We study the two-dimensional reduction of the Michaelis-Menten reaction of enzyme kinetics. First, we prove the existence and uniqueness of a slow manifold between the horizontal and vertical isoclines. Second, we determine the concavity of all solutions in the first quadrant. Third, we establish the asymptotic behaviour of all solutions near the origin, which generally is not given by a Taylor series. Finally, we determine the asymptotic behaviour of the slow manifold at infinity. To this end, we show that the slow manifold can be constructed as a centre manifold for a fixed point at infinity.

We study the planar and scalar reductions of the nonlinear Lindemann mechanism of unimolecular decay. First, we establish that the origin, a degenerate critical point, is globally asymptotically stable. Second, we prove there is a unique scalar solution (the slow manifold) between the horizontal and vertical isoclines. Third, we determine the concavity of all scalar solutions in the nonnegative quadrant. Fourth, we establish that each scalar solution is a centre manifold at the origin given by a Taylor series. Moreover, we develop the leading-order behaviour of all planar solutions as time tends to infinity. Finally, we determine the asymptotic behaviour of the slow manifold at infinity by showing that it is a unique centre manifold for a fixed point at infinity.where k 1 , k −1 , and k 2 are the rate constants.

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In this paper, we will develop an iterative procedure to determine the detailed asymptotic behaviour of solutions of a certain class of nonlinear vector differential equations which approach a nonlinear sink as time tends to infinity. This procedure is indifferent to resonance in the eigenvalues. Moreover, we will address the writing of one component of a solution in terms of the other in the case of a planar system. Examples will be given, notably the Michaelis-Menten mechanism of enzyme kinetics.