One-Dimensional Dynamical Systems

Part 5: Bifurcation

Bifurcation points

The Logistic map has an attracting fixed point for lambda

= 1.5 that becomes repelling as we increase lambda

to 3.1.

Definition (Bifurcation): If the qualitative dynamics of f changes as the parameter lambda is varied, we call this a bifurcation.

Definition (Bifurcation Point): In case of a bifurcation, there is a special value lambda _b for which the following holds: the dynamics for lambda close to but smaller than lambda _b is qualitatively different from the dynamics for lambda close to but larger than lambda _b This value lambda _b is called a bifurcation point.

Not every value of lambda is a bifurcation point. Only when a qualitative change in the dynamics occurs, we say that the system underwent a bifurcation.

Approximately at what value between 1.5 and 3.1 does a bifurcation occur?
At exactly what value between 1.5 and 3.1 does a bifurcation occur? Explain why you know this value is exact.

In the following section we show how to visualise the dynamics as a function of lambda .

Up: Bifurcation