In the mathematics of evolving systems, the concept of a center manifold was originally developed to determine stability of degenerate equilibria. Subsequently, the concept of center manifolds was realised to be fundamental to mathematical modelling. Center manifolds play an important role in bifurcation theory because interesting behavior takes place on the center manifold and in multiscale mathematics because the long time dynamics of the micro-scale often are attracted to a relatively simple center manifold involving the coarse scale variables. Informal example Saturn's rings provide a rough example of the center manifold of the tidal forces acting on particles within the rings. Tidal forces have a characteristic "compress and stretch" action on bodies, with the compressing direction defining the stable manifold, the stretching direction defining the unstable manifold, and the neutral direction being the center manifold. In the case of Saturn, a particle in orbit above or below the rings will cross the rings, and, from the viewpoint of the rings, it will appear to oscillate from above to below the plane and back. Thus, it appears that the rings are "attractive". Friction, via collisions with other particles in the rings, will dampen those oscillations; thus they will decrease. Such converging trajectories are characteristic of the stable manifold: particles in the stable manifold come closer together. Particles within the ring will have an orbital radius that is a random walk: as they meet in close encounters with other particles in the ring, they will exchange energy in those encounters, and thus alter their radius. In this sense, the space where the rings lie is neutral: there are no further forces upwards or downwards (out of the plane of the rings), nor inwards or outwards (changing the radius within the rings). This example is a bit confusing, as, properly speaking, the stable, unstable and neutral manifolds do not divide up the coordinate space; they divide up the phase space. In this case, the phase space has the structure of a tangent manifold: for every point in space (a 3D position), there is the collection of "tangent vectors": all possible velocities a particle might have. Some position-velocity pairs are driven towards the center manifold, others are flung away from it. Those that are in the center manifold are susceptible to small perturbations that generally push them about randomly, and often push them out of the center manifold. That is, small perturbations tend to