Ordinary Differential Equations/Locally linear


We will study autonomous systems

where the components of are functions so that we are able to Taylor expand them to first order. A system of the form
is called locally linear around a critical point of if

Example presenting the methodEdit

We study the damped oscillating pendulum system:

 
where   is called the damping constant and as in the spring problem it is responsible for removing energy.
  1. First we find the critical points. From the previous section we have:
     
  2. Second we Taylor expand the RHS of the system   around arbitrary critical point  :
     
  3. Here   is the Jacobian matrix at   which, for function  , is defined as:
     
  4. The linearization around   for an even integer   is:
     
  5. The eigenvalues of that matrix are:
     
  6. If  , then the eigenvalues are real, distinct, and negative. Therefore, the critical points will be stable nodes.We observe that the basins of attractions for each even-integer critical points are well-separated.
  7. If  , then the eigenvalues are repeated, real, and negative. Therefore, the critical points will be stable nodes.
  8. If  , then the eigenvalues are complex with negative real part. Therefore, the critical points will be stable spiral sinks.
  9. The linearization around   for odd integer   is:
     
  10. The eigenvalues of that matrix are:
     
  11. Therefore, it has one negative eigenvalue   and one positive eigenvalue  , and so the critical points will be unstable saddle points.