# Integral manifold

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A set of points of the phase space ( -space) of the system (*)

filled by the integral curves of this system (cf. Integral curve), defined for all and forming a manifold in -space. The dimension of the section of by the plane is usually called the dimension of the integral manifold . In the definition of an integral manifold, the requirement that it be a manifold is sometimes replaced by the requirement that the set be representable analytically by an equation with a function defined for all in and in some domain and possessing a specific smoothness in , for . The integral manifold is then called -dimensional and of the same smoothness as the function .

Examples. An integral curve of a periodic solution of the system (*), that is, a periodic integral curve; the family of integral curves of the system (*) formed by a family of quasi-periodic solutions of (*), filling an -dimensional torus in the -space when , that is, an -dimensional toroidal integral manifold; etc.

The integral manifolds that have been most extensively studied are the toroidal manifolds, that is, sets that are tori for any fixed . These manifolds are widely encountered in systems of type (*) describing oscillatory processes.

How to Cite This Entry:
Integral manifold. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Integral_manifold&oldid=17392
This article was adapted from an original article by A.M. Samoilenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article