Some global results for nonlinear fourth order eigenvalue problems

In this paper, we consider the nonlinear fourth order eigenvalue problem. We show the existence of family of unbounded continua of nontrivial solutions bifurcating from the line of trivial solutions. These global continua have properties similar to those found in Rabinowitz and Berestycki well-known global bifurcation theorems.     

一个在无穷远点分支出八个极限环的多项式微分系统

本文研究一类高次系统无穷远点的中心条件与极限环分支问题．作者首先推出一个计算系统无穷远点奇点量的线性递推公式，并利用计算机代数系统计算出该系统在无穷远点处的前11个奇点量，从而导出无穷远点成为中心和最高阶细焦点的条件，在此基础上作者首次给出了多项式系统在无穷远点分支出8个极限环的实例。     

Hopf bifurcation at infinity in 3D symmetric piecewise linear systems. Application to a Bonhoeffer–van der Pol oscillator

In this work, a Hopf bifurcation at infinity in three-dimensional symmetric continuous piecewise linear systems with three zones is analyzed. By adapting the so-called closing equations method, which constitutes a suitable technique to detect limit cycles bifurcation in piecewise linear systems, we give for the first time a complete characterization of the existence and stability of the limit cycle of large amplitude that bifurcates from the point at infinity. Analytical expressions for the period and amplitude of the bifurcating limit cycles are obtained. As an application of these results, we study the appearance of a large amplitude limit cycle in a Bonhoeffer–van der Pol oscillator.     

Asymptotic Methods for Bifurcation and Stability Problems

A method of asymptotically determining the bifurcating solutions of a nonlinear eigenvalue problem is described. The method is based on the smallness of a parameter which is different from the usual parameter used in the LyapunovSchmidt procedure. A discussion of the stability and evolution of bifurcating solutions is included. It is shown how the method may be useful for determining secondary bifurcations and turning points on bifurcation curves.     

Bifurcation of stable equilibria and nonlinear flux boundary condition with indefinite weight

We study bifurcation and stability of positive equilibria of a parabolic problem under a nonlinear Neumann boundary condition having a parameter and an indefinite weight. The main motivation is the selection migration problem involving two alleles and no gene flux acrossing the boundary, introduced by Fisher and Fleming, and Henry?s approach to the problem.Local and global structures of the set of equilibria are given. While the stability of constant equilibria is analyzed, the exponential stability of the unique bifurcating nonconstant equilibrium solution is established. Diagrams exhibiting the bifurcation and stability structures are also furnished. Moreover the asymptotic behavior of such solutions on the boundary of the domain, as the positive parameter goes to infinity, is also provided.The results are obtained via classical tools like the Implicit Function Theorem, bifurcation from a simple eigenvalue theorem and the exchange of stability principle, in a combination with variational and dynamical arguments.     

Local Hopf bifurcation and global periodic solutions in a delayed predator-prey system

We consider a delayed predator-prey system. We first consider the existence of local Hopf bifurcations, and then derive explicit formulas which enable us to determine the stability and the direction of periodic solutions bifurcating from Hopf bifurcations, using the normal form theory and center manifold argument. Special attention is paid to the global existence of periodic solutions bifurcating from Hopf bifurcations. By using a global Hopf bifurcation result due to Wu [Trans. Amer. Math. Soc. 350 (1998) 4799], we show that the local Hopf bifurcation implies the global Hopf bifurcation after the second critical value of delay. Finally, several numerical simulations supporting the theoretical analysis are also given.     

Stability,Steady‐State Bifurcations,and Turing Patterns in a Predator–Prey Model with Herd Behavior and Prey‐taxis

In this paper, we consider a predator–prey model with herd behavior and prey‐taxis subject to the homogeneous Neumann boundary condition. First, by analyzing the characteristic equation, the local stability of the positive equilibrium is discussed. Then, choosing prey‐tactic sensitivity coefficient as the bifurcation parameter, we obtain a branch of nonconstant solutions bifurcating from the positive equilibrium by an abstract bifurcation theory, and find the stable bifurcating solutions near the bifurcation point under suitable conditions. We have shown that prey‐taxis can destabilize the uniform equilibrium and yields the occurrence of spatial patterns. Furthermore, some numerical simulations to illustrate the theoretical analysis are also carried out, Turing patterns such as spots pattern, spots–strip pattern, strip pattern, stable nonconstant steady‐state solutions, and spatially inhomogeneous periodic solutions are obtained, which also expand our theoretical results.     

Two new approaches to computing Hopf bifurcation problems

We consider the computation of Hopf bifurcation for ordinary differential equations. Two new extended systems are given for the calculation of Hopf bifurcation problems: the first is composed of differential-algebraic equations with index 1, the other consists of differential equations by using a symmetry inherited from the autonomous system of ordinary differential equations. Both methods are especially suitable for calculating bifurcating periodic solutions since they transform the Hopf bifurcation problem into regular nonlinear boundary value problems which are very easy to implement. The bifurcation solutions become isolated solutions of the extended system so that our methods work both in the subcritical and supercritical case. The extended systems are based on an additional parameter ε; practical experience shows that one gets convergence for ε sufficiently large so that a substantial part of the bifurcating branch can be computed. The two methods are illustrated by numerical examples and compared with other procedures.     

On a limiting system of the triangular SKT competition model with large cross-diffusion and random-diffusion

This paper mainly focus on the limiting system which arises from the study of SKT competition model when both the cross-diffusion rate and the random-diffusion rate tend to infinity. For multi-dimensional domains, the existence of positive steady states bifurcating from a double eigenvalue can be proved by applying the bifurcation argument with some special transformation when the birth rate of one species is near some critical value. Further by virtue of the spectral perturbation argument based on the Lyapunov–Schmidt decomposition method, we prove the spectral instability of such nontrivial positive steady states for the limiting system.     

Stability of bifurcating solutions of the problem about capillary-gravity surface waves in spatial layer of floating fluid

In prolongation of our previous investigations on capillary-gravity surface waves in spatial fluid layers the stability of the bifurcating families of solutions in the horizontal layers of the floating (and without flotation) incompressible heavy capillary fluid is considered. The assumption about layer depth simplifies the proof of the existence of bifurcating solutions at the high dimensions of the linearized operator degeneracy, computation of their asymptotics and as the main subject of this communication the investigation of their stability, relative to perturbations with the same symmetry as bifurcating solutions. Group analysis methods of differential equations are used. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Positive solutions bifurcating from zero solution in a Lotka-Volterra competitive system with cross-diffusion effects

A strongly coupled elliptic system under the homogeneous Dirichlet boundary condition denoting the steady-state system of the Lotka-Volterra two-species competitive system with cross-diffusion effects is considered. By using the implicit function theorem and the Lyapunov-Schmidt reduction method, the existence of the positive solutions bifurcating from the trivial solution is obtained. Furthermore, the stability of the bifurcating positive solutions is also investigated by analyzing the associated characteristic equation.     

种群动力学中椭圆系统在零解处的分支正解

考虑齐次Dirichlet边界条件下具有交错扩散压力的广义Lotka-Volterra两种群竞争反应扩散稳态系统. 首先借助Lyapunov-Schmidt约化方法考虑了系统在零解处小分支正解的存在性, 然后借助标准的线性化方法研究了这些分支正解的稳定性.     

Dominant and recessive solutions at infinity and genera of conjoined bases for discrete symplectic systems

In this paper we introduce the theory of dominant solutions at infinity for nonoscillatory discrete symplectic systems without any controllability assumption. Such solutions represent an opposite concept to recessive solutions at infinity, which were recently developed for such systems by the authors. Our main results include: (i) the existence of dominant solutions at infinity for all ranks in a given range depending on the order of abnormality of the system, (ii) construction of dominant solutions at infinity with eventually the same image, (iii) classification of dominant and recessive solutions at infinity with eventually the same image, (iv) limit characterization of recessive solutions at infinity in terms of dominant solutions at infinity and vice versa, and (v) Reid’s construction of the minimal recessive solution at infinity. These results are based on a new theory of genera of conjoined bases for symplectic systems developed for this purpose in this paper.     

Global Bifurcation on Time Scales

We consider the structure of the solution set of a nonlinear Sturm-Liouville boundary value problem defined on a general time scale. Using global bifurcation theory we show that unbounded continua of nontrivial solutions bifurcate from the trivial solution at the eigenvalues of the linearization, and we show that certain nodal properties of the solutions are preserved along these continua. These results extend the well-known results of Rabinowitz for the case of Sturm-Liouville ordinary differential equations.     

Direction and stability of bifurcation branches for variational inequalities

We consider a class of variational inequalities with a multidimensional bifurcation parameter under assumptions guaranteeing the existence of smooth families of nontrivial solutions bifurcating from the set of trivial solutions. The direction of bifurcation is shown in a neighborhood of bifurcation points of a certain type. In the case of potential operators, also the stability and instability of bifurcating solutions and of the trivial solution is described in the sense of minima of the potential. In particular, an exchange of stability is observed.     

Hopf bifurcation in a three-species system with delays

A kind of three-species system with Holling II functional response and two delays is introduced. Its local stability and the existence of Hopf bifurcation are demonstrated by analyzing the associated characteristic equation. By using the normal form method and center manifold theorem, explicit formulas to determine the direction of the Hopf bifurcation and the stability of bifurcating periodic solution are also obtained. In addition, the global existence results of periodic solutions bifurcating from Hopf bifurcations are established by using a global Hopf bifurcation result. Numerical simulation results are also given to support our theoretical predictions.     

Stability and Hopf bifurcation in a delayed competition system

In this paper, Hopf bifurcation for two-species Lotka–Volterra competition systems with delay dependence is investigated. By choosing the delay as a bifurcation parameter, we prove that the system is stable over a range of the delay and beyond that it is unstable in the limit cycle form, i.e., there are periodic solutions bifurcating out from the positive equilibrium. Our results show that a stable competition system can be destabilized by the introduction of a maturation delay parameter. Further, an explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is derived by using the theory of normal forms and center manifolds, and numerical simulations supporting the theoretical analysis are also given.     

液面的分叉与稳定性分析

本文讨论液体层在内聚力以及液体与外界相互作用下,其表面形状出现的一类分叉现象。利用分叉的基本理论,我们得到了这类现象产生的必要条件。接着,我们给出了在分叉点附近的奇异摄动解。最后,利用极小势能原理讨论了分叉解的稳定性。     

Bifurcation structure of rotating wave solutions of the Fitzhugh-Nagumo equations

The FitzHugh-Nagumo (FHN) equations are model equations for nerve cell behavior. They support traveling wave solutions which depend on certain parameters. In this paper, a two parameter study of rotating wave solutions (i.e. periodic wavetrains) are considered. These solutions arise from bifurcations of stationary equilibria. The local bifurcation equations are analyzed to determine bifurcation directions as functions of the parameters. In addition, dependence on parameters is computed by numerical continuation and properties of the rotating wave solutions are summarized in parameter space. Finally, some of the biological implications are discussed.     

Bifurcation and stability results for an equation arising in population genetics

Summary We consider a semilinear elliptic boundary value problem, which arises in population genetics. Although there is no obvious corresponding linearized problem, we establish, by using Implicit Function Theorem methods, necessary and sufficient conditions forbifurcation to occur from a branch of trivial solutions. The stability of the bifurcating solutions is also investigated.