Invariant submanifold

The common boundary of basins of attraction of these two fixed points is a 1- dimensional line, which is also an invariant submanifold for the renormalization flows. This line has one attractive fixed point ... 

The General Properties of Invariant Submanifolds on Hamiltonian Systems... 

A smooth function / is called a particular integral of a field t if one ( ) of its level surfaces is invariant with respect to the field t (in particular, with respect to the flow induced by the field). If this level has the form Mq = (/ = 0), then the indicated condition will be written as v = (/) (/=o) = 0. In the case of the general position, Af is an invariant submanifold of codimension 1. More generally, a submanifold L of codimension A in Af given by the equations L = (/i = 0,..., / . = 0) is called invariant (with respect to the field v) if u(/ ) l = 0. In particular, if several functionally independent first integrals /i,..., /jb of the system v are given, then each submanifold of the form... 

The Liouville theorem asserts that each invariant compact submanifold in (for an integrable system) is a torus. Nekhoroshev and Kozlov have noticed that one may single out the case where a single isolated invariant submanifold I = (/i =. ../ +fc = 0)inM2 is a torus. To this end, one should require that on the submanifold L we have the relations ... 

It is clear that this proposition is similar to the assertions proved above for the family invariant submanifolds. 

The special form of the invariant submanifold li(k) in Eq. (2.3) is due to the explicit existence of fast-reacting radical species. Well known (Maas and Pope [1], Lam [2]) is the existence of invariant manifolds also in reactions systems without simple separation of radical (fast) and nonradical (slow) chemical species, but with a complex hierarchy of reactions at very different time scales. In this situation we can at least suppose the existence of an invariant submanifold U C in concentration space representing the slow... 

Imagine for simplicity at first a reaction system with kinetic equations (2.2) and with clear separation x = (y, z) of slow (y) and fast (z) chemical species. At least locally we can solve the conditions Eq. (2.3) for the invariant submanifold in the dynamic space resulting in an explicit representation z = (y, k) of the radical concentrations. The kinetic equations reduce themselves into a low-dimensional dynamical system describing the effective kinetics of the system... 

Concerning the dynamics within the invariant submanifold ii(k) a local variation k k + Ak in the parameters is redundant, iff the right hand side in Eq. (3.1) remains unchanged. In a linear approximation we conclude, that the (linear) subspace... 

In the general case, when there is no separation of slow and fast species we have at least local representations Eqs. (2.4) of an invariant submanifold U in concentration space. A local parameter variation k k + Ak. is redundant, iff the corresponding variation of the kinetic equations f is orthogonal to the tangent space Tit, i.e. 

In order to illustrate the theoretical considerations above we analyse the existence of invariant submanifolds in concentration space and of effective parameter manifolds on the basis of the well-known reaction system... 

If we assume the radical species of this reaction as fast components, the invariant submanifold U is characterized by the equations d Bi]/di = d[H]/dt = 0. This simple case allows for the explicit representation of the fast concentrations [Br] and [H] resulting in the following equations for the effective kinetics of this reaction, d Bi2]/dt = d R2]/dt = — d[HBr]/dt, and... 

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