  Full understanding of synchronous behavior in coupled dynamical systems
beyond the identical case requires an explicit construction of the generalized
synchronization manifold, whether we wish to compare the systems, or to
understand their stability. Nonetheless, while synchronization has become an
extremely popular topic, the bulk of the research in this area has been focused
on the identical case, specifically because its invariant manifold is simply
the identity function, and there have yet to be any generally workable methods
to compute the generalized synchronization manifolds for non-identical systems.
Here, we derive time dependent PDEs whose stationary solution mirrors exactly
the generalized synchronization manifold, respecting its stability. We
introduce a novel method for dealing with subtle issues with boundary
conditions in the numerical scheme to solve the PDE, and we develop first order
expansions close to the identical case. We give several examples of increasing
sophistication, including coupled non-identical Van der Pol oscillators. By
using the manifold equation, we also discuss the design of coupling to achieve
desired synchronization.
