Problem: First studied by Otto E$.$ R$$ssler in $1976$ the R$$ssler equations comprise three coupled

ordinary differential equations:$()$

cxzb

td

zd

ayx

td

yd

zy

td

xd

$+=$

$+=$

$=$

$.$

Specifically R$$ssler studied these equations with parameter values: a $=0\mathrm{.}2,$ b $=0\mathrm{.}2$ and c $=5\mathrm{.}7.$

Using either fourth order Runge$-$Kutta or $4-$step Adams$-$Bashforth$/$Moulton with fourth order

Runge$-$Kutta starter develop a MATLAB function M$-$file or Python file to solve the R$$ssler

equations with these parameter values over the time frame $0<=$ t $<=1500$ sec$.$ for arbitrary initial

conditions. Using this code solve the equations for initial conditions: x$(0)=0\mathrm{.}0001,$ y$(0)=0\mathrm{.}0001,$

z$(0)=0\mathrm{.}0001.$

You may visualise the solution by simply plotting y$($t$)$ vs x$($t$)$ for example, but it is much better to

use the plot$3$ command to give a $3$D plot of $($x$($t$),$y$($t$),$z$($t$))$ parameterised by t$.$ With some suitable

rotation you should see a shape which is called the R$$ssler attractor.

A feature of the R$$ssler equations for these parameter values is sensitive dependence on initial

conditions. Solve the equations again but this time with the initial conditions: x$(0)=0\mathrm{.}01,$ y$(0)=$

$0\mathrm{.}0001,$ z$(0)=0\mathrm{.}0001,$ which are rather close to those considered previously. Compare the resulting

solutions by$,$ for example, plotting the partial solutions x$($t$)$ for the first and second sets of initial

conditions on the same axes. You should observe that, although the two solutions start out very

close and although at many times they will again be very close, at various times they have moved

apart rather significant