Consider the following differential equation
 |
|
|
(44) |
with
and
. A periodic solution with period
satisfies the following system
 |
|
|
(45) |
For simplicity the period
is treated as a parameter resulting in the system
 |
|
|
(46) |
If
is its solution then the shifted solution
is also a solution to (46) for any value of
. To select one solution, a phase condition is added to the system. The complete BVP (boundary value problem) is
 |
|
|
(47) |
where
is the derivative of a previous solution. A limit cycle is a closed phase orbit corresponding to this periodic solution.