Math Insight

A graphical approach to finding equilibria of discrete dynamical systems

Overview

One way to determine the equilibria of a discrete dynamical system is to determine the equation the equilibrium must satisfy and then solve that equation.

Here, we introduce a way of finding the equilibria from the graph of the updating function. If we have a dynamical system in function iteration form, \begin{align*} x_{n+1} &= f(x_n)\\ x_0 &= x_0, \end{align*} then, we can find the equilibria simply by plot the graph of $f$. If $f$ intersects the diagonal line where $x_{n+1}=x_n$, then we know the input of $f$ was the same as its output. If the updating function $f$ holds a particular number fixed, then that number is an equilibrium.

Video 1

The first video introduces the graphical approach of finding equilibria by looking at how we can solve equations graphically.

Question from video 1

Video 2

The second video gives you another chance to see both the graphical and analytic approach to finding equilibria with another sample function.