大维度非线性生化振荡器的分析方法

Analytic procedures for large dimention nonlinear biochemical oscillators.

作者信息

Rapp P

出版信息

Biosystems. 1975 Jul;7(1):92-100. doi: 10.1016/0303-2647(75)90047-7.

Abstract

An analytic method is presented which can be used to determine if the following system of nonlinear differential equations has periodic solutions x1 = h(xn)-b1x1 xj = gj-1xj-1-bjxj j = 2, ... n A systematic dual input describing function procedure is given for constructing a function of the reaction constants R, where if R greater than 1 a periodic solution exists and if R smaller than 1 there is no periodic solution. The form of R constructed generalizes immediately to an arbitrarily large dimension. The method generalizes to cover systems displaying hysteresis kinetics, systems subject to chemical noise, and systems containing delay components. The method has been applied to a well known biochemical problem where h(xn)-k/(1 + alphaxnrho). For rho = 1, for all n, there are no stable limit cycles such that xj(t) greater than O, t larger than or equal to O. For rho = 2,n larger than or equal to 8 it is possible to construct a parameter set such that stable oscillations appear.

摘要

提出了一种分析方法，可用于确定以下非线性微分方程组是否具有周期解：

(x_1 = h(x_n) - b_1x_1)

(x_j = g_{j - 1}x_{j - 1} - b_jx_j)，(j = 2, \cdots, n)

给出了一种系统的双输入描述函数方法，用于构建反应常数(R)的函数，其中如果(R\gt1)，则存在周期解；如果(R\lt1)，则不存在周期解。所构建的(R)的形式可立即推广到任意大的维度。该方法可推广到涵盖具有滞后动力学的系统、受化学噪声影响的系统以及包含延迟成分的系统。该方法已应用于一个著名的生化问题，其中(h(x_n) = k / (1 + \alpha x_n^{\rho}))。对于(\rho = 1)，对所有(n)，不存在稳定极限环使得(x_j(t) \gt 0)，(t \geq 0)。对于(\rho = 2)，(n \geq 8)，有可能构建一组参数使得出现稳定振荡。