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The basic dynamic properties of a four dimensional hyperchaotic system are investigated in this paper. More precisely, the stability of equilibrium point of hyperchaotic system is studied by means of nonlinear dynamics theory. We analyses the existence and stability of Hopf bifurcation, and the formulas for determining the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are derived. In addition, a sliding mode controller is designed and controlled the hyperchaotic system to any fixed point to eliminate the chaotic vibration by means of sliding mode method. Finally, the numerical simulations were presented to confirm the effectiveness of the controller. [ABSTRACT FROM AUTHOR]

The dynamical behaviors of a SEIR epidemic model are investigated in this paper. More precisely, we presented a new discrete-time SEIR epidemic model by using the Forward-Euler difference method. And the existence, stability and direction of Hopf bifurcation of the SEIR epidemic model and its discretization model are also studied. In addition, the numerical simulations were presented to illustrate the theoretical analysis. Finally, some comparisons of bifurcation between the continuous-time epidemic system and its discrete-time system are given. [ABSTRACT FROM AUTHOR]