一个二维滞后Logistic映射的分岔与分形

利用理论推导分析了二维滞后Logistic映射周期解的稳定性和分岔,利用相图、分岔图、Lyapunov指数和分维数等计算方法,证明了二维滞后Logistic映射依次经叉形分岔和Hopf分岔通向混沌.对二维滞后Logistic映射的吸引盆及其广义M-J集的研究表明:不同周期轨道的吸引盆形状相似,大小不同,每个吸引盆中周期和非周期区域之间的边界是分形的;广义M集的结构与a,R和有N关,广义J集的结构与a,R,N,和Cx,Cy有关,并且广义M-J集具有分形特征.

Bifurcation and Fractal of Two-Dimensional Lagged Logistic System

The bifurcation of Two-dimensional Lagged Logistic System is analyzed theoretically and numerically. By using phase maps,bifurcation graphics,fractal dimension and Lyapunov exponent,the paper reveals the general features of two-dimensional lagged Logistic system transition from regularity to chaos and the fractal configuration of Periodic attraction basin and general Mandelbrot-Julia sets,the following conclusions are shown:(1)Chaotic patterns of the map may emerge out of fork bifurcation and Hopf bifurcation in turn;(2)shape is similar and size is different among different periodic attractor basins,the boundaries between periodic and non-periodic regions is fractal that indicates the moving end-result of the points in phase plane is predicted impossibly;(3)The boundaries of the general Mandelbrot-Julia sets are fractal,The structures of the general Mandelbrot sets are determined by the control parameters a,R and N ,The structures of the general Julia sets are determined by the control parameters a,R,N,Cx and Cy.