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$\lambda =1$

When $\lambda =1$, a period doubling bifurcation occurs, and periodic points will emerge as seen in Figure 1.

$\displaystyle x_{n+1} = \lambda x_n$     (12)

near the fixed point.

For this case, solving (9) for x yields

$\displaystyle x = b + \frac{1}{2}$     (13)

So it can be seen that the value for $x$ at which this change in stability occurs can be "pushed off" simply by increasing $b$. This can be seen in comparing Figure 2 to Figure 1. In Figure 2, the addition of the $b$ term has necessitated a larger value of $a$ in order for the onset of the period doubling bifurcation.

URA Program Website 2004-12-02