Integral manifold

A set $S_t$ of points of the phase space ($(t,x)$-space) of the system

$$\frac{dx}{dt}=X(t,x),\tag{*}$$

filled by the integral curves of this system (cf. Integral curve), defined for all $t\in\mathbf R$ and forming a manifold in $(t,x)$-space. The dimension of the section of $S_t$ by the plane $t=\text{const}$ is usually called the dimension of the integral manifold $S_t$. In the definition of an integral manifold, the requirement that it be a manifold is sometimes replaced by the requirement that the set $S_t$ be representable analytically by an equation

$$x=f(t,C)$$

with a function $f$ defined for all $t$ in $\mathbf R$ and $C=(C_1,\dots,C_m)$ in some domain $D$ and possessing a specific smoothness in $t$, $C$ for $t,C\in\mathbf R\times D$. The integral manifold is then called $m$-dimensional and of the same smoothness as the function $f$.

Examples. An integral curve of a periodic solution of the system \ref{*}, that is, a periodic integral curve; the family of integral curves of the system \ref{*} formed by a family of quasi-periodic solutions of \ref{*}, filling an $m$-dimensional torus in the $x$-space when $t=0$, that is, an $m$-dimensional toroidal integral manifold; etc.

The integral manifolds that have been most extensively studied are the toroidal manifolds, that is, sets $S_t$ that are tori for any fixed $t\in\mathbf R$. These manifolds are widely encountered in systems of type \ref{*} describing oscillatory processes.

References

[1]	N.N. Bogolyubov, "On certain statistical methods in mathematical physics" , L'vov (1945) (In Russian)
[2]	N.N. Bogolyubov, Yu.A. Mitropol'skii, "The method of integral manifolds in non-linear mechanics" , Proc. Internat. Symp. Non-linear Oscillations , 1 , Kiev (1963) pp. 96–154 (In Russian)
[3]	N.N. Bogolyubov, Yu.A. Mitropol'skii, "The method of integral manifolds in the theory of differential equations" , Proc. 4-th All-Union Math. Congress (1961) , 2 , Leningrad (1964) pp. 432–437 (In Russian)
[4]	N.N. [N.N. Bogolyubov] Bogoliubov, Yu.A. [Yu.A. Mitropol'skii] Mitropoliskii, A.M. [A.M. Samoilenko] Samolenko, "The method of accelerated convergence in non-linear mechanics" , Springer (1976) (Translated from Russian) MR407380
[5]	Yu.A. Mitropol'skii, "Problems of the asymptotic theory of nonstationary vibrations" , D. Davey (1965) (Translated from Russian)
[6]	Yu.A. Mitropol'skii, O.B. Lykova, "Integral manifolds in non-linear mechanics" , Moscow (1973) (In Russian)

Comments

Nowadays integral manifolds are usually called invariant manifolds. Basic theorems on the permanence of invariant manifolds under perturbations are: 1) Fenichel's theorem, in case the Lyapunov exponents (cf. Lyapunov characteristic exponent) in the directions transversal to the manifold are larger in absolute value than those in directions parallel to the manifold, cf. [a1]; and 2) the Kolmogorov–Arnol'd–Moser theorem on persistence of quasi-periodic solutions in perturbations of integrable Hamiltonian systems (cf. Integrable system; Hamiltonian system; [a2]).

References

[a1]	M.W. Hirsch, C. Pugh, M. Shub, "Invariant manifolds" , Springer (1977) MR0501173 Zbl 0355.58009
[a2]	V.I. Arnol'd, "Mathematical methods of classical mechanics" , Springer (1978) pp. Appendix 8 (Translated from Russian) Zbl 0692.70003 Zbl 0572.70001 Zbl 0647.70001
[a3]	J. Guckenheimer, P. Holmes, "Non-linear oscillations, dynamical systems, and bifurcations of vector fields" , Springer (1983) MR709768