Bogdanov–Takens bifurcation of a Holling IV prey–predator model with constant-effort harvesting

A prey–predator model with constant-effort harvesting on the prey and predators is investigated in this paper. First, we discuss the number and type of the equilibria by analyzing the equations of equilibria and the distribution of eigenvalues. Second, with the rescaled harvesting efforts as bifurcation parameters, a subcritical Hopf bifurcation is exhibited near the multiple focus and a Bogdanov–Takens bifurcation is also displayed near the BT singularity by analyzing the versal unfolding of the model. With the variation of bifurcation parameters, the system shows multi-stable structure, and the attractive domains for different attractors are constituted by the stable and unstable manifolds of saddles and the limit cycles bifurcated from Hopf and Bogdanov–Takens bifurcations. Finally, a cusp point and two generalized Hopf points are found on the saddle-node bifurcation curve and the Hopf bifurcation curves, respectively. Several phase diagrams for parameters near one of the generalized Hopf points are exhibited through the generalized Hopf bifurcation.

ON NONHOMOGENEOUS BOUNDARY VALUE PROBLEM FOR THE STATIONARY NAVIER-STOKES EQUATIONS IN A SYMMETRIC CUSP DOMAIN

The nonhomogeneous boundary value problem for the stationary NavierStokes equations in 2D symmetric multiply connected domain with a cusp point on the boundary is studied. It is assumed that there is a source or sink in the cusp point. A symmetric solenoidal extension of the boundary value satisfying the LerayHopf inequality is constructed. Using this extension, the nonhomogeneous boundary value problem is reduced to homogeneous one and the existence of at least one weak symmetric solution is proved. No restrictions are assumed on the size of fluxes of the boundary value.

Higher Codimension Bifurcation Analysis of Predator–Prey Systems with Nonmonotonic Functional Responses

In this paper, we study the dynamic behaviors of a predator–prey system with a general form of nonmonotonic functional response. Through analysis, it is found that the system exists in extinction equilibrium, boundary equilibrium and two positive equilibria, one or no positive equilibrium. Furthermore, the conditions are given such that the boundary equilibrium is a saddle, node or a saddle-node point of codimension 1, 2 or 3. Then, some conditions are obtained so that the unique positive equilibrium of the system is a cusp point of codimension 2, 3 and higher or a saddle-node one of codimension 1 or 3, or a nilpotent saddle-node of codimension 4. When there are two positive equilibria in the system, the equilibrium with a large number of preys is a saddle point. For the other one, the system may undergo Hopf bifurcation. To verify our conclusion, we consider the functional response function in the literature [ Zhu et al., 2002 ; Xiao & Ruan, 2001 ]. Finally, we give a brief discussion and numerical simulation.

Profile Design of the Helical Groove Milling Tool

In the process of milling special helicoid , there is a cusp point between its profile curve and linear esicoaginal junction . Ordinary milling cutter can't processing .If milling tool profiles are calculated and designed by smooth connection, the findings of milling tool profiles will appear profiles jagged or profiles detached phenomena. For separation of milling cutter blade shape, in order to prevent point was cut off, it can be treated with the method of open groove, but the two point easy to wear and affect the durability of the cutting tool. Solving method of transitional profiles of detached section is given.

BIFURCATIONS OF THE GLOBAL STABLE SET OF A PLANAR ENDOMORPHISM NEAR A CUSP SINGULARITY

The dynamics of a system defined by an endomorphism is essentially different from that of a system defined by a diffeomorphism due to interaction of invariant objects with the so-called critical locus. A planar endomorphism typically folds the phase space along curves J0 where the Jacobian of the map is singular. The critical locus, denoted J1, is the image of J0. It is often only piecewise smooth due to the presence of isolated cusp points that are persistent under perturbation. We investigate what happens when the stable set Ws of a fixed point or periodic orbit interacts with J1 near such a cusp point C1. Our approach is in the spirit of bifurcation theory, and we classify the different unfoldings of the codimension-two singularity where the curve Ws is tangent to J1 exactly at C1. The analysis uses a local normal-form setup that identifies the possible local phase portraits. These local phase portraits give rise to different global manifestations of the behavior as organized by five different global bifurcation diagrams.

Absolute and Convective Instabilities of a Collapsible Tube

In this paper we extend a numerical method, developed previously by the author, to compute the eigen modes of collapsible viscoelastic duct convening a fowing fuid. A new technique is developed in order to eliminate the need for recurrence formula used in the old method. This gives a more powerful and tractable method to find the eigen modes of the system. The new method has allowed the identification of a new unstable modes in a collapsible tube. It is found that there is a set of standing non axisymmetric waves representing an absolute instabilities and a set of unstable upstream and downstream propagated waves representing a convective instabilities. Two standing waves have equal frequency in their cusp points. The frequency of the other standing waves, in their cusp points, are a multiple of the frequency of the first wave and that in good agreement with experimental finding available in the literature. It is found that the first absolute unstable mode becomes convective at high Reynolds number while the other standing wave remain absolutely unstable modes for Re higher than 100. The frequency ratio of the absolute unstable modes in their cusp point are preserved for all Reynolds number and that with an good agreement with the experience. The absolute unstable mode which becomes convective at high Reynolds number keeps a monochromatic wave with a frequency equal to the frequency of the second absolute unstable mode, in its cusp point, and that for all Reynolds number higher than the limit of absolute instability of the first mode in surprising agreement with the experimental results. It is founded that the viscosity of the solid stabilizes the standing wavesat different degree. The boundary separating the absolute instabilities zone from the convective instabilities zone are found.