In this paper, stochastic Hopf–Hopf bifurcation of the discrete coupling logistic system with symbiotic interaction is investigated. Firstly, orthogonal polynomial approximation of discrete random function in the Hilbert spaces is applied to reduce the discrete coupling logistic system with random parameter to the deterministic equivalent system. Then, it is concluded that Hopf–Hopf bifurcation exists in the equivalent deterministic system according to the principle of algebraic criteria. Numerical simulations show that the bifurcation critical value varies with the intensity of random parameter, and Hopf–Hopf bifurcation and period-doubling bifurcation behavior exist. In particular, Hopf–Hopf bifurcation can be drift with the change of random intensity, and frequency locking phenomenon occurs in the stochastic system.

本文研究了具有共生相互作用的离散耦合逻辑系统的随机Hopf-Hopf分岔。首先，应用希尔伯特空间中离散随机函数的正交多项式逼近，将具有随机参数的离散耦合逻辑系统简化为确定性等价系统。然后得出结论，根据代数准则的原理，Hopf-Hopf分岔存在于等价的确定性系统中。数值模拟表明，分叉临界值随随机参数强度的变化而变化，并且存在Hopf-Hopf分叉和周期倍增的分叉行为。特别是，Hopf–Hopf分叉会随随机强度的变化而漂移，并且在随机系统中会发生频率锁定现象。