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A singular perturbation result for a system of ordinary differential equations. (English) Zbl 0532.34040

The author proves a flow-invariance result for a system of ordinary differential equations: \(x'(t)=f(t,x(t))\), \(x(0)=x^ 0\), \(x\in {\mathbb{R}}^ n\), \(t\in {\mathbb{R}}_+\) relative to a class of polygonal domains.
He applies his result to study the behavior of the solution \(x(\mu\),\(\cdot)\) of the system \((S_{\mu}) \mu x'(t)=f(t,x(t))\), \(x(0)=x^ 0(\mu)\) when the parameter \(\mu\) converges to \(\omega,\omega \in {\mathbb{R}}_+^ n\) with \(\mu>\omega (\Leftrightarrow \mu_ i>\omega_ i\), \(i=1,2,...,n)\). With convenient hypotheses about the system \((S_{\mu})\) it is proved that the generalized sequence \(x(\mu\),\(\cdot)\) converges to \(x(\omega\),\(\cdot)\) in a decreasing way for the compact topology on \({\mathbb{R}}_+\). This result may be compared to a similar one due to Tikhonov.
An application of this main theorem is done for the system of enzymatic reactions theory \[ x'=-x+(x+a)y,\quad x(0)=x_ 0(\mu),\quad \mu y'=x- (x+b)y,\quad y(0)=(\mu),\quad 0<a<b \] when \(\mu\) converges to 0. In fact the author investigates sufficient conditions about initial data in order that the solutions of \(S(\mu)\) converge to a solution of the degenerate system: \(u'=-x+(x+a)y\), \(0=x-(x+b)y\), \(x(0)=\bar u_ 0\), \(y(0)=\hat y_ 0\). He gives three conditions about \(x_ 0(\mu)\) and \(y_ 0(\mu)\) and remarks that the consequences of the first two of them which can be deduced: \(\bar u_ 0-(xAd-+b)\bar y_ 0=0\), \(0\leq x_ 0(\mu)<b-a \forall \mu>0\) are quite natural. The condition \(u_ 0\in [0,b-a[\) has an experimental justification in many concrete cases. So the author justifies the approximation given in the Michaelis-Menten theory.
Similar applications are suggested for singular perturbations of mechanical type. Moreover a forthcoming paper about \(\mu x'=k(x)\) with k satisfying conditions of functional type is announced.
Reviewer: J.Genet

MSC:

34E15 Singular perturbations for ordinary differential equations
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
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