Bifurcation Analysis and Chaos Control in a Discrete Epidemic System

Abstract

The dynamics of discreteSIepidemic model, which has been obtained by the forward Euler scheme, is investigated in detail. By using the center manifold theorem and bifurcation theorem in the interiorR+2, the specific conditions for the existence of flip bifurcation and Neimark-Sacker bifurcation have been derived. Numerical simulation not only presents our theoretical analysis but also exhibits rich and complex dynamical behavior existing in the case of the windows of period-1, period-3, period-5, period-6, period-7, period-9, period-11, period-15, period-19, period-23, period-34, period-42, and period-53 orbits. Meanwhile, there appears the cascade of period-doubling 2, 4, 8 bifurcation and chaos sets from the fixed point. These results show the discrete model has more richer dynamics compared with the continuous model. The computations of the largest Lyapunov exponents more than 0 confirm the chaotic behaviors of the systemx→x+δ[rN(1-N/K)-βxy/N-(μ+m)x],y→y+δ[βxy/N-(μ+d)y]. Specifically, the chaotic orbits at an unstable fixed point are stabilized by using the feedback control method.