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
= 1.5, because
the slope is between -1 and 1. However, it is repelling for
= 3.1, since
the graph of the derivative map no longer has a slope between -1 and
1. Hence, as we increase
, the fixed
point changes from being attracting to being repelling. This scenario
is called a bifurcation.
Written by Hinke Osinga
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Created: Apr 6 1998 ---
Last modified: Thu Apr 9 14:18:16 1998