Stability of Periodic Solutions Generated by Hopf Points Emanating from a Z＿2－symmetry－breaking Takens－Bogdanov Point

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Consequences of weak Allee effect on prey in the May–Holling–Tanner predator–prey model

In this work, a modified Holling–Tanner predator–prey model is analyzed, considering important aspects describing the interaction such as the predator growth function is of a logistic type; a weak Allee effect acting in the prey growth function, and the functional response is of hyperbolic type. Making a change of variables and time rescaling, we obtain a polynomial differential equations system topologically equivalent to the original one in which the non‐hyperbolic equilibrium point (0,0) is an attractor for all parameter values. An important consequence of this property is the existence of a separatrix curve dividing the behavior of trajectories in the phase plane, and the system exhibits the bistability phenomenon, because the trajectories can have different ω ? limit sets; as example, the origin (0,0) or a stable limit cycle surrounding an unstable positive equilibrium point. We show that, under certain parameter conditions, a positive equilibrium may undergo saddle‐node, Hopf, and Bogdanov–Takens bifurcations; the existence of a homoclinic curve on the phase plane is also proved, which breaks in an unstable limit cycle. Some simulations to reinforce our results are also shown. Copyright © 2015 John Wiley & Sons, Ltd.     

一个具有唯一鞍焦点的三维混沌系统分析

讨论了一个具有唯一鞍焦点的多参数三维混沌系统,该系统包含了Sprott提出的一个最简混沌模型.在特定的条件下得到了Hopf分岔的存在性条件;进一步利用规范型理论获得了决定Hopf分岔方向和分支周期解稳定性的公式,同时利用计算机模拟证实本文的理论分析结果.     

Zero‐Hopf bifurcation in the FitzHugh–Nagumo system

We characterize the values of the parameters for which a zero‐Hopf equilibrium point takes place at the singular points, namely, O (the origin), P₊, and P_? in the FitzHugh–Nagumo system. We find two two‐parameter families of the FitzHugh–Nagumo system for which the equilibrium point at the origin is a zero‐Hopf equilibrium. For these two families, we prove the existence of a periodic orbit bifurcating from the zero‐Hopf equilibrium point O. We prove that there exist three two‐parameter families of the FitzHugh–Nagumo system for which the equilibrium point at P₊ and at P_? is a zero‐Hopf equilibrium point. For one of these families, we prove the existence of one, two, or three periodic orbits starting at P₊ and P_?. Copyright © 2014 John Wiley & Sons, Ltd.     

具有功能性反应函数为的捕食系统

本文给出了具有功能性反应函数为 x的捕食系统x=γx-δ x y-αx2 ,y=-sy+β x y-εy2的全局相图 .得到了两种群持续共存和捕食者种群必将灭绝的条件 .讨论了此系统唯一正平衡点的 Hopf分支 ,并证明了该点可以成为二阶不稳定细焦点 ,从而得到该系统有出现至少三个极限环的可能 .     

Existence of traveling waves in compressible Euler equations with viscosity and capillarity

Given any Lax shock of the compressible Euler dynamics equations, we show that there exists the corresponding traveling wave of the system when viscosity and capillarity are suitably added. For a traveling wave corresponding to a given Lax shock, the governing viscous–capillary system is reduced to a system of two differential equations of first-order, which admits an asymptotically stable equilibrium point and a saddle point. We then develop the method of estimating attraction domain of the asymptotically stable equilibrium point for the compressible Euler equations and show that the saddle point in fact lies on the boundary of this set. Then, we establish a saddle-to-stable connection by pointing out that there is a stable trajectory leaving the saddle point and entering the attraction domain of the asymptotically stable equilibrium point. This gives us a traveling wave of the viscous–capillary compressible Euler equations.     

LIMIT CYCLE PROBLEM OF QUADRATIC SYSTEM OF TYPE （ Ⅲ ）m=0, （ Ⅲ ）

To continue the discussion in （Ⅰ ） and （ Ⅱ ）,and finish the study of the limit cycle problem for quadratic system （ Ⅲ ）m=0 in this paper. Since there is at most one limit cycle that may be created from critical point O by Hopf bifurcation,the number of limit cycles depends on the different situations of separatrix cycle to be formed around O. If it is a homoclinic cycle passing through saddle S1 on 1 ＋ax-y = 0,which has the same stability with the limit cycle created by Hopf bifurcation,then the uniqueness of limit cycles in such cases can be proved. If it is a homoclinic cycle passing through saddle N on x= 0,which has the different stability from the limit cycle created by Hopf bifurcation,then it will be a case of two limit cycles. For the case when the separatrix cycle is a heteroclinic cycle passing through two saddles at infinity,the discussion of the paper shows that the number of limit cycles will change from one to two depending on the different values of parameters of system.     

Effects of Hard Limits on Bifurcation, Chaos and Stability

An SMIB model in the power systems,especially that concering the effects of hard limits onbifurcations,chaos and stability is studied.Parameter conditions for bifurcations and chaos in the absence ofhard limits are compared with those in the presence of hard limits.It has been proved that hard limits can affectsystem stability.We find that (1) hard limits can change unstable equilibrium into stable one;(2) hard limits canchange stability of limit cycles induced by Hopf bifurcation;(3) persistence of hard limits can stabilize divergenttrajectory to a stable equilibrium or limit cycle;(4) Hopf bifurcation occurs before SN bifurcation,so the systemcollapse can be controlled before Hopf bifurcation occurs.We also find that suitable limiting values of hard limitscan enlarge the feasibility region.These results are based on theoretical analysis and numerical simulations,such as condition for SNB and Hopf bifurcation,bifurcation diagram,trajectories,Lyapunov exponent,Floquetmultipliers,dimension of attractor and so on.     

Slow Passage Through the Nonhyperbolic Homoclinic Orbit of the Saddle-Center Hamiltonian Bifurcation

Slowly varying Hamiltonian systems, for which action is a well-known adiabatic invariant, are considered in the case where the system undergoes a saddle center bifurcation. We analyze the situation in which the solution slowly passes through the nonhyperbolic homoclinic orbit created at the saddle-center bifurcation. The solution near this homoclinic orbit consists of a large sequence of homoclinic orbits surrounded by near approaches to the autonomous nonlinear nonhyperbolic saddle point. By matching this solution to the strongly nonlinear oscillations obtained by averaging before and after crossing the homoclinic orbit, we determine the change in the action. If one orbit comes sufficiently close to the nonlinear saddle point, then that one saddle approach instead satisfies the nonautonomous first Painlevé equation, whose stable manifold of the unstable saddle (created in the saddle-center bifurcation) separates solutions approaching the stable center from those involving sequences of nearly homoclinic orbits.     

Existence of traveling waves associated with Lax shocks which violate Oleinik’s entropy criterion

This paper answers to the question whether a shock wave in conservation laws satisfying the Lax shock inequalities but not Oleinik’s entropy criterion is admissible under the vanishing viscosity-capillarity effects. Such a shock appears in van der Waals fluids when a secant line meets the graph of the flux function at four distinct points, and the shock jumps between the two farthest points. The existence of the corresponding traveling waves would justify the admissibility of the shock. For this purpose, we will first show that the corresponding traveling waves satisfy a system of differential equations with two saddle points and two asymptotically stable points. Second, we estimate the domains of attraction of the asymptotically stable equilibrium points, relying on Lyapunov’s stability theory. Third, we investigate the circumstances when an unstable trajectory leaving the saddle point corresponding to the left-hand state of the shock will ever enter the domain of attraction of each of the two asymptotically stable equilibrium points. Finally, we establish the existence of traveling waves associated with a Lax shock but violating the Oleinik’s entropy criterion.     

Limit cycles in a Gause‐type predator–prey model with sigmoid functional response and weak Allee effect on prey

The goal of this work is to examine the global behavior of a Gause‐type predator–prey model in which two aspects have been taken into account: (i) the functional response is Holling type III; and (ii) the prey growth is affected by a weak Allee effect. Here, it is proved that the origin of the system is a saddle point and the existence of two limit cycles surround a stable positive equilibrium point: the innermost unstable and the outermost stable, just like with the strong Allee effect. Then, for determined parameter constraints, the trajectories can have different ω ? limit sets. The coexistence of a stable limit cycle and a stable positive equilibrium point is an important fact for ecologists to be aware of the kind of bistability shown here. So, these models are undoubtedly rather sensitive to disturbances and require careful management in applied contexts of conservation and fisheries. Copyright © 2012 John Wiley & Sons, Ltd.     

Multiple equilibrium points in global attractors for some p-Laplacian equations

In this paper, we continue to study the properties of the global attractor for some p-Laplacian equations with a Lyapunov function F in a Banach space when the origin is no longer a local minimum point but a saddle point of F. By using the abstract result established in our previous work, we prove the existence of multiple equilibrium points in the global attractor for some p-Laplacian equations under some suitable assumptions in the case that the origin is an unstable equilibrium point.     

Anti-control of Hopf bifurcation in the new chaotic system with two stable node-foci

In order to further understand a complex 3D dynamical system showing strange chaotic attractors with two stable node-foci near Hopf bifurcation point, we propose nonlinear control scheme to the system and the controlled system, depending on five parameters, can exhibit codimension one, two, and three Hopf bifurcations in a much larger parameter regain. The control strategy used keeps the equilibrium structure of the chaotic system and can be applied to degenerate Hopf bifurcation at the desired location with preferred stability.     

Global Analysis of Predator–Prey System with Hawk and Dove Tactics

Functional response of the Holling type II is incorporated into a predator–prey model with predators using hawk‐dove tactics to consider combination effects of nonlinear functional response and individual tactics. By mathematical analysis, it is shown that the model undergoes a sequence of bifurcations including saddle‐node bifurcation, supercritical Hopf bifurcation and homoclinic bifurcation. New phenomena are found that include the bistable coexistence of prey and predators in the form of a stable limit cycle and a stable positive equilibrium, the bistable coexistence of prey and predators in a large stable limit cycle that encloses three positive equilibria and a stable positive equilibrium within the cycle, and the bistable coexistence of two stable limit cycles.     

An Infectious Disease Model With Media Coverage and Limited Medical Resources北大核心CSCD

该文建立和分析了一类具有媒体报道效应和有限医疗资源的传染病动力学模型,定义了疾病的基本再生数,分析了平衡点的存在性和稳定性,给出了系统发生前向分支、后向分支和Hopf分支的条件.通过数值模拟发现：提高媒体报道的信息覆盖率或医院对病人的最大容纳量,可以显著降低疾病流行的峰值或稳态时的感染人数;随着参数变化,系统不仅可能会产生后向分支或前向分支,还可能会出现鞍结点分支、Hopf分支以及地方病平衡点稳定性随参数变化而变化等动力学行为.     

Monotonicity of solutions converging to a saddle point equilibrium

If a difference equation has a saddle point equilibrium, then there are solutions which converge to this equilibrium. For second order equations, conditions are given which imply that these solutions are monotone. These results are used to analyze a rational difference equation which possesses a saddle point equilibrium.     

A modified Leslie–Gower predator–prey model with ratio‐dependent functional response and alternative food for the predator

In this work, a modified Leslie–Gower predator–prey model is analyzed, considering an alternative food for the predator and a ratio‐dependent functional response to express the species interaction. The system is well defined in the entire first quadrant except at the origin ( 0 , 0 ) . Given the importance of the origin ( 0 , 0 ) as it represents the extinction of both populations, it is convenient to provide a continuous extension of the system to the origin. By changing variables and a time rescaling, we obtain a polynomial differential equations system, which is topologically equivalent to the original one, obtaining that the non‐hyperbolic equilibrium point ( 0 , 0 ) in the new system is a repellor for all parameter values. Therefore, our novel model presents a remarkable difference with other models using ratio‐dependent functional response. We establish conditions on the parameter values for the existence of up to two positive equilibrium points; when this happen, one of them is always a hyperbolic saddle point, and the other can be either an attractor or a repellor surrounded by at least one limit cycle. We also show the existence of a separatrix curve dividing the behavior of the trajectories in the phase plane. Moreover, we establish parameter sets for which a homoclinic curve exits, and we show the existence of saddle‐node bifurcation, Hopf bifurcation, Bogdanov–Takens bifurcation, and homoclinic bifurcation. An important feature in this model is that the prey population can go to extinction; meanwhile, population of predators can survive because of the consumption of alternative food in the absence of prey. In addition, the prey population can attain their carrying capacity level when predators go to extinction. We demonstrate that the solutions are non‐negatives and bounded (dissipativity and permanence of population in many other works). Furthermore, some simulations to reinforce our mathematical results are shown, and we further discuss their ecological meanings. Copyright © 2017 John Wiley & Sons, Ltd.     

Instability of equilibrium points of some Lagrangian systems

In this work we show that, if L is a natural Lagrangian system such that the k-jet of the potential energy ensures it does not have a minimum at the equilibrium and such that its Hessian has rank at least n−2, then there is an asymptotic trajectory to the associated equilibrium point and so the equilibrium is unstable. This applies, in particular, to analytic potentials with a saddle point and a Hessian with at most 2 null eigenvalues.The result is proven for Lagrangians in a specific form, and we show that the class of Lagrangians we are interested can be taken into this specific form by a subtle change of spatial coordinates. We also consider the extension of this results to systems subjected to gyroscopic forces.     

Bifurcation analysis of a Leslie-type predator–prey system with simplified Holling type IV functional response and strong Allee effect on prey

In this paper, a Leslie-type predator–prey system with simplified Holling type IV functional response and strong Allee effect on prey is proposed. The dissipativity of the system and the existence of all possible equilibria are investigated. The investigation emphasizes the exploring of bifurcation. It is shown that the system exists several non-hyperbolic positive equilibria, such as a weak focus of multiplicities one and two, (degenerate) saddle–nodes and Bogdanov–Takens singularities (cusp case) of codimensions 2 and 3. At these equilibria, it is proved that the system undergoes various kinds of bifurcations, such as saddle–node bifurcation, Hopf bifurcation, degenerate Hopf bifurcation and Bogdanov–Takens bifurcation of codimensions 2 and 3. With the parameters selected properly, there exhibits a limit cycle, a homoclinic loop, two limit cycles, a semistable limit cycle, or the simultaneous occurrence of a homoclinic loop and a limit cycle in the system. Moreover, it is also proved that the system has a cusp of codimension at least 4. Hence, there may exist three limit cycles generated from Hopf bifurcation of codimension 3. Numerical simulations are done to support the theoretical results.