If we consider the following system: $$ \frac{ \mathrm{d} \vec{x}}{ \mathrm{d} t}=\vec{f}(\vec{x})\qquad \text{ with } \vec{x}\in\mathbb{R}^n $$ and assume we know a fixed point $\vec{x^*}$, we can analyse the stability of the system around the fixed point by proceeding to a Taylor Expansion up to the first term to understand if the point is stable or not.
However, if we continue up to the 2nd order terms, we can determine the existence of limit cycles, (by Hopf, Melkinov...) at least in $2$-dimensional systems.
$\implies$What are the existing tools and methods to determine the existence of limit cycles in systems in $\mathbb{R}^n$?
You can refer me to papers or books. Most of the literature I encountered only worries about stability of fixed points up to the linear order (Robert May, Gardner etc).