Fig. 2.

Classically, ODEs are imagined in phase space. Considering variables from a network perspective allows us to consider synchronous nodes, while maintaining a natural correspondence to phase space. If the node space is $\mathbb{R}$, then the phase space of any admissible ODE for the depicted network is ${\mathbb{R}}^3$. Balanced colorings represent patterns of synchrony that correspond to polysynchrony subspaces of the phase space. For example, if all cells are equal for all time, the trajectory of a solution to the ODE is inside the fully synchronous subspace $\Delta$ depicted on the bottom left. If $x_1\left(t\right)=x_3\left(t\right)$ for all time, we can represent the pattern of synchrony with a balanced coloring $\bowtie$ in the network; and the coloring corresponds to the polysynchrony subspace ${\Delta }_{\bowtie }$ depicted on the bottom right