Explanation behind cobweb diagram behaviour?

Question

Why is it that cobweb diagrams converge to a point when the absolute value of the gradient at the point is less than one and why do they diverge in other situations? I find it easier to understand why a cobweb diagram does not converge in situations when the gradient is > 1 but struggle to understand why a diagram would converge in other situations. For example, in this situation if the cobweb diagram begins at a point x=1, is there any intuitive explanation behind why it would converge to the point of intersection? (despite the gradient at the point being greater than 1.)

Answer 1

As Hans Lundmark noted in a comment, I think you are misunderstanding what happens in this particular example -- for all $x < 1.7$, iteration will go to $-\infty$ and all $x > 3.7$ to $+\infty$.

But your initial question is: Why, when the gradient has an absolute value smaller than 1, is a fixed point attractive? This is a consequence of the mean value theorem: Suppose $p$ is fixed and $|f'(p)| < 1.$ Then there is an interval around $p$ for which the gradient is strictly less than 1. Then for any $x$ in that interval: $$\left|\frac{f(x)-f(p)}{x-p}\right| < 1$$ so and since $f(p) = p$ we have $$|f(x)-p| < |x-p|$$ and thus for each successive iteration of $x$ by $f$ gets closer to the fixed point at $p$.

Intuitively, the gradient at a point $p$ measures (approximately, unless the function is linear) how far apart $f(x)$ and $f(p)$ are, relative to the difference between $x$ and $p$, so long a $x$ is close to $p$, since $$f(x) - f(p) \approx f'(p)(x-p).$$ In the particular case that $p$ is a fixed point of $f$, then, the gradient approximates how close $f(x)$ is to $p$ relative to how close $x$ was to $p$ to begin with -- when this gradient is smaller than 1, we see that $f(x)$ is closer to $p$ than $x$ was originally.