The Center Manifold Theorem
The center manifold theorem states that if ƒ is Cr (r times continuously differentiable), then at every equilibrium point there is a unique Cr stable manifold, a unique Cr unstable manifold, and a (not necessarily unique) Cr−1 center manifold.
In example applications, a nonlinear coordinate transform to a normal form (mathematics) can clearly separate these three manifolds. A web service currently undertakes the necessary computer algebra for a range of finite dimensional systems.
There is theory for the existence and relevance of center manifolds in infinite dimensional dynamical systems. The general theory currently only applies when the center manifold itself is of finite dimension. However, some applications, such as to shear dispersion, can justify and construct an infinite dimensional center manifold.
Read more about this topic: Center Manifold
Famous quotes containing the words center, manifold and/or theorem:
“Children can’t be a center of life and a reason for being. They can be a thousand things that are delightful, interesting, satisfying, but they can’t be a wellspring to live from. Or they shouldn’t be.”
—Doris Lessing (b. 1919)
“A large city cannot be experientially known; its life is too manifold for any individual to be able to participate in it.”
—Aldous Huxley (1894–1963)
“To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.”
—Albert Camus (1913–1960)