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1.
Huaizhong Zhao 《应用数学学报(英文版)》1993,9(4):348-366
The existence of the Hopf bifurcation of a complex ordinary differential equation system in the complex domain is studied in this paper by using the complex qualitative theory. In the complex domain, we conclude that the Hopf bifurcation appears for both directions of the parameter. The formulae of the Hopf bifurcation are also given in this paper. 相似文献
2.
In this paper, we consider the discretization of parameter-dependent delay differential equation of the formIt is shown that if the delay differential equation undergoes a Hopf bifurcation at τ=τ*, then the discrete scheme undergoes a Hopf bifurcation at τ(h)=τ*+O(hp) for sufficiently small step size h, where p1 is the order of the Runge–Kutta method applied. The direction of numerical Hopf bifurcation and stability of bifurcating invariant curve are the same as that of delay differential equation. 相似文献
3.
Summary
Some years ago W.Velte [1]and A.Marino [2]have studied the problem of convective motions in a incompressible viscous fluid in dimensions two and three, respectively, focusing their interest in the case of nonuniqueness of the solution of the nonlinear system. They look at it as a bifurcation problem and prove the existence of solutions bifurcating from same line of trivial solutions under the hypothesis that the linearized operator has eigenvalues of odd multiplicities. They observe that the operators involved are not potential operators, thus variational tools can not be applied. In this paper, we prove that, in case of dimension two, for almost every domain , all the eigenvalues of the linearized operator are simple. Our procedure is related to that in [3];the fact that we deal with the system here, necessitates, however, some basic changes.Supported in part by C.N.R. (G.N.A.F.A.). 相似文献
4.
In this paper a mathematical model of AIDS is investigated. The conditions of the existence of equilibria and local stability of equilibria are given. The existences of transcritical bifurcation and Hopf bifurcation are also considered, in particular, the conditions for the existence of Hopf bifurcation can be given in terms of the coefficients of the characteristic equation. The method extends the application of the Hopf bifurcation theorem to higher differential equations which occur in biological models, chemical models, and epidemiological models etc.This project is supported by the National Science Foundation Tian Yuan Terms and LNM Institute of Mechanics Academy of Science.This project is supported by the National and Yunnan Province Natural Science Foundation of China. 相似文献
5.
We consider a family of semilinear evolution equations in Hilbert space of the form with, in general, unbounded operators *A(λ), F(λ·) depending analytically on a real parameter λ. We assume that the origin is a stationary solution, i.e. F(λ,0) = 0, for all λ ε R and that the linearization (with respect to u) at the origin is given by du/dt + A(λ)u = 0. Our essential assumption is the following: A(λ) possesses one pair of simple complex conjugate eigenvalues μ(λ) = Re μ(λ) ± i Im μ(λ) such that Im μ(0) > 0 and for some m ε N or If m = 1 the curves of eigenvalues μ(λ) cross the imaginary axis transversally at ±i Im μ(0). In this case a unique branch of periodic solutions emanates from the origin at λ = 0 which is commonly called Hopf bifurcation. If μ(λ) and the imaginary axis are no longer transversal, i.e. m > 1, we call a bifurcation of periodic solutions, if it occurs, a generalized Hopf bifurcation. It is remarkable that up to m such branches may exist. Our approach gives the number of bifurcating solutions, their direction of bifurcation, and its asymptotic expansion. We regain the results of D. Flockerzi who established them in a completely different way for ordinary differential equations. 相似文献
6.
7.
In this paper, we prove that the linearized elasticity system has
no eigenvalues in two geometric situations: the whole space
and a local perturbation of the half-space. We consider the Lamé coefficients and the
density varying in an unbounded part of the domain. For the whole space,
we use the operations curl and div
to reduce our system to a scalar problem
and use a limiting absorption principle for the reduced scalar equation
given by the partial Fourier transform. For the perturbed half-space, this
decompositions being no longer valid, we give an other method based on a
pseudo-decomposition using the operations div
and curl in the horizontal
direction. In contrast to the whole space case, the reduced problems depend
strongly on the dual Fourier variable which do not enable us to use same
techniques. To study these reduced problems, we use the analytic theory of
linear operators. 相似文献
8.
Gerhard Dangelmayr Werner Güttinger Michael Wegelin 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》1993,44(4):595-638
A generic Hopf bifurcation involving an eight dimensional center eigenspace is considered for systems possessing aD
3 ×D
3-symmetry. This kind of Hopf bifurcation can occur in systems of three interacting groups of oscillators, where each group itself is composed of three individual oscillators. The terminology micro and Macro is introduced here to denote symmetry operations acting on individual oscillators and on the whole groups, respectively. The normal form for the Hopf bifurcation admits 11 distinct periodic solutions with maximal isotropy subgroups. These are classified and their branching-types and stabilities are determined in terms of the cubic and relevant quintic coefficients of the normal form. The symmetry properties of these solutions when only certain Macro variables in the oscillator groups are observed are discussed in the context of the remaining symmetry. Furthermore, the relation of the normal form to the corresponding one for a singleD
3-symmetry is established by restricting the system to four dimensional fixed point subspaces associated with submaximal isotropy subgroups. Based on this information the possibility of quasiperiodic solutions and of a particular class of heteroclinic cycles is discussed.Dedicated to Prof. Klaus Kirchgässner on the occasion of his 60th birthday 相似文献
9.
《Nonlinear Analysis: Real World Applications》2007,8(4):1144-1170
We apply the equivariant degree method to a Hopf bifurcation problem for a symmetric system of neutral functional differential equations, which reflects two symmetrically coupled configurations of the lossless transmission lines. The spectral information of the linearized system is extracted and translated into a bifurcation invariant, which carries structural information of the solution set. We calculate the values of the bifurcation invariant by following the standard computational scheme and using a specially developed package. The computational results, as well as the minimal number of bifurcating branches and their least symmetries are summarized. 相似文献
10.
The purpose of this paper is to study a non-Kolmogrov type prey-predator system. First, we investigate the linear stability of the model by analyzing the associated characteristic equation of the linearized system. Second, we show that the system exhibits the Hopf bifurcation. The stability and direction of the Hopf bifurcation are determined by applying the norm form theory and center manifold theorem. Finally, numerical simulations are performed to illustrate the obtained results. 相似文献
11.
Qiu Weisheng 《组合设计杂志》1994,2(3):117-129
Applying the method that we presented in [19], in this article we prove: “Let G be an elementary abelian p-group. Let n = dn1. If d(≠ p) is a prime not dividing n1, and the order w of d mod p satisfies $ w > \frac{{d^2}}{3} $, then the Second Multiplier Theorem holds without the assumption n1 > λ, except that only one case is yet undecided: w ≤ d2, and $ \frac{{p - 1}}{{2w}} \ge 3 $, and t is a quadratic residue mod p, and t is not congruent to $ x^{\frac{{p - 1}}{{2w}}j} $ (mod p) (1 ≤ j < 2w), where t is an integer meeting the conditions of Second Multiplier Theorem, and x is a primitive root of p.”. © 1994 John Wiley & Sons, Inc. 相似文献
12.
Maria-Cristina Ciocci 《Journal of Difference Equations and Applications》2013,19(13):1119-1135
We investigate the bifurcation of q-periodic points from a fixed point of a 2-parameter family of n-dimensional reversible diffeomorphisms (n ≥ 4). The focus is on the codimension 2 non-semisimple 1:1 resonance case, that is, when the linearization at the fixed point has a pair of double eigenvalues on the unit circle, exp ( ± 2iπp/q) (with gcd(p, q) = 1), with geometric multiplicity 1. Through a modified version of Lyapunov Schmidt reduction, the reduced bifurcation equations are obtained. These are then analysed using reversible normal form theory and (reversible) singularity theory. We obtain the existence of two branches of symmetric q-periodic orbits bifurcating from the fixed point. We also describe the corresponding bifurcation scenario for a family of reversible systems in the case of a two-fold resonance, cfr. [12 5]. 相似文献
13.
Adriana Nenciu 《Algebras and Representation Theory》2004,7(2):159-172
We study the quasitriangular structures for a family of pointed Hopf algebras which is big enough to include Taft's Hopf algebras H
n
2, Radford's Hopf algebras H
N,n,q, and E(n). We give necessary and sufficient conditions for the Hopf algebras in our family to be quasitriangular. For the case when they are, we determine completely all the quasitriangular structures. Also, we determine the ribbon elements of the quasitriangular Hopf algebras and the quasi-ribbon elements of their Drinfel'd double. 相似文献
14.
Takashi Suzuki 《Journal of Differential Equations》1985,58(2):243-256
Using a third order Picard-Fuchs equation we show that a certain two parameter family of planar vectorfields for parameter values in a certain cone has a unique limit cycle, which is born from a Hopf bifurcation and dies in a saddle connection. This removes a superfluous hypothesis in Theorem 3.2, Chapter 13 of S. N. Chow and J. K. Hale (“Methods of Bifurcation Theory,” Springer-Verlag, New York, 1982). 相似文献
15.
In this article we study a controllability problem for an elliptic partial differential equation in which the control is the shape of the domain where the equation holds. The quantity to be controlled is the trace of the solution, with a given right hand side source term, into an open subdomain. The mapping that associates this trace to the shape of the domain is nonlinear. We first consider the linearized problem and show an approximate controllability property. We then address the same questions in the context of a finite difference discretization of the elliptic problem. We prove a local controllability result applying the Inverse Function Theorem together with a ``unique continuation' property of the underlying adjoint discrete system.
Mathematics Subject Classification (1991):35J05, 93B03, 65M06 相似文献
16.
研究了一类具有时滞的捕食—食饵系统,通过分析正平衡点处的特征方程,讨论了系统正平衡点的稳定性;以时滞作为分支参数,应用Hopf分支理论,得到了系统存在Hopf分支的充分条件. 相似文献
17.
Giovanna Carnovale 《Algebras and Representation Theory》2006,9(1):99-120
We show which H
op
-cleft extensions of k for a dual quasi-triangular Hopf algebra (H, r) are H-Azumaya. The result is given in terms of bijectivity of a map defined in terms of the universal r-form r and the 2-cocycle σ, generalizing a well-known result for the commutative and co-commutative case. We illustrate the Theorem with an explicit computation for the Hopf algebras of type E(n).Presented by A. Verschoren 相似文献
18.
In this paper we study a generalized Gause model with prey harvesting and a generalized Holling response function of type III: . The goal of our study is to give the bifurcation diagram of the model. For this we need to study saddle-node bifurcations, Hopf bifurcation of codimension 1 and 2, heteroclinic bifurcation, and nilpotent saddle bifurcation of codimension 2 and 3. The nilpotent saddle of codimension 3 is the organizing center for the bifurcation diagram. The Hopf bifurcation is studied by means of a generalized Liénard system, and for b=0 we discuss the potential integrability of the system. The nilpotent point of multiplicity 3 occurs with an invariant line and can have a codimension up to 4. But because it occurs with an invariant line, the effective highest codimension is 3. We develop normal forms (in which the invariant line is preserved) for studying of the nilpotent saddle bifurcation. For b=0, the reversibility of the nilpotent saddle is discussed. We study the type of the heteroclinic loop and its cyclicity. The phase portraits of the bifurcations diagram (partially conjectured via the results obtained) allow us to give a biological interpretation of the behavior of the two species. 相似文献
19.
As we know, for non-smooth planar systems there are foci of three different types, called focus-focus (FF), focus-parabolic (FP) and parabolic-parabolic (PP) type respectively. The Poincaré map with its analytical property and the problem of Hopf bifurcation have been studied in Coll et al. (2001) [3] and Filippov (1988) [6] for general systems and in Zou et al. (2006) [13] for piecewise linear systems. In this paper we also study the problem of Hopf bifurcation for non-smooth planar systems, obtaining new results. More precisely, we prove that one or two limit cycles can be produced from an elementary focus of the least order (order 1 for foci of FF or FP type and order 2 for foci of PP type) (Theorem 2.3), different from the case of smooth systems. For piecewise linear systems we prove that 2 limit cycles can appear near a focus of either FF, FP or PP type (Theorem 3.3). 相似文献