Fig. 3: Directionality induced (de)synchronization with an alternative symmetrization method.

a Synchronization diagram in the plane (q, σ₁) for a system of Rössler oscillators with x-x cubic coupling. q is the symmetrization parameter, such that we have a directed hypergraph for q = 0, while we get an undirected sturcture for q = 1/3. σ₁ is the coupling strength of the pairwise interactions. The white area indicates the region of stability, while the orange one the region where synchronization is lost. The horizontal dashed lines represent two values of σ₁ for which the system transits from a synchronized to an unsynchronized state as a function of q (green line), and the other way around (blue line). Panels b–e display the locus of eigenvalues of the effective Laplacian matrix \({{{{{{{\mathcal{M}}}}}}}}\) as a function of q, for a hypergraph with N = 20 nodes at two different values of σ₁ (color coding is such that the directed case q = 0 is represented in yellow, and the symmetric one q = 1/3, in blue). In the background, the white area indicates the region where the Master Stability Function (MSF) is negative, the black line the boundary of this region, and the gray area the region with positive MSF. Panels c and e represent a zoom of the area close to the origin of panels b and d, respectively. Panels b and c display a setting where the symmetric topology drives the system unstable, starting from a directed hypergraph for which the synchronization manifold is stable. Panels d and e show a case for which the symmetric topology admits a stable synchronization state, while the directed hypergraph drives to instability. The coupling strength for panels b and c is fixed to σ₁ = 0.195, while for panels d and e to σ₁ = 0.03. In both cases we set the ratio r₂ = σ₂/σ₁ = 0.7, being σ₂ the coupling strength of the three-body interactions.