Linearizations and Eigensystems

First, we need to define teh functions we are interested in.

f1[x1_, x2_]  =  -5 x1  + 2 x2 +  
        x1 x2
f2[x1_, x2_]  =  x1 +  x2  - 4  x2 x1

Now, I am going to find the fixed points so I know where I am interested in performing the linearization.

Solve[{f1[x1, x2] == 0, f2[x1, x2] == 0}, {x1, 
        x2}]

We need the partial derivatives.

df1x1[x1_, x2_]  =  D[f1[x1, x2], x1] 
df1x2[x1_, x2_]  =  D[f1[x1, x2], x2] 
df2x1[x1_, x2_]  =  D[f2[x1, x2], x1] 
df2x2[x1_, x2_] = D[f2[x1, x2], 
        x2]

Here is the linearization about the first fixed point:

x10  = 0;
x20  =  0;

linf1[x1_, x2_]  =  
      f1[x10, x20] +  df1x1[x10, x20]  (x1 - x10)  +  
        df1x2[x10, 
            x20]  (x2 - 
              x20)

linf1[x1_, x2_]  =  
      f2[x10, x20] +  df2x1[x10, x20]  (x1 - x10)  +  
        df2x2[x10, x20]  (x2 - x20)

Here is the linearization baout the second fixed point:

x10  = 7/19;
x20  =  7/9;

linf1[x1_, x2_]  =  
      Simplify[f1[x10, x20] +  df1x1[x10, x20]  (x1 - x10)  +  
          df1x2[x10, 
              x20]  (x2 - 
                x20)]

linf1[x1_, x2_]  =  
      Simplify[f2[x10, x20] +  df2x1[x10, x20]  (x1 - x10)  +  
          df2x2[x10, x20]  (x2 - x20)]

Here is an example of finding the eigenvalues and eigenvectors of a matrix:

matrix  =  {{-5, 2}, {1, 1. }}
MatrixForm[matrix]
Eigenvalues[matrix]
Eigenvectors[matrix]
Eigensystem[matrix]

matrix  =  {{-722, 
          405}, {-361, -81. }}
MatrixForm[matrix]
Eigenvalues[matrix]
Eigenvectors[matrix]
Eigensystem[matrix]

Needs["Graphics`PlotField`"]

field = 
      PlotVectorField[{-5 x1 + 2 x2,  x1 +  x2}, {x1, -1, 
          1}, {x2, -1, 1}, Frame -> True, 
        FrameLabel -> {"x1", "x2"}]
evalsunstab = 
      Graphics[{Arrow[{0, 0}, {-0.99/3, 0.15/3}], 
          Arrow[{0, 
              0}, {-0.30/3, -0.95/3}]}]
evalsstab  =  
      Graphics[{Arrow[{0, 0}, {0.99/3, -0.15/3}], 
          Arrow[{0, 0}, {0.30/3, 0.95/3}]}]
Show[field, evalsstab, evalsunstab, 
      Graphics[Disk[{0, 0}, 0.05]]]