How to Show that Lorenz equations are invariant?

Question

I am struggling a little bit with this question. I know that that the Lorenz equations are: \begin{align} \dot{x} &= \sigma(y-x)\\ \dot{y} &= rx - y- xz\\ \dot{z} &= xy - bz \end{align}

Let us now say that I have the following transformation: \begin{align} x \to -x\\ y \to -y \end{align}

How can I show that the Lorenz equations are invariant under this transformation?

Answer 1

You are switching from $(x,y,z)$ to $(X,Y,Z) = (-x,-y,z)$.

We have $\dot{X} = \dot{(-x)} = -\dot{x} = - \sigma (y-x) = \sigma (x-y) = \sigma (Y-X)$.

$\dot{Y} = -\dot{y} = - (rx-y-xz) = xz+y-rx = rX - Y -XZ$

$\dot{Z} = \dot{z} = xy-bz= XY-bZ$

We have shown that $(X,Y,Z)$ are solution of the same set of differential equations than $(x,y,z)$, which was the thing to prove here.