7. Geometry of Singular Perturbations: Critical Cases

Asymptotic expansions of slow invariant manifolds and reduction of chemical kinetics models

“…The above asymptotic method for constructing slow integral manifolds in implicit form was proposed several years earlier than [18] (see, e.g., [5] and also [10,12,23]) and is much simpler and more effective than the ILDM construction of an implicit equation. For illustrative purposes, we consider a system with scalar variables х and у.…”

“…Now the results of the previous section are used to compute approximations of the one dimensional slow invariant manifold of system (13) and the equation describing the motion on this manifold. In this case, we have Using formulas (12) and the inverse matrix expression where a = (a 4 + a 5 ) -1 and Δ = x + a 1 + a 2 + a 3 ax 2 , yields an approximate slow invariant manifold where The zero approximation of motion on this manifold is described by the equation …”

“…Now we find a representation of slow motions using Eqs. (12). In this case, we have Using formula (12) and the inverse matrix expression where Δ = K s x 2 + K i x 1 + K s K i , we obtain an approximation of the slow invariant manifold:…”

Black swans and canards in two predator – one prey model

“…Among the invariant manifolds we distinguish the invariant surfaces of slow motions whose dimension is equal to that of the slow subsystem, the so-called slow invariant manifolds. Stability or instability of a slow invariant manifold is defined by stability or instability of its zero-order approximation (ε = 0), the so-called slow surface [26][27][28][29].…”

“…Note that in many problems it is more convenient to find a canard in a parametric form [28,29]. Moreover, for the construction of the asymptotic expansions (2.5) it is assumed that the degenerate equation (23) allows one to find the slow surface explicitly.…”