Slow invariant manifold

The corresponding structure of fast-slow time separation in phase space is not necessarily a smooth slow invariant manifold, but may be similar to a "crazy quilt" and may consist of fragments of various dimensions that do not join smoothly or even continuously. 

If the Jacobian of the kinetic system of differential equations has M eigenvalues with negative real parts that are much larger (i.e. more negative) than the other Re (Ay) values, then the solution is quickly attracted onto an (A/j-M)-dimensional surface Q, which is called the slow invariant manifold (SIM) (Fenichel 1979). Denote 7VJ2 and TyF as two subspaces, where the slow subspace TyFi is the space of movement on J2 and the fast subspace TyF contains the directions of fast approaches to the manifold. These spaces can be spanned by the following basis vectors TyF = span(a /= ,..., M) and TyQ = span(a / = M +1,..., N ). Vectors a, form matrices A,. = [ai, a2,..., m and Aj = [Am+i, m+2> avJ- On this basis, the right-hand side of the kinetic ODE can be decomposed as... 

Mengers, J.D., Powers, J.M. One-dimensional slow invariant manifolds for fully coupled reaction and micro-scale diffusion. SIAM J. Appl. Dyn. Syst. 12, 560-595 (2013)... 

Bykov, V., Gol dshtein, V. Fast and slow invariant manifolds in chranical kinetics. Comput. Math. Appl. 65, 1502-1515 (2013)... 

Roussel, M.R., Tang, T. The functional equation truncation method for approximating slow invariant manifolds a rapid method for computing intrinsic low-dimensional manifolds. J. Chem. Phys. 125, 214103 (2006)... 

The quasi-invariant manifold Msiow c M composed of slow states ( i)siow This manifold is quasi-invariant with respect to the time evolu-... 

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... 

Now, all the tools of the standard singular perturbation theory can be applied to the analysis of the system. These tools include its decomposition to slow and fast subsystems and reduction of the near steady-state dynamics to dynamics on invariant slow manifolds (slow curves in the present model). The zero approximation, e = 0 k 00), of the slow invariant curve is given... 

Thus, a trajectory of the system (12.4.7) for small e behaves in the following way during some finite time it comes to a small neighborhood of one of the invariant manifolds Mgq or Mpo, so that its a -coordinate is nearly fixed. Then, it drifts along the invariant manifold so that it corresponds to a slow change of x. As for the initial system (12.4.6) one observes a jump of the 2/-variables to the invariant manifold followed by a finite speed motion in the ar-variable. In addition, if this is a manifold of a fast periodic orbit, then we have a fast circular motion of the y-variables on the manifold, as depicted in Fig. 12.4.7. 

The assumption that an invariant slow manifold of low dimension exists in the state space yields... 

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