非线性颤振系统中既是超临界又是亚临界的Hopf分岔点研究

研究了二元机翼非线性颤振系统的Hopf分岔．应用中心流形定理将系统降维,并利用复数正规形方法得到了以气流速度为分岔参数的分岔方程．研究发现,分岔方程中一个系数不含分岔参数的一次幂,故使得分岔具有超临界和亚临界双重性质．用等效线性化法和增量谐波平衡法验证了所得结果．     

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.     

HOPF BIFURCATION AND OTHER DYNAMICAL BEHAVIORS FOR A FOURTH ORDER DIFFERENTIAL EQUATION IN MODELS OF INFECTIOUS DISEASE

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Global estimates and energy identities for elliptic systems with antisymmetric potentials

We derive global estimates in critical scale invariant norms for solutions of elliptic systems with antisymmetric potentials and almost holomorphic Hopf differential in two dimensions. Moreover, we obtain new energy identities in such norms for sequences of solutions of these systems. The results apply to harmonic maps into general target manifolds and surfaces with prescribed mean curvature. In particular, the results confirm a conjecture of Rivière in the two-dimensional setting.     

Nonlinear waves in complex oscillator network with delay

We investigate the spatio-temporal patterns of Hopf bifurcating periodic solutions in a delay complex oscillator network. Firstly, we calculate the critical values of Hopf bifurcation. Secondly, the bifurcating periodic solutions can take on two cases: one is synchronization or anti-synchronization, and another is the coexistence of two phase-locked, N mirror-reflecting and N standing waves, because the system has group symmetry. Finally, the stability of these nonlinear oscillations is determined using the center manifold theorem and normal form method with the imaginary eigenvalues being simple and double.     

The nonbifurcation of periodic solutions when the variational matrix has a zero eigenvalue

It is assumed that the variational matrix of the 2-dimensional system x′ = F(x, ?) has at least one zero eigenvalue rather than the usual Hopf assumption of two conjugate pure imaginary eigenvalues. It is then shown that genetically, although one may expect a bifurcation of stationary solutions, a bifurcation of periodic solutions will not occur.     

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…     

Solutions of yang-baxter equation in an endomorphism semigroup and quasi-(co)braided almost bialgebras ∗

The aim of this paper is to study the solutions of the Yang-Baxter equation in the endomorphism semigroup of the tensor product of a vector space. As preparation, we introduce the concepts of quasi-braided almost bialgebra (see also [10]) and quasi-cobraided almost bialgebra, and discuss some of their properties. In particular, it is shown that the quasi R-matrix R of every quasi-braided almost weak Hopf algebra is regular under von Neumann's meaning. The solutions of the Yang-Baxter equation in the endomorphism semigroups are constructed respectively from every quasi-braided almost bialgebra and every quasi-cobraided almost bialgebra. As examples, we explain how to build solutions of the Yang-Baxter equation from some weak Hopf algebras and all Clifford monoids. Finally, the FRT construction is given so as to build every solution of the Yang-Baxter equation from a quasi-cobraided bialgebra.     

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

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Periodic solutions of Abel differential equations

For a class of polynomial non-autonomous differential equations of degree n, we use phase plane analysis to show that each equation in this class has n periodic solutions. The result implies that certain rigid two-dimensional systems have at most one limit cycle which appears through multiple Hopf bifurcation.     

Pairing and Quantum Double of Multiplier Hopf Algebras

We define and investigate pairings of multiplier Hopf (*-)algebras which are nonunital generalizations of Hopf algebras. Dual pairs of multiplier Hopf algebras arise naturally from any multiplier Hopf algebra A with integral and its dual Â. Pairings of multiplier Hopf algebras play a basic rôle, e.g., in the study of actions and coactions, and, in particular, in the relation between them. This aspect of the theory is treated elsewhere. In this paper we consider the quantum double construction out of a dual pair of multiplier Hopf algebras. We show that two dually paired regular multiplier Hopf (*-)algebras A and B yield a quantum double which is again a regular multiplier Hopf (*-)algebra. If A and B have integrals, then the quantum double also has an integral. If A and B are Hopf algebras, then the quantum double multiplier Hopf algebra is the usual quantum double. The quantum double construction for dually paired multiplier Hopf (*-)algebras yields new nontrivial examples of multiplier Hopf (*-)algebras.     

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.     

The energy of Hopf vector fields

The 3-dimensional Hopf vector field is shown to be a stable harmonic section of the unit tangent bundle. In contrast, higher dimensional Hopf vector fields are unstable harmonic sections; indeed, there is a natural variation through smooth unit vector fields which is locally energy-decreasing, and whose asymptotic limit is a singular vector field of finite energy. This energy is explicitly calculated, and conjectured to be the infimum of the energy functional over all smooth unit vector fields. Received: 17 March 1999     

Hopf bifurcations to quasi-periodic solutions for the two-dimensional plane Poiseuille flow

This paper studies various Hopf bifurcations in the two-dimensional plane Poiseuille problem. For several values of the wavenumber α, we obtain the branch of periodic flows which are born at the Hopf bifurcation of the laminar flow. It is known that, taking α ≈ 1, the branch of periodic solutions has several Hopf bifurcations to quasi-periodic orbits. For the first bifurcation, calculations from other authors seem to indicate that the bifurcating quasi-periodic flows are stable and subcritical with respect to the Reynolds number, Re. By improving the precision of previous works we find that the bifurcating flows are unstable and supercritical with respect to Re. We have also analysed the second Hopf bifurcation of periodic orbits for several α, to find again quasi-periodic solutions with increasing Re. In this case the bifurcated solutions are stable to superharmonic disturbances for Re up to another new Hopf bifurcation to a family of stable 3-tori. The proposed numerical scheme is based on a full numerical integration of the Navier-Stokes equations, together with a division by 3 of their total dimension, and the use of a pseudo-Newton method on suitable Poincaré sections. The most intensive part of the computations has been performed in parallel. We believe that this methodology can also be applied to similar problems.     

Algebras Graded by Discrete Doi–Hopf Data and the Drinfeld Double of a Hopf Group-Coalgebra

We study Doi–Hopf data and Doi–Hopf modules for Hopf group-coalgebras. We introduce modules graded by a discrete Doi–Hopf datum; to a Doi–Hopf datum over a Hopf group coalgebra, we associate an algebra graded by the underlying discrete Doi–Hopf datum, using a smash product type construction. The category of Doi–Hopf modules is then isomorphic to the category of graded modules over this algebra. This is applied to the category of Yetter–Drinfeld modules over a Hopf group coalgebra, leading to the construction of the Drinfeld double. It is shown that this Drinfeld double is a quasitriangular ${\mathbb{G}}$ -graded Hopf algebra.     

Numerical solution of Hopf bifurcation problems

We present two numerical methods for the solution of Hopf bifurcation problems involving ordinary differential equations. The first one consists in a discretization of the continuous problem by means of shooting or multiple shooting methods. Thus a finite-dimensional bifurcation problem of special structure is obtained. It may be treated by appropriate iterative algorithms. The second approach transforms the Hopf bifurcation problem into a regular nonlinear boundary value problem of higher dimension which depends on a perturbation parameter ?. It has isolated solutions in the ?-domain of interest, so that conventional discretization methods can be applied. We also consider a concrete Hopf bifurcation problem, a biological feedback inhibition control system. Both methods are applied to it successfully.     

An Algebraic Framework for Group Duality

A Hopf algebra is a pair (A, Δ) whereAis an associative algebra with identity andΔa homomorphism formAtoA⊗Asatisfying certain conditions. If we drop the assumption thatAhas an identity and if we allowΔto have values in the so-called multiplier algebraM(A⊗A), we get a natural extension of the notion of a Hopf algebra. We call this a multiplier Hopf algebra. The motivating example is the algebra of complex functions with finite support on a group with the comultiplication defined as dual to the product in the group. Also for these multiplier Hopf algebras, there is a natural notion of left and right invariance for linear functionals (called integrals in Hopf algebra theory). We show that, if such invariant functionals exist, they are unique (up to a scalar) and faithful. For a regular multiplier Hopf algebra (A, Δ) (i.e., with invertible antipode) with invariant functionals, we construct, in a canonical way, the dual (Â, Δ). It is again a regular multiplier Hopf algebra with invariant functionals. It is also shown that the dual of (Â, Δ) is canonically isomorphic with the original multiplier Hopf algebra (A, Δ). It is possible to generalize many aspects of abstract harmonic analysis here. One can define the Fourier transform; one can prove Plancherel's formula. Because any finite-dimensional Hopf algebra is a regular multiplier Hopf algebra and has invariant functionals, our duality theorem applies to all finite-dimensional Hopf algebras. Then it coincides with the usual duality for such Hopf algebras. But our category of multiplier Hopf algebras also includes, in a certain way, the discrete (quantum) groups and the compact (quantum) groups. Our duality includes the duality between discrete quantum groups and compact quantum groups. In particular, it includes the duality between compact abelian groups and discrete abelian groups. One of the nice features of our theory is that we have an extension of this duality to the non-abelian case, but within one category. This is shown in the last section of our paper where we introduce the algebras of compact type and the algebras of discrete type. We prove that also these are dual to each other. We treat an example that is sufficiently general to illustrate most of the different features of our theory. It is also possible to construct the quantum double of Drinfel'd within this category. This provides a still wider class of examples. So, we obtain many more than just the compact and discrete quantum within this setting.     

DYNAMICAL BEHAVIORS FOR A THREE－DIMENSIONAL DIFFERENTIAL EQUATION IN CHEMICAL SYSTEM

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Global dynamics of Nicholsonʼs blowflies equation revisited: Onset and termination of nonlinear oscillations

We revisit Nicholson?s blowflies model with natural death rate incorporated into the delay feedback. We consider the delay as a bifurcation parameter and examine the onset and termination of Hopf bifurcations of periodic solutions from a positive equilibrium. We show that the model has only a finite number of Hopf bifurcation values and we describe how branches of Hopf bifurcations are paired so the existence of periodic solutions with specific oscillation frequencies occurs only in bounded delay intervals. The bifurcation analysis and the Matlab package DDE-BIFTOOL developed by Engelborghs et al. guide some numerical simulations to identify ranges of parameters for coexisting multiple attractive periodic solutions.