Center manifold

In mathematics, the center manifold of an equilibrium point of a dynamical system consists of orbits whose behavior around the equilibrium point is not controlled by either the attraction of the stable manifold or the repulsion of the unstable manifold. The first step when studying equilibrium points of dynamical systems is to linearize the system. The eigenvectors corresponding to eigenvalues with negative real part form the stable eigenspace, which gives rise to the stable manifold. Similarly, eigenvalues with positive real part yield the unstable manifold. In mathematics, the center manifold of an equilibrium point of a dynamical system consists of orbits whose behavior around the equilibrium point is not controlled by either the attraction of the stable manifold or the repulsion of the unstable manifold. The first step when studying equilibrium points of dynamical systems is to linearize the system. The eigenvectors corresponding to eigenvalues with negative real part form the stable eigenspace, which gives rise to the stable manifold. Similarly, eigenvalues with positive real part yield the unstable manifold. This concludes the story if the equilibrium point is hyperbolic (i.e., all eigenvalues of the linearization have nonzero real part). However, if there are eigenvalues whose real part is zero, then these give rise to the center manifold. If the eigenvalues are precisely zero, rather than just real part being zero, then these more specifically give rise to a slow manifold. The behavior on the center (slow) manifold is generally not determined by the linearization and thus is more difficult to study. Center manifolds play an important role in: bifurcation theory because interesting behavior takes place on the center manifold; and multiscale mathematics because the long time dynamics often are attracted to a relatively simple center manifold. Let d x d t = f ( x ) {displaystyle {frac {d{ extbf {x}}}{dt}}={ extbf {f}}({ extbf {x}})} be a dynamical system with equilibrium point x ∗ {displaystyle { extbf {x}}^{*}} . The linearization of the system near the equilibrium point is The Jacobian matrix A {displaystyle A} defines three main subspaces: Depending upon the application, other subspaces of interest include center-stable, center-unstable, sub-center, slow, and fast subspaces.These subspaces are all invariant subspaces of the linearized equation. Corresponding to the linearized system, the nonlinear system has invariant manifolds, each consisting of sets of orbits of the nonlinear system. The center manifold existence theorem states that if the right-hand side function f ( x ) {displaystyle { extbf {f}}({ extbf {x}})} is C r {displaystyle C^{r}} ( r {displaystyle r} times continuously differentiable), then at every equilibrium point there exists a neighborhood of some finite size in which there is at least one of In example applications, a nonlinear coordinate transform to a normal form can clearly separate these three manifolds. A web service currently undertakes the necessary computer algebra for a range of finite-dimensional systems.