Bifurcations and Chaos of a Discrete Mathematical Model for Respiratory Process in Bacterial Culture  

作　　者：Xiang-ling FU Jin DENG 

机构地区：[1]School of Mathematics,Hunan University of Science and Technology [2]Department of Mathematics and Physics,Hunan Institute of Engineering

出　　处：《Acta Mathematicae Applicatae Sinica》2014年第4期871-886,共16页应用数学学报（英文版）

基　　金：Supported by the National Natural Science Foundation of China(10671063 and 10801135);the Scientific Research Foundation of Hunan Provincial Education Department(09C255)

摘　　要：The discrete mathematical model for the respiratory process in bacterial culture obtained by Euler method is investigated. The conditions of existence for flip bifurcation and Hopf bifurcation are derived by using center manifold theorem and bifurcation theory, condition of existence of chaos in the sense of Marotto＇s definition of chaos is proved. The bifurcation diagrams, Lyapunov exponents and phase portraits are given for different parameters of the model, and the fractal dimension of chaotic attractor was also calculated. The numerical simulation results confirm the theoretical analysis and also display the new and complex dynamical behaviors compared with the continuous model. In particular~ we found that the new chaotic attractor, and new types of two or four coexisting chaotic attractors, and two coexisting invariant torus.The discrete mathematical model for the respiratory process in bacterial culture obtained by Euler method is investigated. The conditions of existence for flip bifurcation and Hopf bifurcation are derived by using center manifold theorem and bifurcation theory, condition of existence of chaos in the sense of Marotto＇s definition of chaos is proved. The bifurcation diagrams, Lyapunov exponents and phase portraits are given for different parameters of the model, and the fractal dimension of chaotic attractor was also calculated. The numerical simulation results confirm the theoretical analysis and also display the new and complex dynamical behaviors compared with the continuous model. In particular~ we found that the new chaotic attractor, and new types of two or four coexisting chaotic attractors, and two coexisting invariant torus.

关 键 词：MAP flip bifurcation Hopf bifurcation Marotto＇s chaos CHAOS 

分 类 号：O175[理学—数学]