CHAPTER 13, Linear Transformations and the Henon Map.

13.1 Linear Transformations.

We start with a brief overview of using Mathematica to study matrices. Given a nested list, we can write it as a matrix:

A={{3,-2},{-1,1}};
B={{3,-1},{5,4}};

MatrixForm[A]
MatrixForm[B]

Then we can do the usual operations on this matrix. The first product is the Kronecker product, where corresponding entries of the matrix are multiplied together. The second is the usual matrix product.

MatrixForm[A*B]
MatrixForm[A.B]

To find the eigenvalues and eigenvectors, use:

Eigenvalues[A]
Eigenvalues[B]

Eigenvectors[A]
Eigenvectors[B]

The eigenvalues of A are real with |2+√3|>1, |2-√3|<1, so the corresponding linear map has a fixed point at the origin whcih is a saddle. The second linear map has complex eigenvalues with absolute value greater than 1, so the origin is an unstable fixed point.  Consider the matrix map written in the form f(x,y)=(.3x+.5y,-.8x+.7y). This can be treated graphically as follows, so we see that when iterated, the points (f^n)(1,1) move towards the origin, which is an attracting fixed point

f[{x_,y_}]:={.3x+.5y,-.8x+.7y}

orbit[f_,{a_,b_}]:=(A=NestList[f,{a,b},5];
ListPlot[A])

NestList[f,{1,1},10]

orbit[f,{1,1}]

Exercises 13.1.

Use NestList and an orbit plot as above to conjecture the stability of the fixed point (0,0) for each of the following:
(i)  f(x,y)=(x/3,y/3), (ii) f(x)=(.4 x+.2 y, -,2 x+.4 y), (iii) f(x)=(x+y, x/4+y).

13.2 The Henon Map.

We now examine the Henon Attractor, obtained from iterating the Henon Map. This is the map H:(R^2)->(R^2), H(({{x}, {y}}))=(({{1+y-a (x^2)}, {b x}})), where a and b are constants:

HenonMap[x0_List,a_,b_,n1_,n2_]:=Module[{data},
F[x_List]:={1+x[[2]]-a*(x[[1]])^2,b*x[[1]]};
data=NestList[F,x0,n1];
Column[{ListPlot[data,PlotStyle->RGBColor[0,0,1],
ImageSize->300]}]]

HenonMap[{0,0},1.4,.3,10000,2000]

f[x_]:={1-1.4x[[1]]^2+0.3x[[2]],x[[1]]};
A=NestList[f,{Random[],Random[]},15000];
ListPlot[A]