Unstable subspace

The behavior of orbits of P near a fixed point x can be described in the case where x is a hyperbolic fixed point, that is, when no eigenvalue (multiplier) of the Jacobian of P at x has modulus equal to 1. In this case there exist (local) stable and unstable manifolds M (x) and M (x) (respectively) containing the point x which are tangent to the stable (resp. unstable) subspace of the Jacobian of P at x. (The stable (unstable) subspace... 

In the (a, 6)-parameter plane, find the transition boundary saddle-focus for the origin, and equations for its linear stable and unstable subspaces. Detect the curves in the parameter plane that correspond to the vanishing of the saddle value a of the equilibrium state at the origin. Find where the divergence of the vector field at the saddle-focus vanishes. Plot the curves found in the (a 6)-plane. ... 

This is exactly the autonomous linearized Hamiltonian (7), the dynamics of which was discussed in detail in Section II. One therefore finds the TS dividing surface and the full set of invariant manifolds described earlier one-dimensional stable and unstable manifolds corresponding to the dynamics of the variables A<2i and APt, respectively, and a central manifold of dimension 2N — 2 that itself decomposes into two-dimensional invariant subspaces spanned by APj and AQj. However, all these manifolds are now moving manifolds that are attached to the TS trajectory. Their actual location in phase space at any given time is obtained from their description in terms of relative coordinates by the time-dependent shift of origin, Eq. (42). 

At Da = 0.4 (Fig. 4.29(b)), the two saddle points from the pure vertices move into the composition triangle. The stable node from the 1,4-BD vertex moves to the kinetic azeotrope at x = (0.0328, 0.6935). Pure water and pure THF now become stable nodes. The unstable node between water and THF remains unmoved, and forms two separatrices with the two saddle points. Thereby, the whole composition space is divided into three subspaces which have each a stable node, namely pure water, pure THF and the kinetic azeotrope. 

It is well known that the stable manifold A (x, 0) of a hyperbolic, unstable rest point (x, 0) has Lebesgue measure zero. This follows from Sard s theorem (see Appendix E) and the fact that the stable manifold is the image of a smooth one-to-one map of into K" x K ", where /, is the dimension of the stable subspace of the linearization of (F.l) about (x, 0) and consequently li[Pg.296]

One further observation must be made about the loss of spin symmetry in the Cl vector in the iterative diagonalization of the Hamiltonian. Even very slight deviations from (117), such as might occur from roundoff errors, become magnified in subsequent iterations and cause the iteration procedure to become numerically unstable because precise adherence to (117) is assumed if any of the Ms — 0 simplifications just described. If necessary, these difficulties can be avoided by explicitly enforcing the spin symmetry of any new vector in the subspace expansion. In this respect it is important to modify the diagonal elements of the Hamiltonian in the preconditioner for the subspace iteration method, as already discussed in section 3.2. 

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