### 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.*

#### 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.

*Graphical approach to find equilibria of discrete dynamical systems.*

#### Thread navigation

##### Math 1241, Fall 2020

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##### Math 201, Spring 22

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