# Bifurcations

This chapter studies the transitions (e.g. from a period 4 orbit to a period 8
orbit) also known as *bifurcations*. To begin, restart the applet to return
the parameters to their default values.
## The first bifurcation

Set *a* to *a=2.8* and the initial value to *x0=0.1* and iterate
either by using **Step** or **Start**. The orbit converges to a single
value, i.e. to a *fixed point*. Now increase *a* to *a=3.2* and
again iterate. Although the orbit approaches close to the intersection of the
diagonal with the curve (the fixed point), rather than continuing to converge to
this point, the orbit moves away and eventually converges to one jumping between
two values, i.e. a *period 2* orbit. We say that the fixed point has become
*unstable* to the period two orbit.
This behavior is perhaps more easily understood by looking at the iterate of
*f* i.e. *f*^{ (2)}=f(f(x)), which shows every other iterate of
*f*. Remember that a period 2 orbit of *f* appears as a fixed point of
*f*^{ (2)} (actually there are two fixed points, corresponding to
the two points on the orbit of *f*). You can also convince yourself that a
fixed point of *f* is also a fixed point of *f*^{ (2)}.
So set *nf=1*, reset *a* to *a=2.8* and iterate: the orbit
converges to the single fixed point.

Again increase *a* to *a=3.2*. There are now three fixed points
(intersections of the diagonal with the curve *f*^{ (2)}). The orbit
passes close to the "old" central fixed point, but diverges from here and
approaches one of the other two fixed points (which one depends on the initial
value *x0*). Comparing this and the previous iteration, it might be clear
that the *instability* of the "old" fixed point is associated with the
*slope* of the curve *f*^{ (2)} becoming greater than
unity - a general result. It should also be apparent that as the slope passes
through unity, the two new fixed points grow out from the old one. Returning
to the original function *f*, this corresponds to the absolute value of
the slope becoming greater than unity (the slope actually passes through -1)
and a period two orbit growing out of the unstable fixed point.

## The second bifurcation

Set *a* to *a=3.4* and look at the iterations of *f* (i.e. set
*nf=0*). Eventually the orbit converges to a stable two cycle. Increase
*a* to *a=3.5* and again iterate. The period 2 cycle has gone unstable,
and the orbit converges to a period 4 cycle.
We have seen that the period two cycle is simple (a fixed point!) in
*f*^{ (2)}. Set *nf=1* and *a=3.4* and iterate the orbit.
The orbit passes close to the unstable "old" fixed point and converges to one of
the *f*^{ (2)} fixed points, which are stable. Increase *a* to
*a=3.5*. Now all the fixed points of *f*^{ (2)} are unstable
and the orbit goes to a period 2 cycle of *f*^{ (2)} corresponding
to the period 4 cycle of *f*.

We can vaguely see that the fixed point has gone unstable in the same way the
fixed point of *f* went unstable. To make this even clearer expand the
central region by using the rescale variable *nsc=1* and iterating for
*a=3.4* and *a=3.5*.

Finally, we understood the instability of the fixed point in *f* by looking
at *f*^{ (2)}: similarly it is useful here to look at
*f*^{ (4)} by setting *nf=2*, and iterate for *a=3.4*
and *a=3.5*. The instability of the period 2 cycle is seen to be associated
with the slope of *f*^{ (4)} becoming greater than unity, and again
two new fixed points of *f*^{ (4)} develop.

## Further bifurcations

Of course the period 4 cycle is a fixed point in *f*^{ (4)}. Setting
*nf=2* and further enlarging the central portion with *nsc=2* we see a
parabolic function with a single fixed point - just as before (but now in
*f*^{ (4)} ). Increasing *a* increases the height of the
parabola, the fixed point moves to the right, eventually reaching a point on the
curve where the slope passes through -1 so that the fixed point becomes
unstable.
Clearly the process repeats, and we will find a succession of bifurcations to
more and more complicated *2*^{m} cycles with *m=0,1,2,.....*.
The values of *a* at which these bifurcations occur get closer and closer
together, so that by the time *a* has reached *a*_{c} which
is 3.569946... for the quadratic map, an infinite number of bifurcations has
occurred. The complex "2 to the infinity" cycle orbit corresponds to the onset
of chaos.

The sequence of bifurcations can be studied (either in the quadratic or sine maps)
using the drop down boxes at the bottom of the screen: choose **Quadratic**
or **Sine**, the **-** sign, and **Instability** and then the value of
**m** in the last box gives the *a* for the *2*^{m} cycle
to just go unstable This is best seen looking at *nf=m* amd *nsc=m*.

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Last modified 18 August, 2009

Michael Cross