A limit cycle of the Van der Pohl type is a strange robust thing. It is not a
cycle in the usual Hamiltonian sense of a harmonic oscillator, for its total
mechanical energy is not a constant of the motion. The limit cycle, unlike the
oscillator, evidently interacts with its surroundings similar to a "breathing
system", taking in energy and exhuding energy twice for each complete phase
space cycle.
If a system that can be described by an action is not isolated from its
surroundings, its Pfaff dimension is greater than 2. If the Pfaff dimension is
4, then may execute irreversible motion. However, as the system decays, it may
reach a state of Pfaff dimension 3, where d(A^dA) = 0 but A^dA <>0.
In this situation there exists a closed orbit  the limit cycle.
Such structures are related to the Hopf map, where it can be shown that the Hopf
map as a 3form is tight and has NO limit cycles. However, a closed addition to
the Action of the Hopf map  having a Reeb vector (extremal vector) component 
can lead to an overtwisted 3form, which will support a limit cycle.
 Downloads 

limitcyc.pdf 150k 
limitcyc.tex NA 