Numerical Continuation - Periodic Motions

Periodic Motions

A periodic motion is a closed curve in phase space. That is, for some period .

The textbook example of a periodic motion is the undamped pendulum.

If the phase space is periodic in one or more coordinates, say, with a vector, then there is a second kind of periodic motions defined by

Here is a diagonal matrix of integers that serves as an index of these periodic motions of the second kind.

The first step in writing an implicit system for a periodic motion is to move the period from the boundary conditions to the ODE:

The second step is to add an additional equation, a phase constraint, that can be thought of as determining the period. This is necessary because any solution of the above boundary value problem can be shifted in time by an arbitrary amount (time does not appear in the defining equations—the dynamical system is called autonomous).

There are several choices for the phase constraint. If is a known periodic orbit at a parameter value near, then, Poincaré used

which states that lies in a plane which is orthogonal to the tangent vector of the closed curve. This plane is called a Poincaré section.

For a general problem a better phase constraint is an integral constraint introduced by Eusebius Doedel, which chooses the phase so that the distance between the known and unknown orbits is minimized: