System specific triangulations for the construction of CPA Lyapunov functions

Recently, a transformation of the vertices of a regular triangulation of \begin{document}$ {\mathbb {R}}^n $\end{document} with vertices in the lattice \begin{document}$ \mathbb{Z}^n $\end{document} was introduced, which distributes the vertices with approximate rotational symmetry properties around the origin. We prove that the simplices of the transformed triangulation are \begin{document}$ (h, d) $\end{document} -bounded, a type of non-degeneracy particularly useful in the numerical computation of Lyapunov functions for nonlinear systems using the CPA (continuous piecewise affine) method. Additionally, we discuss and give examples of how this transformed triangulation can be used together with a Lyapunov function for a linearization to compute a Lyapunov function for a nonlinear system with the CPA method using considerably fewer simplices than when using a regular triangulation.