Given a smooth, $2\pi$-periodic function $\kappa(s)$, the associated planar curve $\gamma(s)$ for which $\kappa(s)$ is the (signed) curvature, is uniquely determined up to Euclidean invariance: a canonical parametrization is for example given by $$ \gamma(s) = \left( \int_0^{s} \cos \phi(\sigma)\,\text{d}\sigma,\,\int_0^{s} \sin \phi(\sigma)\,\text{d}\sigma \right),\;\phi(\sigma) = \int_0^\sigma \kappa(\tau)\,\text{d}\tau. \tag{1} $$ The goal is to determine whether $\gamma$ is closed and simple.
In principle, parametrization $(1)$ suffices to check whether $\gamma$ is closed and simple; however, the nested integrals makes this cumbersome in practice. In my case, I have a family of functions $\kappa(s)$ as periodic orbits of a given dynamical system, and I would like to select those $\kappa$-orbits that give rise to a simple closed curve $\gamma$.
A priori, one could consider the total curvature $K = \int_0^{2\pi} \kappa(s)\,\text{d}s$. For a closed curve $\gamma$, the condition $K = 2\pi$ is necessary to avoid self-intersections. However, this condition is unfortunately not sufficient. It is straightforward to construct an example where a homotopy within a curve family for which $K=2\pi$ induces self-intersection:
Are there results from differential geometry that I can use here, is my only option to check the injectivity and periodicity of the explicit parametrization $(1)$?
[Related question for algebraic, non-closed curves: https://mathoverflow.net/questions/170320/conditions-for-a-parametric-curve-to-avoid-self-intersection]
