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

ＤＹＮＡＭＩＣＡＬＢＥＨＡＶＩＯＲＳＦＯＲＡＴＨＲＥＥ－ＤＩＭＥＮＳＩＯＮＡＬＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮＩＮＣＨＥＭＩＣＡＬＳＹＳＴＥＭＬＩＮＹＩＰＩＮＧ（ＳｅｃｔｉｏｎｏｆＭａｔｈｅｍａｔｉｃｓ，ＫｕｎｍｉｎｇＩｎｓｔｉｔｕｔｅｏｆＴ...     

OSCILLATORY AND ASYMPTOTIC BEHAVIORS OF FIRST ORDER FUNCTIONAL DIFFERENTIAL EQUATIONS

In this paper the author discusses the following first order functional differentialequations: x'(t) +integral from n=a to b p(t, ξ)x[g(t, ξ)]dσ(ξ)=0, (1) x'(t) +integral from n=a to b f(t, ξ, x[g(t, ξ)])dσ(ξ)=0. (2)Some suffcient conditions of oscillation and nonoseillafion are obtained, and two asymptolioproperties and their criteria are given. These criferia are better than those in [1, 2], and canbe used to the following equations: x'(t) + sum from i=1 to n p_i(t)x[g_i(t)] =0, (3) x'(t) + sum from i=1 to n f_i(t, x[g_i(t)] =0. (4)     

ON THE EXISTENCE OF PERIODIC SOLUTIONS TO CERTAIN FOURTH ORDER DIFFERENTIAL EQUATION

ＯＮＴＨＥＥＸＩＳＴＥＮＣＥＯＦＰＥＲＩＯＤＩＣＳＯＬＵＴＩＯＮＳＴＯＣＥＲＴＡＩＮＦＯＵＲＴＨＯＲＤＥＲＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮＦｅｎｇＣｈｕｎｈｕａ（冯春华）（ＧｕａｎｇｘｉＮｏｒｍａｌＵｎｉｖｅｒｓｉｔｙ，广西师范大学，邮编：５...     

EXISTENCE OF PERIODIC SOLUTION FOR HIGHER ORDER NONLINEAR DIFFERENTIAL EQUATION

The existence of periodic solutions is proved for the higher order nonlinear differential equation by applying Leray-Schauder principle and Wirtinger's inequality.     

EXISTENCE OF SOLUTIONS OF NONLINEAR TWO AND THREE-POINT BOUNDARY VALUE PROBLEMS FOR FOURTH ORDER NONLINEAR DIFFERENTIAL EQUATIONS

In this paper, the two and three-point boundsry problems (with nonlinear boundary conditions)for the genaral noniinear equations of fourth order are discussed.We have set some grups of the assurnpion coditions and proved the existence of solutins for corresponding boundary value problems under these conditons.     

A NEW DETECTING METHOD FOR CONDITIONS OF EXISTENCE OF HOPF BIFURCATION

ＡＮＥＷＤＥＴＥＣＴＩＮＧＭＥＴＨＯＤＦＯＲＣＯＮＤＩＴＩＯＮＳＯＦＥＸＩＳＴＥＮＣＥＯＦＨＯＰＦＢＩＦＵＲＣＡＴＩＯＮＳＨＥＮＪＩＡＱＩ（沈家骐）；ＪＩＮＧＺＨＵＪＵＮ（井竹君）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＳｈａｎｄｏｎｇＵｎ...     

THE INSTABILITY OF A DIFFERENTIAL EQUATION WITH ODD ORDER

Our work pays much attention to the instability of an nth-order differential equation whose coefficients are not all constants, where n is an odd integer and n ≥ 3. Finally, we obtained a sufficient condition of the instability.     

ON STABILITY OF SOLUTION OF A CERTAIN FOURTH ORDER VECTOR DIFFERENTIAL EQUATIONS

ＯＮ　ＳＴＡＢＩＬＩＴＹ　ＯＦ　ＳＯＬＵＴＩＯＮ　ＯＦ　Ａ　ＣＥＲＴＡＩＮ　ＦＯＵＲＴＨ　ＯＲＤＥＲ　ＶＥＣＴＯＲ　ＤＩＦＦＥＲＥＮＴＩＡＬ　ＥＱＵＡＴＩＯＮＳ　ＯＮＳＴＡＢＩＬＩＴＹＯＦＳＯＬＵＴＩＯＮＯＦＡＣＥＲＴＡＩＮＦＯＵＲＴＨＯＲＤＥＲＶＥ...     

ASYMPTOTIC CHARACTERISTIC AND OSCILLATION OF HIGH ORDER NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATION

ＡＳＹＭＰＴＯＴＩＣＣＨＡＲＡＣＴＥＲＩＳＴＩＣＡＮＤＯＳＣＩＬＬＡＴＩＯＮＯＦＨＩＧＨＯＲＤＥＲＮＥＵＴＲＡＬＦＵＮＣＴＩＯＮＡＬＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮ￥ＷｅｎＸｉａｎｇｃａｉ（温香彩）（ＨｅｎａｎＮｏｒｍａｌＵｎｉｖｅｒｓｉｔｙ...     

BOUNDARY DIFFERENCE-INTEGRAL EQUATION METHOD AND ITS ERROR ESTIMATES FOR SECOND ORDER HYPERBOLIC PARTIAL DIFFERENTIAL EQUATION

Combining difference method and boundary integral equation method,we propose a new numerical method for solving initial-boundary value problem of second order hyperbolic partial differential equations defined on a bounded or unbounded domain in R~3 and obtain the error estimates of the approximate solution in energy norm and local maximum norm.     

Hopf bifurcation for delay differential equation with application to machine tool chatter

In the machining process, unstable self-excited vibrations known as regenerative chatter can occur, causing excessive tool wear or failure, and a poor surface finish on the machined workpiece, hence the relevant measures must be taken to predict and avoid this phenomenon of instability. In this paper, we propose a weakly nonlinear model with square and cubic terms in both structural stiffness and regenerative terms, to represent self-excited vibrations in machining. It is proved that Hopf bifurcation exists when bifurcation parameter equals a critical value, a formula for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions are given by using the normal form method and center manifold theorem. Numerical simulations show excellent agreement with the theoretical results.     

Hopf bifurcation analysis of a general non-linear differential equation with delay

This work represents Hopf bifurcation analysis of a general non-linear differential equation involving time delay. A special form of this equation is the Hutchinson–Wright equation which is a mile stone in the mathematical modeling of population dynamics and mathematical biology. Taking the delay parameter as a bifurcation parameter, Hopf bifurcation analysis is studied by following the theory in the book by Hazzard et al. By analyzing the associated characteristic polynomial, we determine necessary conditions for the linear stability and Hopf bifurcation. In addition to this analysis, the direction of bifurcation, the stability and the period of a periodic solution to this equation are evaluated at a bifurcation value by using the Poincaré normal form and the center manifold theorem. Finally, the theoretical results are supported by numerical simulations.     

Hopf bifurcation and analysis of equilibrium for a third-order differential equation in a model of competition

1. IntroductionOne of the more challenging aspects of mathematical biology is competition modelling.Although the mathematical idea is simple[1], this type of modelling is so difficult to carryout in any generality since there are so maily ways for a population to compete.The simplest form of competition is called eXPloitative competition. This occurs whenone or more populations compete for the same resources such as a common food supplyor a growth-limiting nutriellt. A simple example of this…     

Hopf bifurcation in models for pertussis epidemiology

Pertussis (whooping cough) incidence in the United States has oscillated with a period of about four years since data was first collected in 1922. An infection with pertussis confers immunity for several years, but then the immunity wanes, so that reinfection is possible. A pertussis reinfection is mild after partial loss of immunity, but the reinfection can be severe after complete loss of immunity. Three pertussis transmission models with waning of immunity are examined for periodic solutions. Equilibria and their stability are determined. Hopf bifurcation of periodic solutions around the endemic equilibrium can occur for some parameter values in two of the models. Periods of about four years are found for epidemiologically reasonable parameter values in two of these models.     

Numerical stability analysis and computation of Hopf bifurcation points for delay differential equations

We present a numerical technique for the stability analysis and the computation of branches of Hopf bifurcation points in nonlinear systems of delay differential equations with several constant delays. The stability analysis of a steady-state solution is done by a numerical implementation of the argument principle, which allows to compute the number of eigenvalues with positive real part of the characteristic matrix. The technique is also used to detect bifurcations of higher singularity (Hopf and fold bifurcations) during the continuation of a branch of Hopf points. This allows to trace new branches of Hopf points and fold points.     

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.     

An example of Bautin-type bifurcation in a delay differential equation

In a previous paper we gave sufficient conditions for a system of delay differential equations to present Bautin-type bifurcation. In the present work we present an example of delay equation that satisfies these conditions.     

Global attractivity, oscillation and Hopf bifurcation for a class of diffusive hematopoiesis models

In this paper, we discuss the special diffusive hematopoiesis model

Analysis of the Hopf bifurcation for the family of angiogenesis models

In this paper we study a family of models with delays describing the process of angiogenesis, that is a physiological process involving the growth of new blood vessels from pre-existing ones. This family includes the well-known models of tumour angiogenesis proposed by Hahnfeldt et al. and d?Onofrio-Gandolfi and is based on the Gompertz type of the tumour growth. As a consequence we start our analysis from the influence of delay onto the Gompertz model dynamics. The family of models considered in this paper depends on two time delays and a parameter α∈[0,1] which reflects how strongly the vessels dynamics depends on the ratio between tumour and vessels volume. We focus on the analysis of the model in three cases: one of the delays is equal to 0 or both delays are equal, depending on the parameter α. We study the stability switches, the Hopf bifurcation and the stability of arising periodic orbits for different α∈[0,1], especially for α=1 and α=0 which reflects the Hahnfeldt et al. and the d?Onofrio-Gandolfi models. For comparison we use also the value α=1/2.