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Difference between revisions of "Integral manifold"

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====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> N.N. Bogolyubov,   "On certain statistical methods in mathematical physics" , L'vov (1945) (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> 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)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> 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)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> 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)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> Yu.A. Mitropol'skii,   "Problems of the asymptotic theory of nonstationary vibrations" , D. Davey (1965) (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> Yu.A. Mitropol'skii,   O.B. Lykova,   "Integral manifolds in non-linear mechanics" , Moscow (1973) (In Russian)</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD> <TD valign="top"> N.N. Bogolyubov, "On certain statistical methods in mathematical physics" , L'vov (1945) (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> 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)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> 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)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> 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) {{MR|407380}} {{ZBL|}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> Yu.A. Mitropol'skii, "Problems of the asymptotic theory of nonstationary vibrations" , D. Davey (1965) (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> Yu.A. Mitropol'skii, O.B. Lykova, "Integral manifolds in non-linear mechanics" , Moscow (1973) (In Russian)</TD></TR></table>
  
  
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> M.W. Hirsch,   C. Pugh,   M. Shub,   "Invariant manifolds" , Springer (1977)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> V.I. Arnol'd,   "Mathematical methods of classical mechanics" , Springer (1978) pp. Appendix 8 (Translated from Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> J. Guckenheimer,   P. Holmes,   "Non-linear oscillations, dynamical systems, and bifurcations of vector fields" , Springer (1983)</TD></TR></table>
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> M.W. Hirsch, C. Pugh, M. Shub, "Invariant manifolds" , Springer (1977) {{MR|0501173}} {{ZBL|0355.58009}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> V.I. Arnol'd, "Mathematical methods of classical mechanics" , Springer (1978) pp. Appendix 8 (Translated from Russian) {{MR|}} {{ZBL|0692.70003}} {{ZBL|0572.70001}} {{ZBL|0647.70001}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> J. Guckenheimer, P. Holmes, "Non-linear oscillations, dynamical systems, and bifurcations of vector fields" , Springer (1983) {{MR|709768}} {{ZBL|}} </TD></TR></table>

Revision as of 16:58, 15 April 2012

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.

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
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