An Intuitive Analysis

Consider the map

$$x_{n+1} = x_{n}^2 ,$$ (3.2)

which corresponds to the mapping function $f(x)=x^2$. It is easy to guess, or check with a calculator, what happens as one repeatedly squares a number $x_0 \geq 0$: If the initial value is greater than $1$, repeated iterations give values that increase without bound. On the other hand, an initial value of exactly $1$ remains at $1$, while any intial value less than $1$ will converge to $0$. If we restrict our discussion to finite values, then this example has two fixed points, one at $x^{*} =0$ and another at $x^{*}=1$. If one starts the iteration at exactly any of these two values one remains there, and so the name 'fixed points'. The fixed point at the origin is called stable because a small deviation (perturbation) from an intial value of zero causes future iterations to be attracted back to the origin. One says that the basin of attraction, (that is the range of values that are attracted to a fixed point), of the fixed point at the origin is $[0,1)$. By contrast the fixed point at $x^{*}=1$ is unstable: any small perturbation away from that value give future values that move further away from that point. A mechanical analogy of the two fixed points, stable and unstable, is a ball lying at the bottom of a valley and one lying on a hill top. Both positions of the ball are points of static equilibrium, but one position is clearly stable while the other not. Furthermore the stability of the ball at the bottom of the valley is limited to perturbations that do not cause the ball to go over a nearby peak, and hence in general the basin of attraction of a point of stable equilibrium is of finite extent.