One-Dimensional Dynamical Systems
Part 5: Bifurcation
The dynamics of a function is a qualitative description of the
eventual behavior of all points on the line. This includes information
such as the number and relative placement of fixed and periodic points
and whether they are attracting, repelling, or neutral. If the
function depends on a parameter, the dynamics of this function
depends on the parameter as well.
As the parameter of the Logistic map increases, a series of period-doubling bifurcations occur. The Logistic map becomes chaotic for even larger (certainly for 4). We say that the Logistic map becomes chaotic as it undergoes a series of period-doubling bifurcations. In fact, this is a typical route to chaos, which is not at all restricted to the Logistic map alone. We never gave a definition of chaotic behavior, and we are not going to, but you can think of a dynamical system being chaotic if it has periodic points of any period, which are all repelling.
Before a map can possibly have infinitely many periodic points with distinct periods, it must have periodic points with all periods of the form 2n. If for a particular parameter value a periodic point of period 5 appears, you can imagine how periodic points of periods 10, 20, 40, etc. are born as the parameter is increased. In fact, there is a very strict, and universal, ordering of the periods described in a famous paper by Li and Yorke called Period 3 implies Chaos. You can read more about this in the Geometry Forum article Sharkovskii's Theorem.
Written by Hinke Osinga
Comments to: email@example.com
Created: Apr 6 1998 --- Last modified: Thu Apr 9 14:41:02 1998