x(1-x), {x,0,1.2}, {.15, 16}, AspectRatio->Automatic]

webPlot[(1.05+Sqrt[8])x(1-x), {x,0,1.2}, {.14, 16}, AspectRatio->Automatic]

In each case the orbit approaches a 3-cycle arbitrarily closely. Consider what happens when r=4.

webPlot[4x(1-x), {x,0,1.2}, {.2, 50}, AspectRatio->Automatic]

No matter the choice of the starting point (Subscript[x, 0]), the iterates seem to behave quite randomly. We are seeing a type of chaotic behaviour.

It is useful to do a ManipulatePlot to see how the nature of the periodic points changes as the parameter increases.

Use the ManipulatePlot command below with each of the functions h, g and k to examine the various bifurcation points of the logistic maps. Notice that a 2-cycle is created when  r= 3.

h[r_,x_]:=r*x(1-x)

g[r_,x_]:=h[r,h[r,x]]

k[r_,x_]:=h[r,h[r,h[r,x]]]

Manipulate[Plot[{x,g[r,x]}, {x,0,1},PlotRange->{0,1},AspectRatio->Automatic],{r,0,4}]

2.2 The Tent Family.

The tent map is defined by: T(x)=1-2|x-1/2|, 0≤x≤ 1. We can use the Piecewise command to define this mapping using Mathematica, or we can define it directly using the Abs[x] function. In the cell below we define  (Subscript[T, a])(x)  , 0 ≤ x ≤ 1, when a=1, and then we define (Subscript[T, a])(x)  where  a  is a parameter, 0 ≤ a ≤ 2. The tent map is obtained when a = 2.

t[x_]:=Piecewise[{{x, x<1/2}, {(1-x), 1/2<x}}]

In order to define the iterates of the tent family, we proceed as follows:

Plot[t[x],{x,0,1}]

Execute the ManipulatePlot command below for each of the functions m, n, p, q and r, to investigate the bifurcation points of the tent map:

m[a_,x_]:=a*t[x]

n[a_,x_]:=m[a,m[a,x]]

p[a_,x_]:=m[a,m[a,m[a,x]]]

q[a_,x_]:=m[a,m[a,m[a,m[a,x]]]]

r[a_,x_]:=m[a,m[a,m[a,m[a,m[a,x]]]]]

Manipulate[Plot[{x,n[a,x]}, {x,0,1},PlotRange->{0,1},AspectRatio->Automatic],{a,0,2}]

2.3 The Creation of the 2-Cycle for c(x)=a cos(x)  and  s(x)=a sin(x) .

The logistic maps are examples of one-dimensional dynamical systems defined on an interval [a,b], having a single critical point at x=c, a<c<b, with f(a)=0, f(b)=0, f increasing on [a,c] and decreasing on [c,b]. We call such functions unimodal maps. Other examples of unimodal maps are given by the sine and cosine functions defined on appropriate intervals. Let us look at a ManipulatePlot for this type of example:

s[x_,a_]:=a*Sin[x]

Manipulate[Plot[{s[x,a],x},{x,0,Pi}],{a,0,Pi}]

Manipulate[Plot[{s[s[x,a],a],x},{x,0,Pi}],{a,0,Pi}]

On the interval [0,π], we see that the behavior of s(x)=a sin(x) is a lot like that of the logistic family. We can estimate the value of a where the 2-cycle is created. We can do the same for c(x)=a cos(x). where we see that c(x) has the fixed point x=-π, when a=π.

c[x_,a_]:=a*Cos[x]

Manipulate[Plot[{c[c[x,a],a],x},{x,-Pi-1/2,Pi}],{a,0,Pi}]

webPlot[Pi*Cos[x], {x,-3Pi/2,3Pi/2}, {-2, 10}, AspectRatio->Automatic]

2.4  Finding the Periodic Points of (Subscript[L, μ])(x)=μ x(1-x).

Let us now try to determine the periodic points of (Subscript[L, μ])(x)=μ x(1-x), 0≤x≤ 1, for 0<μ<4 (we use r in place of μ for convenience).

Clear[f,q,r,g,h,k]
f[x_,r_]:=r*x(1-x)

q[x_,r_]:=(f[f[x,r],r]-x)/(f[x,r]-x)

q[x,r]

Simplify[q[x,r]]

Solve[q[x,r]==0,x]

This gives the period-2 points of (Subscript[L, r])(x). Note that (r^2)-2r-3=(r-3)(r+1), so that their is a 2-cycle when r>3. The period-3 points are more problematic. We follow the method of Feng given in Section 2.6 of the textbook.

g[x_,r_]:=(f[f[f[x,r],r],r]-x)/(f[x,r]-x)

g[x,r]

Factor[g[x,r]]

b=7+2a-a^2

h[z_,a_]:=g[-z/a,a]

k[z_,b_]:=(z^3+z^2(3a+1)/2+z(2a+3-b/2)+(a+5)/2-b/2)^2+b(z+1)^2(z+a)^2/4

Simplify[h[z,a]-k[z,b]]

This shows that  h(z,a) and k(z,b) are identically equal for all z. Note that b=1+2a-(a^2)=0, when a=1+√8. 
Now 
 (Subscript[k, b])(z)= (({(z^3) + (z^2)(3 a/2 + 1/2) + z(2 a + 3 - b/2) + (a + 5)/2 - b/2})^2)+ ((b(z +1))^2)(((z+a))^2)
so (Subscript[k, b])(z)>0 for all z, when b>0. This means that for r<1+√8, there are no 3-cycles, and the first 3-cycle appears when r=1+√8. we can confirm these findings by graphing  (Subscript[g, r])(x), when r=1+√8. The graph of (Subscript[g, r])(x), for r<1+√8, lies entirely above the x-axis.

a=1+Sqrt[8]

Plot[g[x,a],{x,0,1}]

Exercises 2.1.

1. Let  (Subscript[f, a])(x)=1-|2x-1((|)^a), 0≤x≤ 1 , 0<a<4. Use a ManipulatePlot to graph the function and its iterates in order to examine where the 2-cycles and 3-cycles arise. Use Mathematica to determine the value of a where the singularity at x=1/2  disappears. (Hint: Use Mathematica to graph the Newton quotient [(Subscript[f, a])(1/2+h)-(Subscript[f, a])(1/2)]/h )]).

2. Using a suitable ManipulatePlot, estimate the value of a for which the 2 and 3-cycles of s(x)=a sin(x) first arise.

3. Estimate the value of a for which the 2-cycles and 3-cycles of f(x)=a (x^2)(1-x)/(1+x) first arise. Use the parameter values 0<a<(5√5+11)/2, thus demonstrating that f(x) is a dynamical system for this range of values .

4. Using a suitable ManipulatePlot, estimate the value of a for which the 3-cycle arises for the tent map.

5. Compare the iterates of the doubling map (defined below) with those of the tent map (i,e., consider a*d[x] as a varies, and its iterates).

d[x_]:=Piecewise[{{x, x<1/2}, {(x-1/2), 1/2<x}}]

6. Discuss the periodic points and the 2-cycles of the map (Subscript[h, c])(x)=(((x+c))^2).

7. If  (Subscript[f, p])(x)=x (e^(x-p)),  x>0, p>0, use a ManipulatePlot to show that a 2-cycle is created when p=2 (see 1(a). One Dimensional Maps, Exercise 1.3 # 4).

8. If f(x)=(x^3)-x+a, use a ManipulatePlot for the parameter a to show that  f(x)  has either one, two or three fixed points, but no 2-cycles.

9. If f(x)=(x^3)-a x, use a ManipulatePlot for the parameter a to investigate the fixed points, the 2-cycles and 3-cycles. Find approximate values of where these occur.