Chemical Systems with Limit Cycles

by Radek Erban and Hye-Won Kang

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Hilbert's 16th problem asks questions about the number of limit cycles that a planar polynomial system of ODEs can have. The solutions of such ODEs are two functions of time, x(t) and y(t), which can be plotted in the x-y plane as shown in the picture for an example which has four stable limit cycles, plotted using the black dashed lines as the four closed curves (periodic solutions). Other solutions of the same ODEs (plotted using different colours) all approach one of the limit cycles.

This example has been constructed using the approach developed in the paper, where we investigate limit cycles in chemical systems. Chemical systems with N=2 chemical species can be described by planar ODE systems. A lower bound on the maximum number of stable limit cycles in such chemical systems has been proven in the paper. This directly implies a lower bound on Hilbert's number H(n) denoting the maximum number of limit cycles for general planar n-degree polynomial ODE systems. In the paper, we also study more general systems with N>2 chemical species. We construct chemical systems with K stable limit cycles, where K can be arbitrarily large.

About the authors:

Radek Erban is a Professor of Mathematics at University of Oxford. Hye-Won Kang is an Associate Professor at University of Maryland.