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invariant manifold : ウィキペディア英語版
invariant manifold

In dynamical systems, a branch of mathematics, an invariant manifold is a topological manifold that is invariant under the action of the dynamical system.〔Hirsh M.W., Pugh C.C., Shub M., Invariant Manifolds, Lect. Notes. Math., 583, Springer, Berlin — Heidelberg, 1977
〕 Examples include the slow manifold, center manifold, stable manifold, unstable manifold, subcenter manifold and inertial manifold.
Typically, although by no means always, invariant manifolds are constructed as a 'perturbation' of an invariant subspace about an equilibrium.
In dissipative systems, an invariant manifold based upon the gravest, longest lasting modes forms an effective low-dimensional, reduced, model of the dynamics.
〔A. J. Roberts. The utility of an invariant manifold description of the evolution of a dynamical system. SIAM J. Math. Anal., 20:1447–1458, 1989. http://locus.siam.org/SIMA/volume-20/art_0520094.html〕
==Definition==
Consider the differential equation dx/dt = f(x),\ x \in \mathbb R^n,
with flow x(t)=\phi_t(x_0) being the solution of the differential equation with x(0)=x_0.
A set S \subset \mathbb R^n is called an ''invariant set'' for the differential equation if, for each x_0 \in S, the solution t \mapsto \phi_t(x_0), defined on its maximal interval of existence, has its image in S. Alternatively, the orbit
passing through each x_0 \in S lies in S. In addition, S is called an ''invariant manifold'' if S is a manifold.
〔C. Chicone. Ordinary Differential Equations with Applications, volume 34 of Texts in Applied Mathematics. Springer, 2006, p.34〕

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