The logistic map: stability of orbits
- Author:
- a.zampa
This applet shows stability properties of orbits of order 1 (fixed points) and 2 of the logistic map, explaining why the Feigenbaum diagram bifurcates even if the fixed points do not disappear.
As we have seen in the previous activity, if is a fixed point of a map and is the slope of the tangent line to the graph of at , then writing , , and approximating with the tangent line for values of "sufficiently near" , the iteration reduces to , giving the iterates : this is the equation of the Malthus' model with as value of the parameter .
When dynamics correspond to extinction, so , i.e. : the fixed point is an attractor in the sense that neighboring points are transformed, by iteration, in points even closer.
When dynamics correspond to demographic explosion , so : the fixed point is a repulsor in the sense that neighboring points are transformed, by iteration, in points further away.
It turns out that fixed points possess basins of attraction or repulsion depending on their type, and a point that is in a basin of repulsion is gradually moved away by the iteration, until ending up in a basin of attraction and thus be "captured" by its attractor.
The same applies to orbits of order greather than 1, since these can be described as fixed points of suitable maps.