论文标题
具有极限周期的化学系统
Chemical systems with limit cycles
论文作者
论文摘要
化学反应网络（CRN）的动力学通常是通过与多项式右侧的普通微分方程（ODE）的系统建模的，这些系统描述了所涉及的化学物质浓度的时间演变。鉴于{\ mathbb n} $中的任意大整数$ k \，我们表明存在一个CRN，因此其ODE模型至少具有至少$ k $稳定的极限周期。只要化学物质的数量与$ k $线性增长，可以用最多二阶的反应来构建这样的CRN。对于具有$ k $稳定极限循环的CRN以及最多的二阶或第七阶动力学的CRN，提出了化学物种数量最少和化学反应数量最少的界限。我们还表明，当化学反应的顺序与$ k $线性增长时，只有两种化学物质的CRN可以具有$ k $稳定的极限周期。
The dynamics of a chemical reaction network (CRN) is often modelled under the assumption of mass action kinetics by a system of ordinary differential equations (ODEs) with polynomial right-hand sides that describe the time evolution of concentrations of chemical species involved. Given an arbitrarily large integer $K \in {\mathbb N}$, we show that there exists a CRN such that its ODE model has at least $K$ stable limit cycles. Such a CRN can be constructed with reactions of at most second order provided that the number of chemical species grows linearly with $K$. Bounds on the minimal number of chemical species and the minimal number of chemical reactions are presented for CRNs with $K$ stable limit cycles and at most second order or seventh order kinetics. We also show that CRNs with only two chemical species can have $K$ stable limit cycles, when the order of chemical reactions grows linearly with $K$.
