## Mathematical definition

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.