Орталық манифольд: Математикалық модельдеу және Сатурн сақиналары

Mathematical concept
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.

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.

However, some applications, such as to dispersion in tubes or channels, require an infinite dimensional center manifold. The most general and powerful theory was developed by Aulbach and Wanner. They addressed non autonomous dynamical systems in infinite dimensions, with potentially infinite dimensional stable, unstable and center manifolds. Further, they usefully generalised the definition of the manifolds so that the center manifold is associated with eigenvalues such that , the stable manifold with eigenvalues , and unstable manifold with eigenvalues They proved existence of these manifolds, and the emergence of a center manifold, via nonlinear coordinate transforms. Potzsche and Rasmussen established a corresponding approximation theorem for such infinite dimensional, non autonomous systems.

All the extant theory mentioned above seeks to establish invariant manifold properties of a specific given problem. In particular, one constructs a manifold that approximates an invariant manifold of the given system. An alternative approach is to construct exact invariant manifolds for a system that approximates the given system called a backwards theory. The aim is to usefully apply theory to a wider range of systems, and to estimate errors and sizes of domain of validity. This approach is cognate to the well established backward error analysis in numerical modeling.

As the stability of the equilibrium correlates with the "stability" of its manifolds, the existence of a center manifold brings up the question about the dynamics on the center manifold. This is analyzed by the center manifold reduction, which, in combination with some system parameter μ, leads to the concepts of bifurcations.

The Wikipedia entry on slow manifolds gives more examples.