Singularities for projections of integral manifolds with applications to control and observation problems

An integral manifold for a system of differential equations is a manifold such that any solution of the equations which has a point on it is entirely contained on it. The method of averaging establishes the existence of such a manifold for a system which is a perturbation of an autonomous system with a periodic orbit. The existence of the manifold is established here under more general hypotheses, namely for perturbations which are ‘integrally small’. The method differs from the original method of Bogolyubov and Mitropolskii and operates directly with the individual solutions. This is made possibly by the use of an appropriate norm, and is equivalent to solving the partial differential equation which occurs in work by Moser and Sacker by the method of characteristics rather than by the introduction of an artificial viscosity term. Moreover, detailed smoothness properties of the manifold are obtained. For periodic perturbations the integral manifold is a torus and these smoothness properties are just sufficient to permit the application of Denjoy's theorem.

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Generic integral manifolds for weight two period domains

Transactions of the American Mathematical Society ◽

10.1090/s0002-9947-04-03485-3 ◽

2004 ◽

Vol 356 (6) ◽

pp. 2241-2249

Author(s):

James A. Carlson ◽

Domingo Toledo

Keyword(s):

Integral Manifolds

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Nonlocal integral manifolds and decoupling of nonlinear parabolic systems

Lecture Notes in Mathematics - Global Analysis - Studies and Applications IV ◽

10.1007/bfb0085949 ◽

1990 ◽

pp. 101-108

Author(s):

V. A. Sobolev

Keyword(s):

Parabolic Systems ◽

Nonlinear Parabolic Systems ◽

Nonlinear Parabolic ◽

Integral Manifolds

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Two Conjectures of G.D. Birkhoff

International Astronomical Union Colloquium ◽

10.1017/s0252921100073061 ◽

1999 ◽

Vol 172 ◽

pp. 439-440

Author(s):

Christopher K. Mccord ◽

Kenneth R. Meyer

Keyword(s):

Angular Momentum ◽

Integral Manifold ◽

Center Of Mass ◽

Angular Momentum Vector ◽

Momentum Vector ◽

Algebraic Set ◽

Special Values ◽

Integral Manifolds ◽

Energy Center ◽

Three Body

The spatial (planar) three-body problem admits the ten (six) integrals of energy, center of mass, linear momentum and angular momentum. Fixing these integrals defines an eight (six) dimensional algebraic set called the integral manifold, 𝔐(c, h) (m(c, h)), which depends on the energy level h and the magnitude c of the angular momentum vector. The seven (five) dimensional reduced integral manifold, 𝔐R(c, h) (mR(c, h)), is the quotient space 𝔐(c, h)/SO2 (m(c, h)/SO2) where the SO2 action is rotation about the angular momentum vector. We want to determine how the geometry or topology of these sets depends on c and h. It turns out that there is one bifurcation parameter, ν = −c2h, and nme (six) special values of this parameter, νi, i = 1, …, 9.At each of the special values the geometric restrictions imposed by the integrals change, but one of these values, ν5, does not give rise to a change in the topology of the integral manifolds 𝔐(c, h) and 𝔐R(c, h). The other eight special values give rise to nine different topologically distinct cases. We give a complete description of the geometry of these sets along with their homology. These results confirm some conjectures and refutes several others.

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Efficient decomposition of singularly perturbed systems

Mathematical Modelling of Natural Phenomena ◽

10.1051/mmnp/2019023 ◽

2019 ◽

Vol 14 (4) ◽

pp. 410 ◽

Cited By ~ 3

Author(s):

Vladimir Sobolev

Keyword(s):

Enzyme Kinetics ◽

Singularly Perturbed ◽

Singularly Perturbed Systems ◽

Differential Systems ◽

Integral Manifolds ◽

Perturbed Systems ◽

Method Of Integral Manifolds

The problem of the decomposition of singularly perturbed differential systems by the method of integral manifolds is studied and the application of the method to the problems of enzyme kinetics is considered.

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