Asymptotics of Solutions of Differential Equations with Respect to a Small Parameter

Tihonov theory and center manifolds for inhibitory mechanisms in enzyme kinetics

“…In a work in preparation, we are trying to apply these techniques to several mechanisms of phosphorylation and dephosphorylation, including cycles of single and double phosphorylation. (Olomouc, Czech Republic) for their translations of the papers [10][11][12] and some precious clarifications concerning some passages of the papers.…”

“…A systematic study of the qualitative aspects of such singular perturbation problems can be found in a series of papers by Tihonov [10][11][12], written in Russian.…”

“…They are both developed setting the derivative of the intermediate complexes equal to zero. The theoretical justification for this assumption is given by Tihonov's Theorem 2.1 [10][11][12][13]. In Tihonov's framework, the assumptionĊ i = 0 is equivalent to impose the aforementioned "appropriate parameter" (the perturbation parameter, usually denoted by ) equal to zero (see, for example, [14,15].…”

“…It should be remarked that we still lack the theoretical investigation of the validity of the tQSSA in the case of successive reactions -where more parameters appear -but nevertheless it can be generalized to them. In particular, Theorem 2.1, as stated in [11], represents the generalization of Tihonov's Theorem [10,13] [20], the possibility to choose the required parameter has been related to the decoupling of the eigenvalues of the Jacobian matrix of the differential system. In [19] the authors focused on chains of reactions, obtaining a block form for the resulting Jacobian matrix.…”

PI/PID Control for Nonlinear Systems via Singular Perturbation Technique

“…Substitution of (8) into (5) The main qualitative property of the singularly perturbed systems is that: if the equilibrium point of the FMS is stable (exponentially stable), then there exists µ ⋆ > 0s u c ht h a tf o ra l l µ ∈ (0, µ ⋆ ), the trajectories of the singularly perturbed system approximate to the trajectories of the SMS (Hoppensteadt, 1966;Klimushchev & Krasovskii, 1962;Litkouhi & Khalil, 1985;Tikhonov, 1948;1952). This property is important both from a theoretical viewpoint and for practical applications in control system analysis and design, in particular, that will be used throughout the discussed below design methodology for continuous-time or sampled-data nonlinear control systems.…”