The Study Of A Class Of Discrete Singular System's Bifurcation And Chaotic Problem

The paper mainly studies the dynamics of a discrete-time singular biological economi-cal system in the cone R+3. The system models on the base of relationship between predator and prey, after joined the economic indicator, performed for the form of differential-algebraic system which combined with two differential equation and a algebraic equation. Using nu-merical calculation of Euler method let of differential-algebraic system discretized. When δ varying in some range, using new normal form of discrete singular systems, center mainfold theorem and bifurcation theory the fixed point's flip bifurcation and Neimark-Sacker bifur-cation are analyzed. Numerical simulations are presented not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviors, such as cascades of period-doubling bifurcation in orbits of period2,4,8and chaotic sets. The Lya-punov exponents are numerically computed to confirm the chaos set. All these results reveal far richer dynamics of the discrete model compared with the continuous model.First chapter as an introduction, modeling process of the discrete singular system, its real meaning, the current research status and the main work of this paper are addressed.In the second chapter, we discuss the existence and stability of fixed points for the system (2.4) in the closed positive cone R+3.In the third chapter, we show that there exist some values of parameters such that the system (2.4) undergoes the flip bifurcation and the Neimark-S acker bifurcation in the interior of R^.And we present numerical simulations for illustrate our results.Finally, the conclusion is showed in the fourth chapter, and some problems we need study later are given.