Hamiltonian Systems: Showing Limit Cycles Impossible

[Nusc]

http://books.google.ca/books?id=Pd8...ook_result&ct=result&resnum=1&ved=0CAkQ6AEwAA


Let ~ denote vector.
For a fixed point.
(~x0 is asymptotically stable if there exists
a neighbourhood U of ~x0 such that if ~x(t) obeys Hamilton's equations and ~x(0) in U, then lim
t->inf
~x(t) = ~x0.)

can you give me a precise definition for periodic solutions (limit cycles) in this context?

 

[Eynstone]

http://books.google.ca/books?id=Pd8...ook_result&ct=result&resnum=1&ved=0CAkQ6AEwAA


Let ~ denote vector.
For a fixed point.
(~x0 is asymptotically stable if there exists
a neighbourhood U of ~x0 such that if ~x(t) obeys Hamilton's equations and ~x(0) in U, then lim
t->inf
~x(t) = ~x0.)

can you give me a precise definition for periodic solutions (limit cycles) in this context?

A solution x is periodic iff there exists T >0 with x(t+T) = x(t) for all t.