反应网络中不稳定性和分岔的符号化搜索
Symbolic hunt of instabilities and bifurcations in reaction networks
AI总结：
本文通过符号化处理反应网络雅可比矩阵，利用子选择刻画特征多项式系数符号变化，提供稳定性继承与纯虚特征值条件，并证明米氏动力学下可实现任意符号实现。
摘要：
在大型参数系统中定位分岔仍然是一个挑战，通常需要将严格准则与非正式直觉相结合。基于这一动机，我们对具有一般动力学的反应网络的雅可比矩阵进行符号化处理。具体而言，我们将反应速率的任何非零偏导数视为自由的正符号。主要工具是子选择（Child-Selections）：将物种$m$关联到其作为反应物参与的反应$j$的单射映射。首先，我们采用柯西-比内（Cauchy-Binet）分析，并用子选择来结构化地表达雅可比矩阵特征多项式的任何系数。特别地，我们完全刻画了任何系数的符号变化。其次，我们证明了雅可比矩阵的（不）稳定性可由子选择所识别的更简单子矩阵的（不）稳定性继承。第三，我们为雅可比矩阵的纯虚特征值提供了充分条件，暗示了霍普夫分岔和振荡行为。所有条件均以由子选择所识别的整数化学计量子矩阵的符号表示，且不需要任何赫尔维茨型计算。最后，我们关注具有米氏（Michaelis-Menten）动力学的系统，并证明通过适当选择动力学常数，可以在固定平衡点实现雅可比矩阵的任何符号实现。
英文摘要：
The localization of bifurcations in large parametric systems is still a challenge where the combination of rigorous criteria and informal intuition is often needed. With this motivation, we address symbolically the Jacobian matrix of reaction networks with general kinetics. More specifically, we consider any nonzero partial derivative of a reaction rate as a free positive symbol. The main tool are the Child-Selections: injective maps that associate to a species $m$ a reaction $j$ where $m$ participates as reactant. Firstly, we employ a Cauchy-Binet analysis and we structurally express any coefficient of the characteristic polynomial of the Jacobian in terms of Child-Selections. In particular, we fully characterize sign-changes of any of the coefficients. Secondly, we prove that the (in)stability of the Jacobian is inherited from the (in)stability of simpler submatrices identified by the Child-Selections. Thirdly, we provide sufficient conditions for purely imaginary eigenvalues of the Jacobian, hinting at Hopf bifurcation and oscillatory behavior. All conditions are in terms of signs of integer stoichiometric submatrices identified by the Child-Selections {and do not require any Hurwitz-type computaton}. Finally, we focus on systems endowed with Michaelis-Menten kinetics and we show that any symbolic realization of the Jacobian matrix can be achieved at a fixed equilibrium by a proper choice of the kinetic constants.
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