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My Strogatz book introduces the supercritical pitchfork bifurcation as having normal form $\dot{x} = rx - x^3$.

It goes on to say that this equation is invariant under the change of variables $x \rightarrow -x$. It defines this as:

if we replace $x$ by $-x$ and then cancel the resulting minus signs on both sides of the equation, we get eqn. (1) back again.

So this seems like it is just describing any old odd function. I feel like I am missing some nuance here. I don't think he is trying to talk about odd functions, but that's all I'm getting out of his definition.

What I did notice is that the fixed points for the two equations are the same, However, it looks to me like stability will be reversed... so same fixed points, but not same behavior. So I have a couple questions:

  1. Is an odd equation equivalent to an invariant system?

  2. Does "oddness" have any higher order extensions? If so, would the equivalence hold?

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    $\begingroup$ If you are okay with more rigorous notion, there are so called equivariant systems. Suppose you have a system $\dot{x} = f(x), \; x \in \mathbb{R}^n$ and an action of group $G$ on $\mathbb{R}^n$. A system is said to be equivariant with respect to this action iff for any solution $\gamma(t)$ and any $g \in G$ follows that $g \cdot \gamma(t)$ is also a solution. In your case you have $g(x) = -x$: check that for any $\gamma(t)$ follows that $-\gamma(t)$ is also a solution. It's better to say that the set of solutions is invariant with respect to action of group. $\endgroup$ – Evgeny Nov 11 at 13:45

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