destabilize points in the center manifold: the center manifold behaves like a saddle point, or rather, an extended collection of saddle points. There are dramatic counterexamples to this idea of instability at the center manifold; see Lagrangian coherent structure for detailed examples. A much more sophisticated example is the Anosov flow on tangent bundles of Riemann surfaces. In that case, one can write a very explicit and precise splitting of the tangent space into three parts: the unstable and stable bundles, with the neutral manifold wedged in the middle between these two. This example is elegant, in the sense that it does not require any approximations or hand-waving: it is exactly solvable. It is a relatively straightforward and simple example for those acquainted with the general outline of Lie groups and Riemann surfaces. 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 exponentially quickly, 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. 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 systems dynamics near the equilibrium. However, if the equilibrium has eigenvalues whose real part is zero, then the corresponding (generalized) eigenvectors form the center eigenspace—for a ball, the center eigenspace is the entire set of unforced rigid body dynamics. 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. Analogously, nonlinearity or forcing in the system perturbs the stable and unstable eigenspaces to a nearby stable manifold and nearby unstable manifold. These three