Analytic procedures for large dimention nonlinear biochemical oscillators

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.