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:
“Actually being married seemed so crowded with unspoken rules and odd secrets and unfathomable responsibilities that it had no more occurred to her to imagine being married herself than it had to imagine driving a motorcycle or having a job. She had, however, thought about being a bride, which had more to do with being the center of attention and looking inexplicably, temporarily beautiful than it did with sharing a double bed with someone with hairy legs and a drawer full of boxer shorts.”
—Anna Quindlen (b. 1952)
“Odysseus saw the sirens; they were charming,
Blonde, with snub breasts and little neat posteriors,”
—John Streeter Manifold (b. 1915)
“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)