| The dynamic system has its complicated side. The bifurcation is a common and important nonlinear phenomenon, and there's something with other nonlinear phenomena (Like chaos, mutation, fractal and so on). Thus the bifurcation research plays an important role in the nonlinear science.The main work of this paper is as follows:1. Make the Leslie-Holling differential systems with Holling type- II and type-III functional response discrete by the forward Euler scheme, and analyze the existence and stability of the fixed points of the new discrete dynamic systems. Then discuss the Flip and the Neimark-Sacker bifurcation by the center fold, which can reduce the dimension, and the formal methods. Finally draw all kinds of bifurcation figures, the maximum lyapunov exponent figures and the phase figures by Matlab and compute the important values as the bifurcation indices by Mathema-tica. The main result is that system (2.4) exists period-5,6,8,9,10,12,14,17, 20,24,26,42,67 orbits, cascade of period-doubling bifurcation including period 2,4,8,16 orbits, quasi-periodic orbits and the strange attractor. And system (2.5) exists period-7,14,21,63,70 orbits, cascade of period-doubling bifurcation including period-2,4 orbits, quasi-periodic orbits and the strange attarctor.2. Apply the bifurcation theories by Kuznetsov to the discrete predator-prey system with Holling type-III. Then analyze the dynamic qualities of the system, mainly for the Neimark-Sacker bifurcation and its direction problem. Finally mimic the important results by Matlab to test the Tightness. The main result is that system (3.9) exists an attracting invariant closed curve and complicated dynamic quality. |
