Abstract

A computational approach to detecting Andronov–Hopf bifurcations in polynomial systems of ordinary differential equations depending on parameters is proposed. It relies on algorithms of computational commutative algebra based on the Groebner bases theory. The approach is applied to the investigation of two models related to the double phosphorylation of mitogen-activated protein kinases, a biochemical network that occurs in many cellular pathways. For the models, we analyze the roots of the characteristic polynomials of the Jacobians in a steady state and prove that Andronov–Hopf bifurcations are absent for biochemically relevant values of parameters. We also performed a search for algebraic invariant subspaces in the systems (which represent “weak” conservations laws) and find all subfamilies admitting linear invariant subspaces. The search is done using the Darboux method. That, is we look for Darboux polynomials and cofactors as polynomials with undetermined coefficients and then determine the coefficients using the algorithms of the elimination theory.

中文翻译：

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一些生化模型的定性研究

摘要

提出了一种根据参数检测常微分方程多项式系统中Andronov-Hopf分支的计算方法。它依赖于基于Groebner基理论的计算可交换代数算法。该方法适用于研究与促分裂原活化蛋白激酶的双重磷酸化有关的两个模型，这是一种发生在许多细胞途径中的生化网络。对于模型，我们分析了稳态条件下雅各宾派特征多项式的根，并证明对于参数的生化相关值不存在Andronov-Hopf分叉。我们还搜索了系统中的代数不变子空间（表示“弱”守恒律），并找到了所有允许线性不变子空间的子族。使用Darboux方法进行搜索。那就是我们寻找Darboux多项式和辅助因子作为具有不确定系数的多项式，然后使用消除理论的算法确定系数。