One Dimensional Maps - Demo 2

Instability of the fixed point

For a=3.2 the orbit no longer converges to a single point, but eventually oscillates between one of two values. (Hitting Clear after the iteration has been running some time will clear the plot of the transient, and keep the iteration going to uncover the long time behavior.)

Starting from an initial value near the fixed point at x=(a-1)/a shows that the fixed point is now unstable - the iterations move x away from this point.

You can check that almost any initial condition leads to an orbit that eventually converges to the period 2 limit cycle by clicking on the running plot to start from various new initial values.

It is instructive to look at the behavior in terms of the second iterate map function F2=F(F(x)) (i.e. the order 2 composition given by setting Compose to 2). To make the behavior clearer, the value of a is increased to a=3.2 in the following.

Note that the slope of F2 is greater than unity (the slope of the diagonal) at the fixed point value of F(x) at x=0.69, showing that this point is indeed unstable. (The slope of F2 at the fixed point is the square of the slope of F here, by the chain rule.) On the other hand the magnitudes of the slopes of F2 at the other two intersection points are less than unity, so that these points are stable fixed points of F2.

A fixed point of F2 may be a fixed point of F, or may correspond to a period 2 orbit of F (since then second iterates repeat). You can check that starting from almost any initial condition, the orbit will converge to one of the two stable fixed points of F2 - which one depends on the initial condition, and just corresponds to sampling the period 2 cycle of F at either of its two values.

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Last modified Tuesday, December 7, 1999
Michael Cross