Center Manifold Theorem

 

It is important to keep in mind that we always speak of invariant manifolds based at a point. This point is usually a fixed point of a flow or a periodic point of a map. In the linear setting, the invariant manifold is just a linear vector space. In the nonlinear setting we can also define invariant manifolds that are not linear subspaces but are still manifolds. That is, locally they look like a copy of . These invariant manifolds are a direct generalization of the invariant subspaces of the linear problem. They are the most important geometric structure used in the analysis of a nonlinear dynamical system.

The way to generalize the notion of an invariant manifold from the linear to the nonlinear setting is straightforward. In both the linear and nonlinear settings, the stable manifold is the collection of all orbits that approach a point . Similarly, the unstable manifold is the collection of all orbits that depart from . The fact that this notion of an invariant manifold for a nonlinear system is well defined is guaranteed by the center manifold theorem [1]:

Center Manifold Theorem for Flows . Let be a smooth vector field on with and . The spectrum (set of eigenvalues) of A divides into three sets , , and , where

Let , , and be the generalized eigenspaces of , , and . There exist smooth stable and unstable manifolds, called and , tangent to and at , and a center manifold tangent to at . The manifolds , , and are invariant for the flow. The stable manifold and the unstable manifold are unique. The center manifold need not be unique.

is called the stable manifold , is called the center manifold , and is called the unstable manifold . A corresponding theorem for maps also holds and can be found in reference [1] or [6]. Numerical methods for the construction of the unstable and stable manifolds are described in reference [8].

Always keep in mind that flows and maps differ: a trajectory of a flow is a curve in while the orbit of a map is a discrete sequence of points. The invariant manifolds of a flow are composed from a union of solution curves; the invariant manifolds of a map consist of a union of a discrete collection of points (Fig. 4.11).

  
Figure 4.11: Invariant manifolds of a saddle for a two-dimensional map.

The distinction is crucial when we come to analyze the global behavior of a dynamical system. Once again, we reiterate that the unstable and stable invariant manifolds are not a single solution, but rather a collection of solutions sharing a common asymptotic past or future.

An example (from Guckenheimer and Holmes [1]) where the invariant manifolds can be explicitly calculated is the planar vector field

This system has a hyperbolic fixed point at the origin where the linearized vector field is

The stable manifold of the linearized system is just the y-axis, and the unstable manifold of the linearized system is the x-axis (Fig. 4.12(a)). Returning to the nonlinear system, we can solve this system by eliminating time:

where c is a constant of integration. It is now easy to see (Prob. 4.20) that (Fig. 4.12(b))

  
Figure 4.12: (a) Invariant manifolds (at the origin) for the linear approximation. (b) Invariant manifolds for the original nonlinear system.