One of the interesting phenomena that can happen in nonlinear systems with three or more variables

is $$chaos$.$ This refers to a situation in which a very small difference in initial conditions leads to

unpredictable differences in the solution curves. This was first observed in the early $1960$s$,$ when

systems of differential equations were applied to weather prediction. There was a certain computer

simulation that was being used, in which certain initial conditions were put in$.$ Later, they wanted

to reproduce that simulation and re$-$entered the same initial values, only to get completely different

results. It turned out that the system they used the first time used a precision of six decimal places,

but only printed out three decimal places in the output. When they tried to re$-$enter the data the

second time, they only used the three decimal places from the printout, and that was the cause of the

difference. The person studying this system, E$.$ N$.$ Lorenz, went on to analyze this phenomenon, and

created a simplified system that exhibits the same chaotic behavior. This system, the Lorenz System

is:

dx

dt $=\backslash$sigma x $+\backslash$sigma y

dy

dt $=\backslash$rho x $$ y $$ xz

dz

dt $=\backslash$beta z $+$ xy

Where the parameter values are: $\backslash$beta $=8/3,\backslash$sigma $=10,$ and $\backslash$rho $=28.$ The $3-$dimensional solution curves to

this system make the famous butterfly shape, illustrating what Lorenz termed $$The Butterfly Effect$$

in a paper in the $1970$s$.$ It looks like:

In this project we will look at the values of one of the variables in the solution curve with slightly

different initial values.

$($a$)(2$ pts$)$ Copy the associated code from the file lorenzgraph.txt $($Note: download the file before

copying$)$ and paste it into the Sage Math Cell linked to the assignment on Canvas. The delta in

this case represents the amount we are modifying the initial values. At first, set delta $=0$ and

evaluate the code. It will take a few minutes to generate its output, which will be a graph of the

x$-$values of the solution curve. Describe the shape of the graph. The two values, one positive,

and one negative, around which the values appear to oscillate correspond to the x$-$coordinates of

the centers of the two $$butterfly wings$$ from the three$-$dimensional solution curve. Estimate these

x$-$values from the graph.

$($b$)(2$ pts$)$ Copy the associated code from the file lorenztable.txt $($Note: download the file before

copying$)$ and paste it into the Sage Math Cell. The delta in this case also represents the amount

we are modifying the initial values. At first, set delta $=0$ and evaluate the code. This code

generates a table of the x$-$values of the solution. Confirm that this table corresponds to the

behavior you observed from the graph. Now modify the value of delta to be $0\mathrm{.}001.$ This represents

a tiny change in the initial values. Compare the values of the two tables for small values of t$,$ and

then for larger values. At what point do the solutions start to be substantially different? Run

the code from lorenzgraph.txt again, changing the delta to $0\mathrm{.}001.$ At what point do the graphs

diverge?

$($c$)(1$ pt$)$ Repeat the previous analysis, but with delta $=0\mathrm{.}01.$ What is the effect of this change?

$($d$)(2$ pts$)$ The values of the around which the solutions oscillate correspond to equilibrium points

in the system of equations. Setting the three differential equations to zero, algebraically solve

for these two nonzero equilibrium points. Test them by putting them in as initial values in the

lorenztable.txt code $($change the values of xinit, yinit and zinit$).$