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On the periodic solutions of a differential system in \(\mathbb R^3\). (Sur les solutions périodiques d’un système différentiel de \(\mathbb R^3\).) (French. English summary) Zbl 1085.34026

A conjectured model of an oscillatory chemical reaction is studied qualitatively. If such a model is to be informative to applied readers, the preliminary data are: 1) initial chemical elements; 2) what was observed (measured), how, and at what time-scale; 3) likely chemical model, including intermediate products; 4) if spatial properties are negligible, the corresponding ordinary differential equation (ODE) reaction model; 5) justification of simplifying assumptions (a ‘reasonably’ reliable model of the Briggs-Rauscher reaction consists of over fifty ODEs, containing linear, quadratic and cubic forms with mostly unknown coefficients). No such data are provided by the authors, and no references are given to any.
The model studied is of third order and comprises merely a single reaction constant \(k\), and a single nonlinear function \(f\) and is assumed to be Lipschitzian. The existence proof is based on the Poincaré-Bendixson criterion, expressed in a ‘modernized’ terminology, and supplemented by the notion of a spectral radius (of eigenvalues). It is shown that periodic orbitally stable solutions exist in a finite interval of \(k\).

MSC:

34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
92E20 Classical flows, reactions, etc. in chemistry
34D99 Stability theory for ordinary differential equations
34C60 Qualitative investigation and simulation of ordinary differential equation models
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