5 - Applications
We consider the following system
It is easy to verify that the system is Hamiltonian and that
The associated Jacobian matrix is
First case :
The fixed points are
and so A_0 is a center. Moreover,
A_1, A_2 and A_3 have the same eigenvalues and are saddles.
Second case :
We have now the system
The only fixed point is A_0=(0,0). The matrix M is now identically zero and A_0 is a degererate saddle. The Hamiltonian in this case is written
Exercice: Try to do the corresponding program.