The Lorentz equations for fluid convection in a two$-$dimensional layer heated from

below are

dx

d t $=\backslash$sigma $($y $$ x$)$

dy

d t $=$ x $(\backslash$tau $$ z$)$ y

dz

d t $=$ xy $\backslash$beta z

Variables and constants are defined as $[$MF$03$; pla$97]$

Variables :

x rate of convective overturning

y horizontal temperature difference

z departure from a linear vertical temperature gradient

Constants :

$\backslash$sigma Prandtl number

$\backslash$tau Rayleigh number

$\backslash$beta a constant with no special name

$$ One typical example of the values of the constants above is $\backslash$sigma $=10,\backslash$tau $=28,$ and

$\backslash$beta $=8/3[$MF$03$; pla$97]$

The Dependence of Chaotic Equations on Initial Conditions $[$HV$19$; p$.314]$

$$ Lorentz equations are classified as chaotic

d for such types of equations, even the small changes in initial conditions may produce

significantly different results

What is Required?

$$ Use simulink to plot all variables $($x$,$ y$,$ and z$)$ for t from $0$ to $10$

$$ Use the following initial conditions and constants

Initial Conditions :

x$02$

y$03\mathrm{.}5$

z$0+21$

Constants :

$\backslash$sigma $10$

$\backslash$tau $28$

$\backslash$beta $8/3$

$$ Change x$0$ to $2\mathrm{.}04$ and check your results again by superimposing the two plots

Additional Requirements

Showing results in scope

$$ Use the scope block in Simulink to view the outputs

$$ From Style options in the scope, customize the plot by changing colors and adding

markers

Showing results in plots

$$ Use To Workspace block in Simulink to export your results to MATLAB workspace

d save your data as array

$$ With plot command and using TEX stream modifiers

d provide axes labels for xvec $=$ x$,$ y$,$ z

and t showing them in italic font whose size is $30$

points

d provide a title for the plot called $$Solution of Lorentz Equations$$ whose size is $37$ points

Additional Requirements

Customize the lines

$$ Use line handles to customize the following properties

LineWidth LineStyle Color

DisplayName

Marker MarkerSize MarkerFaceColor MarkerEdgeColor

Customize axes

$$ Enable grid lines using grid on and minor grid lines using grid minor

$$ Use axes handles to customize the following properties

GridAlpha FontSize

MinorGridAlpha MinorGridLineStyle

Additional Requirements

Ensure neat work

$$ All blocks in the Simulink file must be well organized and suitably sized

$$ All initial conditions and constants must be specified in the workspace of MATLAB

d initial conditions and constants must be stored in structure format

$$ Your code must be well organized easy to read

$$ You should be able to obtain all desired results with one click in MATLAB workspace

which simulates the system using sim command