There is a systematic approach to the computation of quasi-steady state reductions, employing the classical theory of Tikhonov and Fenichel, rather than the commonly used ad-hoc method. In the present paper we discuss the relevant case that the local slow manifold (in the asymptotic limit) is a vector subspace, give closed-form expressions for the reduction and compare these to the ones obtained by the customary method. As it turns out, investment of more theory pays off in the form of simpler reduced systems. Applications include a number of standard models for reactions in biochemistry, for which the reductions are extended to the fully reversible setting. In a short final section we illustrate by example that a QSS assumption may be erroneous if the hypotheses for Tikhonov's theorem are not satisfied.

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We determine the limit sets of a system modelling suicide substrate kinetics, and show that a result by Tatsunami et al. (Biochim Biophys Acta 662:226-235, 1981), derived under additional quasi-steady state assumptions, holds generally.

We discuss real and complex polynomial vector fields and polynomially nonlinear, input-affine control systems, with a focus on invariant algebraic varieties. For a given real variety we consider the construction of polynomial ordinary differential equationṡ x = f (x) such that the variety is invariant and locally attracting, and show that such a construction is possible for any compact connected component of a smooth variety satisfying a weak additional condition. Moreover we introduce and study natural controlled invariant varieties (NCIV) with respect to a given input matrix g, i.e. varieties which are controlled invariant sets ofẋ = f (x) + g(x)u for any choice of the drift vector f . We use basic tools from commutative algebra and algebraic geometry in order to characterize NCIV's, and we present a constructive method to decide whether a variety is a NCIV with respect to an input matrix. The results and the algorithmic approach are illustrated by examples.