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This paper focuses on the vibration periodic property of Duffing oscillator with generalized initial conditions. Firstly, the undamped case is solved by Ji-Huan He’s frequency formulation; Secondly, the formulation is extended to the damped case. Numerical verification shows that the frequency formulation is mathematically simple and physically insightful and practically applicable. This paper paves a novel way for engineers to use the formulation to study nonlinear vibration system with ease and reliability. [ABSTRACT FROM AUTHOR]

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 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]

A complex mode-locking (entrainment) topology underlying the continuous stirred tank reactor reaction model subjected to impulsive perturbations is identified. Employing high-resolution stability diagrams, we exhibit the global structure of mode-locking oscillations and describe how they are interconnected and how their complexity unfolds with control parameters varying. The scenarios shown in the bi-parametric planes revealed that the skeleton of Arnold's tongues is organized according to the symmetric Stern--Brocot sum tree. Moreover, the mode-locking organization is controlled by an invariant torus (a pair of frequencies) initiated from Hopf bifurcations. Interestingly, the mode-locking order is unfolded in an elusive way, that is, in perfect agreement with the reciprocal of the Stern--Brocot sum tree. The findings reported here contribute to providing a description and classification of mode-locking oscillations for the impulsive system. [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]