### Mathematical definition

The Period Doubling bifurcation curve is defined by the following system

|  |  | (64)

this is exactly the system defining limit cycles but with one extra constraint where is defined as the solution component of the system

|  |  | (65)

which is exactly the same system as was used to detect the Period Doubling bifurcation.

Instead of using this functional both systems can be combined in one larger system

|  |  | (66)

The first method (using system (64) and (65)) is implemented in the curve definition file `perioddoubling`. The other method is implemented in `perioddoubling2`. We will not use it further and it should not be confused with the continuation of the period doubled curve that branches off in a PD point on a limit cycle curve.