8. The general solution of a linear system.
Consider the linear dynamical system
We look for solutions on the form
If we instert these to the system we get
This equation system has nontrivial solutions if and
that is, if and only if
This is the characteristic
equation of the system and the solutions its eigenvalues.
If we now for each eigenvalues solve the equation system
we obtain the eigenvectors
Then we have the general solution of our original
dynamical system as
where C1 and C2 are arbitrary constants.
Example 8: Assume that the matrix A is
This matrix has characteristic equation
with the eigenvalues
and corresponding eigenvectors
This means that the general solution of the
corresponding dynamical system is
Example 9: Solve the system
Solution: We consider the matrix
and its characteristic equation
The eigenvalues thus are
with corresponding eigenvectors
This means that the dynamical system has the general
These are all complex solutions.
We are actually only interested in the real solutions. With
help of Euler's
formula we get
If we now pick arbitrary real constants D1 and D2
we get the general real solution
part Back to chapter 9: Dynamical systems, chaos,
stability and bifurcations.