§3.9 Two dimensional linear flow
A two-dimensional linear system is a system of the form
x  ax  by
y  cx  dy
(9.1)
where a, , b, c, d are parameters. (9.1) can be written in vector form
X  AX
where
a b 
 x
 and X   
A  
c d 
 y
(9.2)
Such a system is a linear in the sense that if X 1 (t ) and X 2 (t ) are solutions of (9.2) ,
then is any linear combination c1 X 1 (t )  c 2 X 2 (t ) . Assume det A  0 , then X  0 is
the unique equilibrium of (9.2) . The solution of X  AX can be visualized as
trajectories moving on the ( x, y ) plane, in this context called phase plane.
For the general solutions of (9.2) we seek trajectories of the form
X (t )  e t v
(9.3)
where v  0 is some fixed vector to be determined and   C to be determined.
Substitute (9.3) into (9.2) , we obtain e t v  e t Av . Cancelling the nonzero scalar
factor e t yields
Av  v
(9.4)
i.e. ( , v) is an eigenpair of 2  2 matrix A . It is easy to find that the eigenvalues
1 ,  2 of A are
1 
   2  4
2
, 2 
   2  4
2
(9.5)
where
  trace( A)  a  d ,   det A  ad  bc
If 1   2 then the corresponding eigenvectors v1 and v 2 are linearly independent.
Any initial condition X 0 can be written as a linear combination of eigenvectors, say,
X 0  c1V1  c 2V2 . This observation allows us to write down the general solution X (t )
as
X (t )  c1e 1tV1  c 2 e 2tV2
(9.6)
There are following cases for various 1 and  2 .
Case 1: 1 ,  2 are real and  2  1
Case 1a (Stable node)  2  1  0
Let L1 , L2 be the line generated by V1 , V2 respectively. Since  2  1  0 ,
X (t )  c1e 1tV1 as t   and the trajectories are tangent to L1
Case 1b (Unstable node) 0   2  1
Then X (t )  c1e 1tV1 as t  
Case 1c (Saddle point)  2  0  1 . In this case, the origin 0 is called a saddle point
and L1 , L2 are unstable manifold and stable manifold.
Case 2: 1 ,  2 are complex.
Let 1    i , 2    i and V1  U  iV and V2  U  iV be complex
eigenvectors. Then
x(t )  ce ( i ) tV1  c e ( i )tV1  2 Re(ce ( i )tV1 )
Let c  ae i . Then
x(t )  2aet (u cos( t   )  v sin( t   )) .
Let P and Q be the line generated by U , V respectively.
Case 2a (Center)   0 ,   0 .
Case 2b (Stable focus, spiral)   0 ,   0 .
Case 2c (Unstable focus, spiral)   0 ,   0 .