一类生化反应方程的分枝

BIFURCATIONS FOR A EQUATION OF BIOCHEMICAL REACTION

文[1]建立了一类生化反应方程E_(α,β),该文借助于计算机作了极限环的一些数值计算。[2]讨论了E_(σ,β)的极限环的不存在性,存在性和唯一性。本文弄清楚了E_(α,β)(α≥0,β>0)的各种分枝,包括高阶奇点分枝,各阶Hopf分枝,同宿轨道分枝,半稳定环分枝。如果还假定至多只有2个极限环,刚半稳定环分枝还是唯一的(详见定理A)。

In the paper [1], a equation Eα,β of biochemical reaction was established and some numeral computations for the limit cycles were obtained. In [2], the non-existence, existence and unique-ness of the limit cycles of the E0,β were considered. In this paper, we obtain the all bifurcations of Eα,β(a≥0,β≥0) including the higher order singular piont bifurcations, Hopf bifurcations, ho-moclinic orbit bifurcations and semi-stability limit cycles bifurcations. The semi-stability limit cycles bifurcation is unique if the system has at most two limit cycles (see Theorem A).