## Periodic points

Now we will investigate the existance and behavior of period two points.

In order to investigate the behavior of period 2 points, it is neccessary to create a new map based on the original equation (5).

For a period two point,

 (14) (15)

where and are given by (6).

Plugging these in and dropping subscripts yields the equations to be satisfied at a period 2 point:

 (16) (17)

Plugging (17) into (16) yields an equation which is quartic in . However, this expression can be simplified since it is known that a period 1 point will also solve this equation. In other words, this equation will be of the form

 (18)

and can be separated into a factor which represents period 1 points and a factor which represents period 2 points. Doing so yields the following expression for the period 2 points:

 (19)

which can be solved for x:

 (20)

Examining the radical term reveals that there won't be a real solution to this expression unless

 (21)

In other words, the delay term requires a larger forcing term, thus pushing off the onset of the emergence of periodic points when viewed in terms of the the control parameter . This is directly observable from the following figure which depicts the onset of the bifurcation point in terms of the control parameter and the delay term:

In the figure above, the numerical values calculated are shown as + signs and (21) is shown as a solid line. Note as approaches 1.00, the predescribed condition from (21) causes non-real solutions.