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 description
Saturn's rings capture much center-manifold geometry. Dust particles in the rings are subject to tidal forces, which act characteristically to "compress and stretch". The forces compress particle orbits into the rings, stretch particles along the rings, and ignore small shifts in ring radius. The compressing direction defines the stable manifold, the stretching direction defining the unstable manifold, and the neutral direction is the center manifold.
While geometrically accurate, one major difference distinguishes Saturn's rings from a physical center manifold. Like most dynamical systems, particles in the rings are governed by second-order laws. Understanding trajectories requires modeling position and a velocity/momentum variable, to give a tangent manifold structure called phase space. Physically speaking, the stable, unstable and neutral manifolds of Saturn's ring system do not divide up the coordinate space for a particle's position; they analogously divide up phase space instead.
The center manifold typically behaves as an extended collection of saddle points. Some position-velocity pairs are driven towards the center manifold, while others are flung away from it. Small perturbations that generally push them about randomly, and often push them out of the center manifold. There are, however, dramatic counterexamples to instability at the center manifold, called Lagrangian coherent structures. The entire unforced rigid body dynamics of a ball is a center manifold.
A much more sophisticated example is the Anosov flow on tangent bundles of Riemann surfaces. In that case, the tangent space splits very explicitly and precisely into three parts: the unstable and stable bundles, with the neutral manifold wedged between.
Definition
The center manifold of a dynamical system is based upon an equilibrium point of that system. A center manifold of the equilibrium then consists of those nearby orbits that neither decay nor grow exponentially quickly.
Mathematically, the first step when studying equilibrium points of dynamical systems is to linearize the system, and then compute its eigenvalues and eigenvectors. The eigenvectors (and generalized eigenvectors if they occur) corresponding to eigenvalues with negative real part form a basis for the stable eigenspace. The (generalized) eigenvectors corresponding to eigenvalues with positive real part form the unstable eigenspace.
Algebraically, let be a dynamical system, linearized about equilibrium point . The Jacobian matrix defines three main subspaces: the center subspace, spanned by generalized eigenvectors whose eigenvalues satisfy (more generally, ); the stable subspace, spanned by generalized eigenvectors whose eigenvalues satisfy (more generally, ); the unstable subspace, spanned by the generalized eigenvectors whose eigenvalues satisfy (more generally, ). Depending upon the application, other invariant subspaces of the linearized equation may be of interest, including center-stable, center-unstable, sub-center, slow, and fast subspaces.
If the equilibrium point is hyperbolic (that is, all eigenvalues of the linearization have nonzero real part), then the Hartman-Grobman theorem guarantees that these eigenvalues and eigenvectors completely characterise the system's dynamics near the equilibrium. However, if the equilibrium has eigenvalues whose real part is zero, then the corresponding (generalized) eigenvectors form the center eigenspace. Going beyond the linearization, when we account for perturbations by nonlinearity or forcing in the dynamical system, the center eigenspace deforms to the nearby center manifold.
If the eigenvalues are precisely zero (as they are for the ball), rather than just real-part being zero, then the corresponding eigenspace more specifically gives rise to a slow manifold. The behavior on the center (slow) manifold is generally not determined by the linearization and thus may be difficult to construct.