Adamec, Ladislav Kinetical systems – local analysis. (English) Zbl 0938.34041 Appl. Math., Praha 43, No. 2, 111-117 (1998). The author deals with systems of the form \(\dot {y}= A G(y),\) where \(A=C-C',\) \(C=(c_{i j})\) and \(C'=(c'_{i j})\) are \(n\times m\)-matrices whose elements are nonnegative integers such that \(0<L=\operatorname {rank}(A)<n\) and \(\det (A(1,\dots ,L))\neq 0\) holds for the corresponding principal submatrix \(A(1,\dots ,L)\) of \(A,\) \(G\) is an \(m\)-vector-valued function defined on \(\mathbb R\times \mathbb R^n\) by \[ G_j(t,y) := -r_j(t) \prod _{i=1}^n y_i^{c_{i j}} +d_j(t) \prod _{i=1}^n y_i^{c'_{i j}}, \quad j=1,\dots ,m, \] and \(r_j,d_j:\mathbb R\mapsto [0,\infty),\) \(j=1,\dots ,m,\) are continuous functions such that \[ \prod _{i=1}^{L}\left [\frac {d_i(t)}{r_i(t)}\right ]{z_{i j}} =\frac {d_j(t)}{r_j(t)}, \quad j=1,\dots ,m, \] and \(\operatorname {col}_j(A)=\sum _{i=1}^L \operatorname {col}_i(A) z_{i j}\) holds for \(j=1,\dots ,m.\) Such systems are called detailed balanced kinetic systems and are often used in chemistry and biology. It is assumed that there exists at least one \(d\times n\)-matrix \(U\) of the rank \(d=n-L\) with nonnegative elements \(u_{i j}\) and such that \(\sum _{i=1}^d u_{i j}>0\) for \(j=1,\dots ,n\) and there exists at least one vector \(b\in \mathbb R^d\) with positive elements such that the linear equation \(Uc=b\) possesses a solution \(c\in \mathbb R^d\) with nonnegative elements and one of such matrices \(U\) and one of such vectors \(b\) are chosen fixed. The author then gives sufficient conditions assuring that any stationary point \(y\in H=\{y: Uy=b\}\) of the given detailed balanced system is asymptotically stable with respect to \(H\). Reviewer: M.Tvrdý (Praha) Cited in 1 Document MSC: 34D05 Asymptotic properties of solutions to ordinary differential equations 80A30 Chemical kinetics in thermodynamics and heat transfer Keywords:ordinary differential equations; asymptotic properties; chemical kinetics PDFBibTeX XMLCite \textit{L. Adamec}, Appl. Math., Praha 43, No. 2, 111--117 (1998; Zbl 0938.34041) Full Text: DOI EuDML References: [1] L. Adamec: Kinetical systems. to appear. · Zbl 0938.34041 · doi:10.1023/A:1023262917306 [2] J. Kurzweil: Ordinary Differential Equations. Elsevier, Amsterodam-Oxford-New York-Tokyo, 1986. · Zbl 0667.34002 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.