Fixed Points and Cobwebs

In this section we develop some tools for analyzing one-dimensional maps of the form x , - /(x ), where f is a smooth function from the real line to itself. 

When we say map, do we mean the function f or the difference equation Following common usage, we ll call both of them maps. If you re disturbed by this, you must be a pure mathematician. .. or should consider becom- 

To determine the stability of x, we consider a nearby orbit x = x 4-Tj and ask whether the orbit is attracted to or repelled from x. That is, does the devia- 

Suppose we can safely neglect the 0(7/ )terms. Then we obtain the linearized, +1 = with eigenvalue or multiplier A = The solution of 

Try Example 10.1.1 on a hand calculator by pressing the x button over and over. You ll see that for sufficiently small Xg, the convergence to x =0 is extremely rapid. Fixed points with multiplier A = 0 are called superstable because perturbations decay like 7/ 7/g , which is much faster than the usual 7/ A 7, at an ordinary stable point. 

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