Study on the Compatibility of Multi-Bifurcations by Simulations of Pattern Formation

Bifurcation is considered the main mathematical mechanism for the formation of spatial self-organizing patterns in many studies. When bifurcation was initially defined, only the transition of the system from stable state to unstable state or from uniform state to non-uniform state was concerned, without considering the compatibility of each bifurcation. The coupling phenomenon between bifurcations can be deduced from the characteristics of bifurcation pattern and pattern type. In this paper, we choose a classical wind-sand vegetation model as an example to study the compatibility between bifurcations. Firstly we transformed the vegetation-sand model into a discrete model to study its fixed point and stability analysis. Then we researched the dynamics of the three bifurcations of Flip bifurcation, Neimark-Sacker bifurcation, Turing bifurcation and their mutual coupling. The numerical simulation of the parameters of the three kinds of bifurcations is carried out, and the bifurcation points are simulated on the basis of bifurcation conditions. The simulation results show that both Flip bifurcation, Neimark-Sacker bifurcation and Turing bifurcation can produce complex vegetation patterns, and the coupling effects of any two types of bifurcation can also produce self-organization. The results of this paper will provide support for the improvement of bifurcation definition in the future work.