零点不可导时锥映射的歧点与正特征元的全局结构

研究了半序Banach空间中一类非线性锥映射歧点的存在性与正特征元的全局结构.与已知文献不同的是,不要求算子在零点沿着锥Frechet可微. 作为应用,讨论了一类椭圆型偏微分方程边值问题正解的歧点与全局结构.     

Fundamental solutions and asymptotic numerical methods for bifurcation analysis of nonlinear bi‐harmonic problems

New algorithms, combining asymptotic numerical method (ANM) and method of fundamental solutions, are proposed to compute bifurcation points on branch solutions of a nonlinear bi‐harmonic problem. Three methods, mainly based on asymptotic developments framework, are then proposed. The first one consists in exploiting the ANM step accumulation close to the bifurcation points on a solution branch, the second method allows the introduction of an indicator that vanishes at the bifurcation points, and finally the first real root of the Padé approximant denominator represents the third bifurcation indicator. Two numerical examples are considered to analyze the robustness of these algorithms.     

Comparison of the bifurcation curves of a two-variable and a three-variable circadian rhythm model

Two dynamical systems describing the circadian fluctuation of two proteins (PER and TIM) in cells are compared. A simplified model with two variables has already been investigated. Detailed study of the possible bifurcation has been carried out. Periodic solutions of the differential equations with 24-h period have been obtained numerically. Here the general, more realistic model having three variables is investigated. The possible phase portraits and local bifurcations are studied in detail. The saddle-node and Hopf-bifurcation curves are determined in the plane of two parameters by using the parametric representation method. Using these curves the number and the type of the stationary points can be determined. The relation of the two bifurcation curves and the Takens–Bogdanov bifurcation points are also studied. The bifurcation curves are compared to those obtained for the simplified two-variable system.     

On a bifurcation governed by hysteresis nonlinearity

We consider autonomous systems with a nonlinear part depending on a parameter and study Hopf bifurcations at infinity. The nonlinear part consists of the nonlinear functional term and the Prandtl--Ishlinskii hysteresis term. The linear part of the system has a special form such that the close-loop system can be considered as a hysteresis perturbation of a quasilinear Hamiltonian system. The Hamiltonian system has a continuum of arbitrarily large cycles for each value of the parameter. We present sufficient conditions for the existence of bifurcation points for the non-Hamiltonian system with hysteresis. These bifurcation points are determined by simple characteristics of the hysteresis nonlinearity.     

Concurrent homoclinic bifurcation and Hopf bifurcation for a class of planar Filippov systems

This paper investigates both homoclinic bifurcation and Hopf bifurcation which occur concurrently in a class of planar perturbed discontinuous systems of Filippov type. Firstly, based on a geometrical interpretation and a new analysis of the so-called successive function, sufficient conditions are proposed for the existence and stability of homoclinic orbit of unperturbed systems. Then, with the discussion about Poincaré map, bifurcation analyses of homoclinic orbit and parabolic–parabolic (PP) type pseudo-focus are presented. It is shown that two limit cycles can appear from the two different kinds of bifurcation in planar Filippov systems.     

Smooth bifurcation for variational inequalities based on the implicit function theorem

We present a certain analog for variational inequalities of the classical result on bifurcation from simple eigenvalues of Crandall and Rabinowitz. In other words, we describe the existence and local uniqueness of smooth families of nontrivial solutions to variational inequalities, bifurcating from a trivial solution family at certain points which could be called simple eigenvalues of the homogenized variational inequality. If the bifurcation parameter is one-dimensional, the main difference between the case of equations and the case of variational inequalities (when the cone is not a linear subspace) is the following: For equations two smooth half-branches bifurcate, for inequalities only one. The proofs are based on scaling techniques and on the implicit function theorem. The abstract results are applied to a fourth order ODE with pointwise unilateral conditions (an obstacle problem for a beam with the compression force as the bifurcation parameter).     

Discretizing the fold bifurcation—A conjugacy result

In the vicinity of fold bifurcation points, the time-h exact and the stepsize-h discretized dynamics are shown to be equivalent via a two-parameter family of conjugacies. The problem of optimal conjugacy estimates remains open. Dedicated to the memory of Professor Miklós Farkas     

Dynamics and bifurcation study on an extended Lorenz system

In this paper, we study dynamics and bifurcation of limit cycles in a recently developed new chaotic system, called extended Lorenz system. A complete analysis is provided for the existence of limit cycles bifurcating from Hopf critical points. The system has three equilibrium solutions: a zero one at the origin and two non-zero ones at two symmetric points. It is shown that the system can either have one limit cycle around the origin, or three limit cycles enclosing each of the two symmetric equilibria, giving a total six limit cycles. It is not possible for the system to have limit cycles simultaneously bifurcating from all the three equilibria. Simulations are given to verify the analytical predictions.     

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.     

Global dynamics,forbidden set,and transcritical bifurcation of a one-dimensional discrete-time laser model

We study the dynamical properties about fixed points, the existence of prime period and periodic points, and transcritical bifurcation of a one-dimensional laser model in ${\mathbb{R}}^{+}$. For the special case, we explore the global dynamics about fixed points, boundedness of positive solution, construction of invariant rectangle, existence of prime period-2 solution, construction of forbidden set, the existence of a prime period and periodic points, and transcritical bifurcation of the discrete-time laser model. Finally, theoretical results are illustrated using numerical simulations.     

Numerical detection of symmetry breaking bifurcation points with nonlinear degeneracies

A numerical tool for the detection of degenerated symmetry breaking bifurcation points is presented. The degeneracies are classified and numerically processed on -D restrictions of the bifurcation equation. The test functions that characterise each of the equivalence classes are constructed by means of an equivariant numerical version of the Liapunov-Schmidt reduction. The classification supplies limited qualitative information concerning the imperfect bifurcation diagrams of the detected bifurcation points.

     

Double Hopf bifurcation and chaos in Liu system with delayed feedback

In this paper, we consider the stability of equilibria, Hopf and double Hopf bifurcation in Liu system with delay feedback. Firstly, we identify the critical values for stability switches and Hopf bifurcationusing the method of bifurcation analysis. When we choose appropriate feedback strength and delay, two symmetrical nontrivial equilibria of Liusystem can be controlled to be stable at the same time, and the stable bifurcating periodic solutions occur in the neighborhood of the two equilibria at the same time. Secondly, by applying the normal form method and center manifold theory,the normal form near the double Hopf bifurcation, as well as classifications of local dynamics are analyzed. Furthermore, we give the bifurcation diagram to illustrate numerically that a family of stable periodic solutions bifurcated from Hopf bifurcation occur in a large region of delay and the Liu system with delay can appear the phenomenon of ``chaos switchover''.     

奇异三阶积分边值问题正解的全局分歧

研究带Riemann-Stieltjes积分边值条件的奇异三阶积分边值问题正解的全局分歧结构.首先,利用相关文献,获得了此类问题的格林函数并推证其满足的性质,同时可获得此类问题等价于一个全连续算子方程;其次,在满足所给的条件时,利用Krein-Rutmann定理建立了此类问题对应的线性问题存在简单的主特征值;最后,当非线性项在零和无穷远处满足非渐进线性增长条件、参数满足不同范围的值时,利用Dancer全局分歧定理、Zeidler全局分歧定理和序列集取极限的方法,建立了此类问题正解的全局结构,进而获得了正解的存在性.     

HARMONIC AND SUBHARMONIC BIFURCATION IN THE BRUSSEL MODEL WITH PERIODIC FORCE

1.IntroductionABrusselatorisoneofthebestexaminedmodelchemicalreactionswhichconsistsoffourstepsItisshowninFig.1schematicallyandisrepreselltedbythefollowingsetofequationsffevedFebruary6,1995.*~workissupportedbytheNationalNaturalScienceFOundationmanYuan"TermsinChina.ThemodelweadoptistheoneduetoPrigogine,Lefever,andNicolis(Brusselator)t'.Fig.1.'TheschematicdiagramofBrusselmodel(AdditionalcirculararrowsrepreseDttheexistenceofautocatalysis.)Herexandystandfortheconcentrationsofreferencereacta…     

On the dynamics of a degenerate parabolic equation: global bifurcation of stationary states and convergence

We study the dynamics of a degenerate parabolic equation with a variable, generally non-smooth diffusion coefficient, which may vanish at some points or be unbounded. We show the existence of a global branch of nonnegative stationary states, covering both the cases of a bounded and an unbounded domain. The global bifurcation of stationary states, implies-in conjuction with the definition of a gradient dynamical system in the natural phase space-that at least in the case of a bounded domain, any solution with nonnegative initial data tends to the trivial or the nonnegative equilibrium. Applications of the global bifurcation result to general degenerate semilinear as well as to quasilinear elliptic equations, are also discussed. Mathematics Subject Classification (1991) 35B40, 35B41, 35R05     

Hopf bifurcation in an activator–inhibitor system with network

A network is introduced to describe the activator–inhibitor system, where the network structure represents the movement directions of molecule random walk. We show that a Hopf bifurcation occurs in the activator–inhibitor system by the linear stability analysis. By an extension of the center manifold approach, we also prove that the Hopf bifurcation is stable and its direction is backward.     

一类非线性系统的周期扰动Hopf分支

研究了小周期扰动对一类存在Hopf分支的非线性系统的影响.特别是应用平均法讨论了扰动频率与Hopf分支固有频率在共振及二阶次调和共振的情形周期解分支的存在性.表明了在某些参数区域内,系统存在调和解分支和次调和解分支,并进一步讨论了二阶次调和分支周期解的稳定性.     

Neimark-Sacker bifurcation of a semi-discrete hematopoiesis model

In this paper, we derive a semi-discrete system for a nonlinear model of blood cell production. The local stability of its fixed points is investigated by employing a key lemma from [23, 24]. It is shown that the system can undergo Neimark-Sacker bifurcation. By using the Center Manifold Theorem, bifurcation theory and normal form method, the conditions for the occurrence of Neimark-Sacker bifurcation and the stability of invariant closed curves bifurcated are also derived. The numerical simulations verify our theoretical analysis and exhibit more complex dynamics of this system.     

A K-theoretical invariant and bifurcation for a parameterized family of functionals

Let F:={fx:x∈X} be a family of functionals defined on a Hilbert manifold and smoothly parameterized by a compact connected orientable n-dimensional manifold X, and let be a smooth section of critical points of F. The aim of this paper is to give a sufficient topological condition on the parameter space X which detects bifurcation of critical points for F from the trivial branch. Finally we are able to give some quantitative properties of the bifurcation set for perturbed geodesics on semi-Riemannian manifolds.     

一类带非单调转化率的捕食-食饵模型的全局分歧

主要研究了一类带非单调转化率的捕食-食饵模型,分别以生长率a和b为分歧参数,运用度理论和分歧理论讨论了这类模型在齐次第一边界条件下全局分歧结构．