Question

There exist smooth - but not analytic - closed curves without self-intersections. I just would like to see a simple example of such a curve.

Answer 1

Let $\psi:\mathbb{R}\to\mathbb{R}$ be a $C^\infty$ function with compact support in $[0,\pi]$ and such that $\psi(t)>0$ if $0 < t < \pi$. Then $$ \gamma(t)=\psi(t)(\cos t,\sin t),\quad 0 < t <\pi,\quad \gamma(0)=\gamma(\pi)=(0,0) $$ is such a curve. You can take for instance $$ \psi(t)=\exp\Bigl(-\frac{1}{t^2(\pi^2-t^2)}\Bigr). $$

Answer 2

Consider the curve

$$\gamma:\quad \phi\ \mapsto\ \Bigl(1+\exp{-1\over \pi^2 -\phi^2}\Bigr)\ (\cos\phi,\sin\phi)\qquad(-\pi< \phi< \pi)$$

with filled-in point $(-1,0)$.