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.

Introduction to finding equilibria of discrete dynamical systems graphically.

Video 2

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

Graphical approach to find equilibria of discrete dynamical systems.