  We formulate a general Kuramoto model on weighted simplicial complexes where
phases oscillators are supported on simplices of any order $k$. Crucially, we
introduce linear and non-linear frustration terms that are independent of the
orientation of the $k+1$ simplices, providing a natural generalization of the
Sakaguchi-Kuramoto model. In turn, this provides a generalized formulation of
the Kuramoto higher-order parameter as a potential function to write the
dynamics as a gradient flow. With a selection of simplicial complexes of
increasingly complex structure, we study the properties of the dynamics of the
simplicial Sakaguchi-Kuramoto model with oscillators on edges to highlight the
complexity of dynamical behaviors emerging from even simple simplicial
complexes. We place ourselves in the case where the vector of internal
frequencies of the edge oscillators lies in the kernel of the Hodge Laplacian,
or vanishing linear frustration, and, using the Hodge decomposition of the
solution, we understand how the nonlinear frustration couples the dynamics in
orthogonal subspaces. We discover various dynamical phenomena, such as the
partial loss of synchronization in subspaces aligned with the Hodge subspaces
and the emergence of simplicial phase re-locking in regimes of high
frustration.
