Center manifold
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.
Dust particles in the rings are subject to tidal forces, which act characteristically to "compress and stretch".
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.
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.
[1] 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.
A center manifold of the equilibrium then consists of those nearby orbits that neither decay nor grow exponentially quickly.
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 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.
[3] 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.
[5] The center manifold existence theorem states that if the right-hand side function
times continuously differentiable), then at every equilibrium point there exists a neighborhood of some finite size in which there is at least one of[6] In example applications, a nonlinear coordinate transform to a normal form can clearly separate these three manifolds.
However, some applications, such as to dispersion in tubes or channels, require an infinite-dimensional center manifold.
Potzsche and Rasmussen established a corresponding approximation theorem for such infinite dimensional, non-autonomous systems.
[13] All the extant theory mentioned above seeks to establish invariant manifold properties of a specific given problem.
The aim is to usefully apply theory to a wider range of systems, and to estimate errors and sizes of domain of validity.
[14] [15] This approach is cognate to the well-established backward error analysis in numerical modeling.
This is analyzed by the center manifold reduction, which, in combination with some system parameter μ, leads to the concepts of bifurcations.
Another example shows how a center manifold models the Hopf bifurcation that occurs for parameter
, the delay differential equation is then approximated by the system In terms of a complex amplitude
, but the cubic nonlinearity then stabilises nearby limit cycles as in classic Hopf bifurcation.
