Three-species food-chain models, in which the prey population exhibits group defense, are considered. Using the carrying capacity of the environment as the bifurcation parameter, it is shown that the model without delay undergoes a sequence of Hopf bifurcations. In the model with delay it is shown that using a delay as a bifurcation parameter, a Hopf bifurcation can also occur in this case. The...

The predator-prey model with stage structure for predator is generalized in the context of ecoepidemiology, where the prey population is infected by a microparasite and the predator completely avoids consuming the infected prey. The intraspecific competition of infected prey is considered. All the equilibria are characterized and the existence of a Hopf bifurcation at the coexistence equilibriu...

An efficient method, called the perturbation-incremental scheme (PIS), is proposed to study, both qualitatively and quantitatively, delay-induced weak or high-order resonant double Hopf bifurcation and the dynamics arising from the bifurcation of nonlinear systems with delayed feedback. The scheme is described in two steps, namely the perturbation and the incremental steps, when the time delay ...

In this paper, the kinetics of a class delayed reaction-diffusion mussel-algae system under Neumann boundary conditions with half-saturation constant is studied. The global existence and priori bounds as well positive equilibrium are obtained. affects stability may result in Turing instability. When exceeds certain critical value, globally asymptotically stable which means that larger forces mu...

This paper studies the global bifurcation and Hopf bifurcation of one kind of complicated financial system with different parameter combinations. Conditions on which bifurcation happens, and the critical system structure when the system transforms from one kind of topological structure to another are studied as well. The criterion for identifying Hopf bifurcation under different parameter combi...

This paper studies the, codimension-3, Boundary-Hopf-Fold (BHF) bifurcation of planar Filippov systems. Filippov systems consist of at least one discontinuity boundary locally separating the phase space to disjoint components with different dynamics. Such systems find applications in several fields, for example, mechanical and electrical engineering, and ecology. The BHF bifurcation appears in ...

In this paper, by using the Hopf’s bifurcation theorem we will discuss the existence of small amplitude periodic solutions of the equation ẍ(t)+ f(x(t))ẋ(t) + g(x(t − r)) = 0, taking as bifurcation parameter c either d or r. We assume that r > 0, f ∈ C, f(0) = c > 0, g(0) = 0 and ġ(0) = d > 0.

The forced response curve is a graph showing the amplitude of the response of a periodically forced system as the forcing frequency is varied. Zhang and Golubitsky (SIADS, 10(4) (2011) 1272–1306) classified the existence and multiplicity of periodic responses to small amplitude periodic forcing of a system near a Hopf bifurcation point, where the forcing frequency, ωF , is close to the Hopf fre...

Time-delayed feedback control was introduced by Pyragas in 1992 as a general method for stabilizing an unstable periodic orbit of a given continuous-time dynamical system. The analysis of Pyragas control focused on its application to the normal form of a subcritical Hopf bifurcation, and it was initially concerned with stabilization near the Hopf bifurcation. A recent study considered this norm...