Smooth attractive invariant manifold

Starting with any (x, y), a trajectory of system (12.4.8) converges typically to an attractor of the fast system corresponding to the chosen value of x. This attractor may be a stable equilibrium, or a stable periodic orbit, or of a less trivial structure — we do not explore this last possibility here. When an equilibrium state or a periodic orbit of the fast system is structurally stable, it depends smoothly on x. Thus, we obtain smooth attractive invariant manifolds of system (12.4.8) equilibrium states of the fast system form curves Meq and the periodic orbits form two-dimensional cylinders Mpo, as shown in Fig. 12.4.6. Locally, near each structurally stable fast equilibrium point, or periodic orbit, such a manifold is a center manifold with respect to system (12.4.8). Since the center manifold exists in any nearby system (see Chap. 5), it follows that the smooth attractive invariant manifolds Meqfe) and Mpo( ) exist for all small e in the system (12.4.7) [48]. 

Theorem 12.3. (Afraimovichr-Shilnikov [3, 6]) If the global unstable set of the saddle-node L is a smooth compact manifold a torus or a Klein bottle) at fi = Oy then a smooth closed attractive invariant manifold 7 (fl torus or a Klein bottle, respectively) exists for all small fi. 

The main problem in the solution of non-linear ordinary and partial differential equations in combustion is the calculation of their trajectories at long times. By long times we mean reaction times greater than the time-scales of intermediate species. This problem is especially difficult for partial differential equations (pdes) since they involve solving many dimensional sets of equations. However, for dissipative systems, which include most applications in combustion, the long-time behaviour can be described by a finite dimensional attractor of lower dimension than the full composition space. All trajectories eventually tend to such an attractor which could be a simple equilibrium point, a limit cycle for oscillatory systems or even a chaotic attractor. The attractor need not be smooth (e.g., a fractal attractor in a chaotic system) and is in some cases difficult to compute. However, the attractor is contained in a low-dimensional, invariant, smooth manifold called the inertial manifold M which locally attracts all trajectories exponentially and is easier to find [134,135]. It is this manifold that we wish to investigate since the dynamics of the original system, when restricted to the manifold, reduce to a lower dimensional set of equations (the inertial form). The inertial manifold is, therefore, a useful notion in the field of mechanism reduction. 

Let us consider next the bifurcation of the saddle-node periodic orbit L in the case where the unstable manifold is a Klein bottle, as depicted in Fig. 12.3.1, i.e. when the essential map has degree m = -1. By virtue of Theorem 12.3, if is smooth, then a smooth invariant attracting Klein bottle persists when L disappears. In its intersection with a cross-section So, the flow on the Klein bottle defines a Poincare map of the form (see (12.2.26))... 

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