I would like to prove that if $$ M = \{(x,y)\in\mathbb R^2: y^2 - 4x^2(1-x^2) = 0\} $$ and $$ P:(0,2\pi)\to M, \quad \theta \mapsto (\sin \theta,\sin 2\theta), $$ $$ Q:(-\pi,\pi)\to M, \quad \theta \mapsto (\sin \theta,\sin 2\theta), $$ are both bijective (this is easy) but $P^{-1}$ and $Q^{-1}$ do not give $M$ the same differentiable structure. What does this mean?
Also, I think that both structures are dipheomorphic, since $M\to M$ given by $P\circ \beta \circ Q^{-1}$, where $\beta(t) = t+\pi$ is differentiable. Is this right?
Edit: $P^{-1}\circ Q$ is $\theta\in(0,2\pi)\mapsto \pi-\theta\in(-\pi,\pi)$ and is well defined.
A differentiable structure on a manifold is, by definition, a maximal $C^{\infty}$-atlas, i.e. a collection of charts (such as $P$ and $Q$ on your example) being maximal with respect to the condition that they are $C^{\infty}$-compatible. This means that their change of coordinates are $C^{\infty}$.
If you have two possibly non-maximal atlases on a manifold you can ask if the charts of one of them are $C^{\infty}$-compatible with respect to the charts of the other one. If all of them turn out to be compatible, then the union of both atlases is a new atlas on the manifold.
However, if you find a chart in one of them which is not compatible with on of the charts of the other atlas, there is no way of finding a bigger atlas that contain both of them. Therefore this gives the idea that, although both atlases serve to give a certain set a differentiable structures, these are not compatible.
In your example, the changes of coordinates are $P^{-1}\circ Q:(-\pi,\pi)\to(0,2\pi)$ and $Q^{-1}\circ P:(0,2\pi)\to(-\pi,\pi)$. These compositions are bijective since $P$ and $Q$ are biyective, However e.g. they are not continuous, for example at $\pi/2$.