Exact reduction of synchronized systems in higher-dimensional spaces

Exact reduction by partial integration has been extensively investigated for the Kuramoto model by means of the Watanabe-Strogatz transform. This is the simplest of higher-dimensional reductions that apply to a hierarchy of models in spaces of any dimension, including Riccati systems. Linear fractional transformations enable the system equations to be expressed in an equivalent matrix form, where the variables can be regarded as time-evolution operators. This allows us to perform an exact integration at each node, which reduces the system to a single matrix equation, where the associated time-evolution operator acts over all nodes. This operator has group-theoretical properties, as an element of SU(1,1)∼SO(2,1) for the Kuramoto model, and SO(d,1) for higher-dimensional models on the unit sphere Sd-1. Parameterization of the group elements using subgroup properties leads to a further reduction in the number of equations to be solved and also provides explicit formulas for mappings on the unit sphere, which generalize the Möbius map on S1. Exact dimensional reduction also applies to another class of much less-studied models on the unit sphere, with cubic nonlinearities, for which the governing equations can again be transformed into an equivalent matrix form by means of the unit map. Exact integration at each node proceeds as before, where now the time-evolution operator lies in SL(d,R). The matrix formulation leads to exact solutions in terms of the matrix exponential for trajectories that asymptotically approach fixed points. As examples, we investigate partially integrable models with combined pairwise and higher-order interactions.