I am confused about one statement in Strogatz's textbook, Nonlinear Dynamics and Chaos:
This equation ($\dot{x} = rx - x^3$) is invariant under the change of variables $x\to -x$. (56)
How is this so, since replacing $x$ by $(-x)$ gives $\dot{x} = r(-x) - (-x)^3 = -\dot{x}$? Am I missing something trivial?
Does invariance instead mean that if I let $u = -x$, then $\dot{u} = -\dot{x} = -rx + x^3$, so $\dot{x} = rx - x^3$?
Replacing $x$ with $-x$ means you replace the $x$ inside the derivative as well, thus for $x := -x$ it is :
$$\dot{(-x)} = r(-x) - (-x)^3 \Leftrightarrow -\dot{x} = -rx + x^3 \Leftrightarrow \dot{x} = rx - x^3 $$