The Hopf Index and Limit Cycles

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 3-form 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 3-form, which will support a limit cycle.


Copyright © 1995-2009, CSDC Inc. All rights reserved.
Last update 01/23/2009
to HomePage