One-Dimensional Dynamical Systems

Part 5: Bifurcation

Qualitative change of dynamics

The theorem of Hartman and Grobman gives us the relationship between the slope of the graph at a fixed point and whether this fixed point is attracting, repelling, or neutral. For the Logistic map, we have already seen that the fixed point other than 0 is attracting for lambda

= 1.5, because the slope is between -1 and 1. However, it is repelling for lambda

= 3.1, since the graph of the derivative map no longer has a slope between -1 and 1. Hence, as we increase lambda

, the fixed point changes from being attracting to being repelling. The exact moment this change takes place is called a bifurcation.

Recall that p is a fixed point of f if f(p) = p. Verify that the Logistic family f(x) = x (1 - x) has fixed points x = 0 and x = .

Compute the slope of the graph of the Logistic map at the fixed point 0 in terms of . What is the slope for the other fixed point?
For what values of are the fixed points of the Logistic family attracting? Repelling? Neutral? Test your values using the computer.

Up: Bifurcation

Qualitative change of dynamics
Bifurcation points
Orbit diagram