Definitions:
Maps in our case are self mapping. Orbit: ${x,f(x), f^2(x), \dots}$ Initial Value: x Fixed Point: f(x) = x Stable and Unstable Fixed Points: Points near do/don’t converge to the points
Cobweb Plot: A smooth graph of the orbits.
Epsilon Neighborhood: $\operatorname{N_\epsilon}(p) = { x \in \mathbb{R} : |x-p| < \epsilon }$
Stable Fixed Point: Points near are moved closer to the fixed point.
Unstable Fixed Point: Points near move away.
Sink/ Attracting Fixed Point: There exists an epsilon neighborhood such that $\lim_{k \to \infty} f^k(x) = p$ for all x in the neighborhood.
Source/Repelling Fixed Point: The exists an epsilon neighborhood such that the limit will be our of the epsilon neighborhood.
Smooth Function: Infinetely Continuously Differentiable
Linear equations only have one fixed points x = 0. This is maybe true, but we haven’t defined maps fully. We might have a y intercept. Then, solving for the fixed pt would be
$$x = mx + b \implies x - mx = b \implies x(1-m) = b \implies x = \frac{b}{1-m}$$
If $m = 1$, we have $x = x + b$ which implies b = 0.
Theorem 1.5: Theorem 1.5 Let f be a (smooth) map on $\mathbb{R}$, and assume that p is a fixed point of f.
- If |f ‘(p)| < 1, then p is a sink.
- If |f ‘(p)| > 1, then p is a source
Idea: $\frac{|f(x) - f(p)|}{|x-p|} < a$ The idea is closer and further from the neighborhood.
Exercise T1.1 $$\frac{|f(x)-f(p)|}{|x-p|} < a$$ $$\frac{|f’(x)-f’(p)|}{|x-p|} < \frac{|f’(x)-|}{|x-p|}$$
Subquestions: If we had the cobweb plot of a certain function, are there hard problems that can be solved given we have the cobweb plot or the fixed pointss?