Systems of matrix Riccati equations, linear fractional transformations, partial integrability and synchronization

Abstract

We partially integrate a system of rectangular matrix Riccati equations which describe the synchronization behavior of a nonlinear complex system of N globally connected oscillators. The equations take a restricted form in which the time-dependent matrix coefficients are independent of the node. We use linear fractional transformations to perform the partial integration, resulting in a system of reduced size which is independent of N, generalizing the well-known Watanabe-Strogatz reduction for the Kuramoto model. For square matrices, the resulting constants of motion are related to the eigenvalues of matrix cross ratios, which we show satisfy various properties such as symmetry relations. For square matrices, the variables can be regarded as elements of a classical Lie group, not necessarily compact, satisfying the matrix Riccati equations. Trajectories lie either within or on the boundary of a classical domain, and we show by numerical example that complete synchronization can occur even for the mixed case. Provided that certain unitarity conditions are satisfied, we extend the definition of cross ratios to rectangular matrix systems and show that again the eigenvalues are conserved. Special cases are models with real vector unknowns for which trajectories lie on the unit sphere in higher dimensions, with well-known synchronization behavior, and models with complex vector wavefunctions that describe synchronization in quantum systems, possibly infinite-dimensional.