Papers by Keyword: Hopf Bifurcation

Abstract: A delayed predator-prey model with stage structure for predator and ratio dependent response function is considered. By calculating characteristic equations and analyzing characteristic roots, the sufficient conditions for local stability of all the equilibria and Hopf bifurcation are obtained. Moreover, We use an iteration technique and comparison arguments to derive the sufficient conditions of the global stability of the boundary and positive equilibrium.

Abstract: In this paper, A mathematical model of two species with stage structure and distributed delays is investigated, the necessary and sufficient of the stable equilibrium point are studied. Further, by analyze the associated characteristic equation, it is founded that Hopf bifurcation occurs when τ crosses some critical value. The direction of Hopf bifurcation as well as stability of periodic solution are studied. Using the normal form theory and center manifold method.

Abstract: In this paper, a modified optically injected semiconductor lasers model is studied in detail. More precisely, we study the stability of the equilibrium points and basic dynamic properties of the autonomous system by means of nonlinear dynamics theory. The existence of Hopf bifurcation is investigated by choosing the appropriate bifurcation parameter. Furthermore, formulas for determining the stability and the conditions for generating Hopf bifurcation of the equilibria are derived. Then, a numerical example is given.

Abstract: Based on the study of a dual component system with elastic constraints, the stability and local bifurcations of the soft-impacts system, such as piecewise property and singularity, was analyzed by using the Poincaré map and Runge-Kutta numerical simulation method. The routes from periodic motions to chaos, via Hopf bifurcation and period-doubling bifurcation, were investigated exactly. In the large constraint stiffness case, the period-doubling and Hopf bifurcation exist in the two-degree-of-freedom system with elastic constraints and clearances. The clearances of the system, stiffness and damping coefficient of the elastic constraints is the main reasons for influencing the chaotic motion. The steady 1-1-1 period orbits or 2-1-1 period orbits will exist within a wideband frequency range and the value of velocity will be higher when appropriate system parameters are chosen.

Abstract: A dimensionless nonlinear state-space model was established considering the structural particularities of digital hydraulic cylinder, and dynamic bifurcation characteristics of the system were analyzed and validated based on MATCONT. The results show that, when the piston diameter, valve orifice area gradient, ball-screw pitch and the maximum desired speed are not designed appropriately, digital hydraulic cylinder is prone to Hopf bifurcation. Limit cycles that the stable and the unstable neutralize each other at subcritical Hopf bifurcation points, causing the system tracking outputs divergence from continuous oscillation to increasing oscillation. Losing synchronism of stepper motor is essentially a form of the instability caused by system supercritical Hopf bifurcation.

Abstract: The paper studies three-dimensional food-chain model with variable consumption rate in Chemostat. Assume the prey population's consumption rate of the nutrients is quadratic function, and the predator's consumption rate of the prey population is linear function. Use qualitative theory of ordinary differential equation to analyze the equilibrium solution of the model, especially the existence and stability of positive equilibrium solutions and Hopf bifurcation solutions. Finally,several numerical simulations illustrating the theoretical analysis are also given.

Abstract: In this paper, a predator–prey model with discrete and distributed delays is investigated. the direction of Hopf bifurcation as well as stability of periodic solution are studied. The method which we used is the normal form theory and center manifold. At last, an example showed the feasibility of results.

Abstract: Hopf bifurcation occurs in most of dynamics systems when the influence from the past state varies. In modeling population dynamics, it is more reasonable taking into account the time delays. In this paper, a stage-structured predator-prey system with delay is considered. The existence of Hopf bifurcations at the positive equilibrium is established by analyzing the distribution of the characteristic values. An explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by using the normal form and the center manifold theory. Numerical simulations to support the analytical conclusions are carried out.

Abstract: In this paper, a predatorprey model with discrete and distributed delays is investigated. The necessary and sufficient of the stable equilibrium point for this model is studied. Further, analyzed the associated characteristic equation. And, it is found that Hopf bifurcation occurs when τ crosses some critical value. Last, an example showed the feasibility of results.

Abstract: To make the theoretical analysis of the microbial continuous culture more close to the experimental results, we consider a chemostat model with Watt type functional response and variable yield. The existence of limit cycles and Hopf bifurcation is investigated, which is useful in the further study of the oscillatory behaviors of the microbial growth in the vessel. The conditions for the global asymptotical stability of the model are obtained by Dulac criterion.