$3$ The Logistic Map

The examples we saw above are both ordinary differential equations with at least three different components that need to be solved for. Because of the existence and uniqueness theory for ordinary differential equations, you need at least three components for chaos to occur. Chaos can be seen in simpler system but these must be discrete time approximations of the differential equation. Below we are going back to our Forward Euler approximation for the logistic equation, but here we are going to make extremely bad choices for the size of our time step. This will lead to chaotic behavior. The first code is going to produce something called a bifurcation diagram. Your goal is to understand what this code is doing and what the plot it produces means. The second code just solves for the time trajectory of our logistic map. For this code, you should adjust the time step $\backslash ($ h $\backslash )$ for values going from near $0$ to larger than $3$ and see what you notice.

: You can download the following files from D$2$L as LogisticFE.m and BadJob.m$,$ respectively.

$```$

clear all; close all; clc;

T$=5000.$;

for h$=1\mathrm{.}95$:$\mathrm{.}0005$:$3.$

$```$

$```$

t$=0.$;

x$=$x$0$;

tsave $(1)=$h;

xsave $(1)=$x$0$;

index$=1$;

while tT

x$=$x$+$h$*$x$*(1-$x$)$;

t$=$t$+$h;

index$=$index$+1$;

tsave$($index$)=$h;

xsave$($index$)=$x;

end

if t$>=1500.$

plot$($h$,$x$,\text{'}0\text{'},$'MarkerSize',$1)$; hold on;

end

end

$```$

$```$

clear all; close all; clc;

h$=2$;

T$=1500.$;

x$0=0\mathrm{.}1$;

t$=0.$;

x$=$x$0$;

index$=1$;

while tT

x$=$x$+$h$*$x$*(1-$x$)$;

t$=$t$+$h;

index$=$index$+1$;

plot$($t$,$x$,\text{'}0\text{'},$'MarkerSize',$1)$; hold on;

end

$```$

Explain what the code LogisticFE.m is plotting. This is called a bifurcation diagram and helps us understand the general behavior of differential equations as we vary parameters in the system. We will look a little more at