A Coordinate-independent Center Manifold Reduction

We give a method for performing the center manifold reduction that eliminates the need to transform the original equations of motion into eigencoordinates To achieve this we write the center manifold as an embedding rather than as a graph over the center subspace Introduction The center manifold reduction is a technique for eliminating non essential degrees of freedom in bifurcation problems The low dimensional equations of motion on the center manifold or their projection onto the center subspace tell us about the ow in the vicinity of the bifurcation point This paper provides an alternative to the graph construction commonly used to identify the center manifold for example Instead we construct the center manifold as an embedding The technique eliminates the need to transform the original equations of motion into eigencoordinates thus emphasizing that the center manifold is a geometric object whose speci cation does not require particular coordinates The Graph Construction Consider a vector eld f X R R with an equilibrium at the origin Let Df be the linear part of the vector eld at this equilibrium The center subspace E is the space spanned by the generalized eigenvectors of Df corresponding to eigenvalues with zero real part The center manifold is tangent to E at X and is invariant under the ow of f In the usual center manifold reduction one transforms the equations of motion into eigen coordinates One then writes the center manifold as a graph over the center subspace the latter having been linearly decoupled from the other degrees of freedom by the coordinate transformation In the eigen coordinates the system of di erential equations has the form p x Bx F x y y Cy G x y where B has eigenvalues with real part equal to zero and C has eigenvalues with negative real part The functions F and G and their rst derivatives vanish at the origin The center manifold is written as a graph W c f x y j y h x g h Dh in the neighborhood of the origin The boundary conditions h and Dh insure that the center manifold passes through the equilibrium and is tangent to E at the equilibrium Since the center manifold is invariant under the ow one can substitute y h x into the second equation of and obtain Dh x Bx F x h x Ch x G x h x This is solved for h x by expanding in a power series about the origin With the center manifold identi ed the vector eld on y h x is projected onto the center subspace x Bx F x h x The stability of the equilibrium for the full system is given by the stability for the reduced system The Center Manifold as an Embedding The above procedure requires an initial transformation to eigencoordinates For high dimensional systems this block diagonalization is prohibitive for hand cal culation and one turns to a machine implementation Unfortunately diagonalizing large algebraic matrices can be di cult for symbolic computation systems We give an alternative center manifold reduction that dispenses with the need to perform the initial coordinate transformation The procedure requires only knowl edge of the vectors spanning the center subspace For systems where Df has zero eigenvalues the center subspace is just the kernel of Df In our experience sym bolic computation packages are able to nd kernels of matrices with little di culty We assume that the unstable subspace is empty though this is not necessary for the construction We assume that the original system of equations X f X has an equilibrium at the origin f For simplicity we assume that Df has a one dimensional kernel spanned by vr It is straightforward to extend the procedure to multi dimensional kernels Before proceeding we need to establish some notation We denote by Df a the action of Df on the vector a The result is a vector with components