Stochastic Hopf-Hopf bifurcation of two-species discrete coupling logistic system with symbiotic interaction

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.