Combined higher-order interactions of mixed symmetry on the sphere

We consider systems of N particles interacting on the unit sphere in d-dimensional space with dynamics defined as the gradient flow of rotationally invariant potentials. The Kuramoto model on the sphere is a well-studied example of such a system but allows only pairwise interactions. Using the Kuramoto model as a guide, we construct n-body potentials from products and sums of rotation invariants, namely, bilinear inner products and multilinear determinants, which lead to a wide variety of higher-order systems with differing synchronization characteristics. The connectivity coefficients, which determine the strength of interaction between any set of n distinct nodes, have mixed symmetries, which follow from those of the symmetric inner product and the antisymmetric determinant. We investigate n-body systems in detail for n⩽6, both as isolated systems and in combination with lower-order systems, and analyze their properties as functions of the coupling constants. We show by example that in many cases, multistable states appear only when we forbid self-interactions within the system.