Specifying equation under variable transformation

[k0um0njin]

Homework Statement

For the following dynamical system in the attached picture
What is the appropriate way to specify the equation is invariant? Thanks in advance.

Homework Equations

No relevant equations

The Attempt at a Solution

Firstly I integrated each of the three equations and the results
x = 10yt - 10xt- -①
y = rxt - yt - xzt -②
z = xyt - 8/3(zt) -③

From equation ③, I got z = (txy)/ (1 + 8/3t) and
then, I substitute the equation of z into the equation ②. Until this point, am I doing it right?

 

[fzero]

The most straightforward way to verify an invariance is by defining new coordinates and checking that the new variables satisfy the same equations as the old variables. Since the claimed invariance is $(x,y,z)\rightarrow (-x,-y,z)$, you should define

$X = -x, ~ Y=-y,~Z=z.$

As an aside, your integration of the equations is incorrect. Since the $x,y,z$ are functions of $t$, solving the differential equations is more complicated than what you've done, which ignored this dependence.