Web diagrams and fixed points

In our web diagrams, we always draw the graph of the function f(x) and the graph of the diagonal line y=x. The intersection of those two curves represent an extremely important aspect of the dynamical system of iterating f(x). These are, of course, the fixed points f(x)=x. Throughout this course, we have often dealt with dynamical systems that are contractive, thereby guaranteeing that they converge to a fixed point (or equilibrium point) of the system. An affine map which is contractive satisfies

with c<1. In one dimension, the affine maps are f(x)=ax+b. The difference between 0<a<1 and a>1 is easy to discern from the web diagram. In the contractive case, we see a stair stepping into the fixed point, while in the expanding case the staircase steps out to infinity.

 
Figure 57: Staircase to the fixed point: and the starting point is . The graph of y=f(x) is in blue, the line y=x is in green, and the web of iterates is in red.

 
Figure 58: Stairway to Heaven: f(x)=2x and the starting point is .

When the slope a is negative, we see a different behavior in the iterates of f(x)=ax+b. It is still true that the iterates converge to the fixed point if |a|<1 and diverge to infinity if |a|=1. However, the negative sign causes an oscillation that appears as a spiral in the web diagram.

 
Figure 59: Spiral to the fixed point: f(x)=-0.5 x and .

 
Figure 60: Spiral to infinity: f(x)=-2 x and .

For nonlinear functions f(x), the same sort of contractive behavior is seen near certain fixed points, but there is a host of other possibilities as well. If we assume the function is differentiable as well as continuous, we can quantify the notion of contraction near a fixed point. Suppose p is a fixed point, i.e. f(p)=p, and suppose x is near p. If f resembles a contraction near p, then |f(x)-p| should be smaller than |x-p|. The Mean Value Theorem of calculus gives us a way to make this precise. This basic theorem says that for any two distinct points a and b there is a point (the traditional letter) strictly in between a and b such that

This means the slope of the tangent line to y=f(x) at equals the slope of the line through the two points (a,f(a)) and (b,f(b)). Applied to our situation, we have

for some point strictly between x and p. If we are given that near p, then we can conclude that f(x) contracts near p.

To conclude that the iterates actually tend to p in the limit, we need slightly stronger conditions.

Theorem: Suppose f(x) is continuously differentiable in an interval [a,b]. Assume that f(p)=p for some , and that |f'(p)|<1. Then there is an open interval contained in [a,b] and containing p such that for all we have

To prove this theorem, start by choosing c such that |f'(p)|< c< 1 (take note of the strict inequalities). Then the fact that the derivative is continuous means there is an interval containing p such that |f'(x)| <c for all . The Mean Value Theorem then implies that for all . Iterating this inequality, we obtain as .

If there is an interval containing a fixed point p of f(x) with the property described in the Theorem above, we say p is an attractive fixed point of f(x) (or stable fixed point). Our theorem says that if |f'(p)|<1, then p is an attractive fixed point. The set of all x such that as is called the stable set of p and is denoted by .

If |f'(p)|>1, we have the opposite phenomenon; namely the iterates move away from p, at least when we start near p. In that case, p is called a repelling fixed point. If , the situation is murkier. Thus, all fixed points with are lumped together as the hyperbolic fixed points.

Let's look at . The equation has three solutions . We may compute f'(0)= -1 and . Thus, the points are repelling fixed points, while the nature of x=0 is not apparent. Some experimentation suggests that x=0 is an attractive fixed point, and its stable set is . See this figure. The reader is invited to wrestle the truth out of this function.

  
Figure 61: Web diagram for , . The iterates settle into an exceedingly slow death spiral into the fixed point x=0.

It is possible to determine the behavior of the iterates from certain aspects of the graph of y=f(x), without using the derivative. Suppose we have a region as shown in this figure. There is a fixed point p, and to the left of the fixed point on the interval [a,p) we have x<f(x)<=p. The stair-step pattern shown makes it clear that the iterates all march directly to p. We can argue this precisely as follows. Start with . Then and . Continuing to iterate, we have

A bounded increasing sequence has a limit, which must be a fixed point of f(x), since f is continuous. The only fixed point of f in that interval is p by hypothesis.

  
Figure 62: A web diagram in a ``bight'' of the curve