Bifurcation analysis in a diffusive Logistic population model with two delayed density-dependent feedback terms

The present paper is concerned with a diffusive population model of Logistic type with an instantaneous density-dependent term and two delayed density-dependent terms and subject to the zero-Dirichlet boundary condition. By regarding the delay as the bifurcation parameter and analyzing in detail the associated eigenvalue problem, the local asymptotic stability and the existence of Hopf bifurcation for the sufficiently small positive steady state solution are shown. It is found that under the suitable condition, the positive steady state solution of the model will become ultimately unstable after a single stability switch (or change) at a certain critical value of delay through a Hopf bifurcation. However, under the other condition, the positive steady state solution of the model will become ultimately unstable after multiple stability switches at some certain critical values of delay through Hopf bifurcations. In addition, the direction of the above Hopf bifurcations and the stability of the bifurcating periodic solutions are analyzed by means of the center manifold theory and normal form method for partial functional differential equations. Finally, in order to illustrate the correction of the obtained theoretical results, some numerical simulations are also carried out.     

The shape of the strongest column with a follower load

An elastic column subjected to a follower load loses stability, as the load increases, through undamped oscillations around its equilibrium position (flutter). For a column with fixed volume and varying but similar cross section, the existence of a critical load is shown rigorously. The necessary conditions for making the critical load and the fundamental frequency of the column stationary with respect to variations in the column shape are derived. A numerical method for finding the optimal shape is given.     

The optimal elbow angle for acrobatics ‐ stability analysis of the elbow with a load

We consider a human elbow with a load and study the existence and stability of equilibrium positions without the use of reflexes. We observe that for each elbow angle there is a critical mass above which the equilibrium becomes unstable. We determine the optimal elbow angle, i.e., the angle such that the corresponding critical mass is as high as possible.     

多重时滞富营养化生态模型的稳定性与分支分析

本文研究了多重时滞富营养化生态模型的稳定性与分支问题.利用特征值方法,分别研究了具有单时滞和双时滞模型的线性稳定性.发现当模型中的时滞经过一系列临界值时,模型在平衡点附近经历了Hopf分支和Hopf-zero分支,并给出Hopf分支和Hopf-zero分支存在的充分条件.最后数值模拟验证了理论结果.     

A cylindrical plate with a weakly fixed curvilinear edge made of a transversally isotropic material

Free oscillations and stability under an axial compression of a thin cylindrical plate with a weakly fixed rectilinear edge made of a transversally isotropic material with low stiffness with respect to transverse displacements are considered. The curvilinear edges of the plate are assumed to be hingedly supported. The oscillation frequencies and the critical load for a plate with a free or weakly fixed edge are smaller than those for a shell closed in the circumferential direction. The shapes of oscillations and the forms of stability loss localized near the weakly fixed edge and damped at a distance from it are considered. The Timoshenko-Reissner model is used. Localized forms are analyzed by using a system of equations for Timoshenko-Reissner shallow shells, which is derived for this purpose. The main special feature of this system is that it contains a separate equation describing a solution with large variability. For the example of the stability problem under consideration, the error involved in the system of equations for Timoshenko-Reissner shallow shells is studied. The critical load values obtained with the use of the Kirchhoff-Love and Timoshenko-Reissner models are compared.     

Bulging of a cylindrical shell with viscoelastic filler during axial compression

The problem of static stability during axial compression of an orthotropic cylindrical shell reinforced by a continuous filler is solved. The filler is considered to be a three-dimensional isotropic linear-viscoelastic body, fixed at its outer surface to the shell. Transverse shear in the shell is taken into account. The loss of stability of the structure is related to the time-dependent development of initial defects, following application of a load, which is smaller than the instant critical load.     

Dynamics and control of a system of two non‐interacting preys with common predator

In this paper, we consider a Holling type model, which describes the interaction between two preys with a common predator. First, we give some sufficient conditions for the globally asymptotic stability and prove that local stability implies global stability. Then, we present a set of sufficient conditions for the existence of a positive periodic solution with strictly positive components. Finally, the optimal control strategy is developed to minimize the number of predator and maximize the number of preys. We also show the existence of an optimal control for the optimal control problem and derive the optimality system. The technical tool used to determine the optimal strategy is the Pontryagin Maximum Principle. Finally, the numerical simulations of global stability and the optimal problem are given as the conclusion of this paper. Copyright © 2011 John Wiley & Sons, Ltd.     

Explicit stability conditions for integro-differential equations with periodic coefficients

New explicit stability conditions are derived for a linear integro-differential equation with periodic operator coefficients. The equation under consideration describes oscillations of thin-walled viscoelastic structural members driven by periodic loads. To develop stability conditions two approaches are combined. The first is based on the direct Lyapunov method of constructing stability functionals. It allows stability conditions to be derived for unbounded operator coefficients, but fails to correctly predict the critical loads for high-frequency excitations. The other approach is based on transforming the equation under consideration in such a way that an appropriate ‘differential’ part of the new equation would possess some reserve of stability. Stability conditions for the transformed equation are obtained by using a technique of integral estimates. This method provides acceptable estimates of the critical forces for periodic loads, but can be applied to equations with bounded coefficients only. Combining these two approaches, we derive explicit stability conditions which are close to the Floquet criterion when the integral term vanishes. These conditions are applied to the stability problem for a viscoelastic bar compressed by periodic forces. The effect of material and structural parameters on the critical load is studied numerically. © 1998 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

一类不可压广义neo-Hookean球体的空穴分岔问题的定性研究

研究了一类不可压的广义neo-Hookean材料组成的球体的空穴分岔问题,该类材料可以看作是带有径向摄动的均匀各向同性不可压的neo-Hookean材料,得到了球体内部空穴生成的条件．与均匀各向同性的neo-Hookean球体的情况相比,证明了当摄动参数属于某些区域时,从平凡解局部向左分岔的空穴分岔解上存在一个二次转向分岔点,空穴生成时的临界载荷会比无摄动的材料的临界载荷小．用奇点理论证明了,空穴分岔方程在临界点附近等价于具有单边约束条件的正规形．用最小势能原理分别讨论了空穴分岔解的稳定性和实际稳定的平衡状态．     

THE HETEROCLINIC CYCLE IN THEMODEL OF COMPETITION BETWEENn SPECIES AND ITS STABILITY

1.IntroductionConsidertheanalogousGause-Lotka-Volterra(GLV)differentialequationsforncompetingpopulationsMayandLeonard[ll,Hcf'bauerandSigmundl21havestudiedthissystemforthecasen=3separately.Bothofthemnotedthatinsuchsystemprobablythereexistsaheterocliniccycle.Biologically,thestudyoftheheterocliniccycleisinterestingsinceitsstabilityiscloselyrelatedtothepermanenceproblem.Ontheotherhand,inrecelltyearsmathematicianshavestudiedwithinterestthestabilityofaheterocltalccycleandthebifurcationofit.Inth…     

Travelling waves in model reacting flows with reversible kinetics

The authors study the existence of wavefront-type travellingwave solutions in the Fickett-Majda model of viscous reactiveflow when the chemistry is modelled by a reversible chemicalreaction. The problem is reduced to proving the existence ofa heteroclinic orbit, in a two-dimensional phase space, connectingtwo critical points that represent the equilibrium states atplus and minus infinity. Reactions in which the forward reactioncan be either endothermic or exothermic are examined, and itis shown that compression waves must be accompanied by a shiftin the equilibrium composition in the endothermic direction,while rarefac-tions are accompanied by a shift in the exothermicdirection. Although compression waves occur on a larger parameterdomain, there are regimes where the rarefactions appear. Finally,the stability of an equilibrium state under small perturbationsis discussed.     

Limit cycles in a generalized Gause-type predator–prey model

We study the problem of the existence of limit cycles for a generalized Gause-type predator–prey model with functional and numerical responses that satisfy some general assumptions. These assumptions describe the effect of prey density on the consumption and reproduction rates of predator. The model is analyzed for the situation in which the conversion efficiency of prey into new predators increases as prey abundance increases. A necessary and sufficient condition for the existence of limit cycles is given. It is shown that the existence of a limit cycle is equivalent to the instability of the unique positive critical point of the model. The results can be applied to the analysis of many models appearing in the ecological literature for predator–prey systems. Some ecological models are given to illustrate the results.     

The heteroclinic cycle in the model of competition betweenn species and its stability

This paper gives a sufficient condition for the existence of heteroclinic cycle in the model of competition betweenn species and a criterion for determining the stability of the heteroclinic cycle. The results given in this paper extend the results obtained by May and Leonard in and by Hofbauer and Sigmund in. A conjecture on the permanence of the model and a open problem on the stability of the heteroclinic cycle for the critical case are given at the end of this paper. This research is supported by the National Natural Science Foundation of China     

Die Knicklast des einseitig eingespannten Stabes mit gleichmässig verteilter,tangentialer Längsbelastung

Summary The paper deals with the application ofGalerkin's method and the calculus of finite differences to a non-conservative problem of elastic stability: the critical load of a bar which is built in at the bottom and compressed by distributed tangential loads is approximately calculated.     

模糊不确定时滞系统的保成本控制

讨论了基于T-S模型的不确定时滞系统的保成本控制问题.文章采用并行补偿状态反馈控制方法和时滞相关稳定性分析方法,通过引入一个带调节因子的Lyapunov-Krasovskii泛函,利用线性矩阵不等式的形式给出了状态反馈控制器存在的充分条件.当调节因子取不同值时,最小保成本值和反馈增益也是不同的,不同的反馈增益导致不同的动态性能,因此,可以通过选取合适的调节因子来优化闭环系统的动态性能. 最小保成本值可以看作调节因子的函数,因此,可以通过求解一个凸优化问题来求得最小的保成本值和最优的调节因子,文章给出了一个求解最小保成本值的算法.并利用仿真示例验证了所给方法的有效性.     

Buckling of a nonlinear elastic rod

The buckling of a pin-ended slender rod subjected to a horizontal end load is formulated as a nonlinear boundary value problem. The rod material is taken to be governed by constitutive laws which are nonlinear with respect to both bending and compression. The nonlinear boundary value problem is converted to a suitable integral equation to allow the application of bounded operator methods. By treating the integral equation as a bifurcation problem, the branch points (critical values of load) are determined and the existence and form of nontrivial solutions (buckled states) in the neighborhood of the branch points is established. The integral equation also affords a direct attack upon the question of uniqueness of the trivial solution (unbuckled state). It is shown that, under certain conditions on the material properties, only the trivial solution is possible for restricted values of the load. One set of conditions gives uniqueness up to the first branch point.     

Algebraic topological methods for the supercritical Q-curvature problem

We study the problem of existence of conformal metrics with prescribed Q-curvature on closed four-dimensional Riemannian manifolds. This problem has a variational structure, and in the case of interest here, it is noncompact in the sense that accumulations points of some noncompact flow lines of a pseudogradient of the associated Euler–Lagrange functional, the so-called true critical points at infinity of the associated variational problem, occur. Using the characterization of the critical points at infinity of the associated variational problem which is established in [42], combined with some arguments from Morse theory, some algebraic topological methods, and some tools from dynamical system originating from Conley's isolated invariant sets and isolated blocks theory, we derive a new kind of existence results under an algebraic topological hypothesis involving the topology of the underling manifold, stable and unstable manifolds of some of the critical points at infinity of the associated Euler–Lagrange functional.     

Dirichlet零边值问题的正解存在性

由于一些本质困难,N=3被称为具Sobolev临界指数2*的Dirichlet问题-△u=λu+|u|2*-2u,x∈ΩRN;u(x)0,x∈Ω;u=0,x∈Ω的临界维数.众所周知,N=3时,上述问题存在古典(正)解的一个充分条件是Ω为R3上的小球以及14λ1λλ1.本文考虑Ω是R3中更一般的有界光滑区域,得出了一正解存在性结论,从而肯定了沈尧天在文中提及的一个未解决的问题.     

Regularization for a class of quasi-variational-hemivariational inequalities

The aim of the present paper is to study the solvability and regularization for a class of multivalued quasi-variational–hemivariational inequalities in reflexive Banach spaces. By applying the Kluge fixed point theorem and the Minty technique, we prove the solvability of the considered multivalued quasi-variational–hemivariational inequality, based on which some convergence results are obtained by introducing its regularization problem with the help of regularization operator. The applicability of the obtained abstract results is established by a mathematical model of a frictional contact problem with a class of elastic material, where the existence and stability results for the weak solution of contact problem are studied.     

Stable Ground States for the Relativistic Gravitational Vlasov–Poisson System

We consider the three dimensional gravitational Vlasov–Poisson (GVP) system in both classical and relativistic cases. The classical problem is subcritical in the natural energy space and the stability of a large class of ground states has been derived by various authors. The relativistic problem is critical and displays finite time blow up solutions. Using standard concentration compactness techniques, we however show that the breaking of the scaling symmetry allows the existence of stable relativistic ground states. A new feature in our analysis which applies both to the classical and relativistic problem is that the orbital stability of the ground states does not rely as usual on an argument of uniqueness of suitable minimizers—which is mostly unknown—but on strong rigidity properties of the transport flow, and this extends the class of minimizers for which orbital stability is now proved.