Degenerate Hopf bifurcations control the extended Sprott E system with only one stable equilibrium.

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具時滞的单种群模型和SIS模型的稳定性和分支分析

第4章 SIS 传染病模型 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.1 引言 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.2 稳定性和 Hopf 分支的存在性. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.3 Hopf 分支的方向和周期解的稳定性 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.4 数值模拟. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.5 本章小结. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
第2章预备知识 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1 几何准则. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Hopf 分支存在条件 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Bifurcation in epidemic models

(7)
(8)
Bifurcations in an SIRS Model (cont.)
where p= α(d + ν) , k d +γ m= , d +ν A = N0 q= k , d +ν
γ . d +ν
Saddle-Node Bifurcation
(0, 0) is an equilibrium. This is a disease free equilibrium. It is always stable. According to the formula from van den Driessche and Watmough (Math. Biosci,2002), we can calculate the reproduction number of this model and obtain R0 = 0. we should know further details of dynamical behaviors: When is the disease persistent? When does the disease die out?
SN Continue (cont.)
I ’ = I2 (A − I − R)/(1 + p I2) − m I R’=qI−R A = 11.5 p = 0.2 q = 4.2 m=5
7
6
5
4 R 3 2 1 0 0 0.5 1 1.5 I 2 2.5 3
Hopf Bifurcation
E1 is a saddle, impossible for a Hopf bifurcation, E2 is a node or a focus, it is possible to have a Hopf bifurcation, a necessary condition is that the trace of the Jacobian matrix at E2 is zero. The sign of the trace is determined by

Hopf bifurcation

Hopf bifurcationFrom Wikipedia, the free encyclopedia (Redirected from Andronov-Hopf bifurcation )Jump to: navigation , search In the mathematical theory of bifurcations , a Hopf or Poincaré–Andronov–Hopf bifurcation, named after Henri Poincaré, Eberhard Hopf , and Aleksandr Andronov , is a local bifurcation in which a fixed point of a dynamical system loses stability as a pair of complex conjugate eigenvalues of the linearization around the fixed point cross the imaginary axis of the complex plane . Under reasonably generic assumptions about the dynamical system, we can expect to see a small-amplitude limit cycle branching from the fixed point.For a more general survey on Hopf bifurcation and dynamical systems in general, see [1][2][3][4][5].Contents[hide ]● 1 Overview r 1.1 Supercritical / subcritical Hopf bifurcationsr 1.2 Remarks r1.3 Example ● 2 Definition of a Hopf bifurcation ● 3 Routh–Hurwitz criterionr 3.1 Sturm seriesr 3.2 Propositions ● 4 Example● 5 References●6 External links [edit ] Overview[edit ] Supercritical / subcritical Hopf bifurcationsThe limit cycle is orbitally stable if a certain quantity called the first Lyapunov coefficient is negative, and the bifurcation is supercritical. Otherwise it isunstable and the bifurcation is subcritical.The normal form of a Hopf bifurcation is:where z , b are both complex and λ is a parameter. WriteThe number α is called the first Lyapunov coefficient.●If α is negative then there is a stable limit cycle for λ > 0:whereThe bifurcation is then called supercritical.●If α is positive then there is an unstable limit cycle for λ < 0. The bifurcation is called subcritical.[edit ] Remarks The "smallest chemical reaction with Hopf bifurcation" was found in 1995 in Berlin, Germany [6]. The same biochemical system has been used in order to investigate how the existence of a Hopf bifurcation influences our ability to reverse-engineer dynamical systems [7].Under some general hypothesis, in the neighborhood of a Hopf bifurcation, a stable steady point of the system gives birth to a small stable limit cycle . Remark that looking for Hopf bifurcation is not equivalent to looking for stable limit cycles. First, some Hopf bifurcations (e.g. subcritical ones) do not imply the existence of stable limit cycles; second, there may exist limit cycles not related to Hopf bifurcations.[edit ] ExampleThe Hopf bifurcation in the Selkov system(see article). As the parameters change, a limitcycle (in blue) appears out of an unstableequilibrium.Hopf bifurcations occur in the Hodgkin–Huxley model for nerve membrane, the Selkov model of glycolysis , the Belousov–Zhabotinsky reaction , the Lorenz attractor and in the following simpler chemical system called the Brusselator as the parameter B changes:The Selkov model isThe phase portrait illustrating the Hopf bifurcation in the Selkov model is shown on the right. See Strogatz, Steven H. (1994). "Nonlinear Dynamics and Chaos" [1], page 205 for detailed derivation.[edit ] Definition of a Hopf bifurcationThe appearance or the disappearance of a periodic orbit through a local change in the stability properties of a steady point is known as the Hopf bifurcation. The following theorem works with steady points with one pair of conjugate nonzero purely imaginary eigenvalues . It tells the conditions under which this bifurcation phenomenon occurs.Theorem (see section 11.2 of [3]). Let J 0 be the Jacobian of a continuous parametric dynamical system evaluated at a steady point Z eof it. Suppose that all eigenvalues of J 0 have negative real parts except one conjugate nonzero purely imaginary pair. A Hopf bifurcation arises when these two eigenvalues cross the imaginary axis because of a variation of the system parameters.[edit ] Routh–Hurwitz criterionRouth–Hurwitz criterion (section I.13 of [5]) gives necessary conditions so that a Hopf bifurcation occurs. Let us see how one can use concretely this idea [8].[edit ] Sturm series Let be Sturm series associated to a characteristic polynomial P . They can be written in the form:The coefficients c i,0 for i in correspond to what is called Hurwitz determinants [8]. Their definition is related to the associated Hurwitz matrix .[edit ] PropositionsProposition 1. If all the Hurwitz determinants c i ,0 are positive, apart perhaps c k,0 then the associated Jacobian has no pure imaginary eigenvalues.Proposition 2. If all Hurwitz determinants c i ,0 (for all i in are positive, c k " 1,0 = 0 and c k" 2,1 < 0 then all the eigenvalues of the associated Jacobian have negative real parts except a purely imaginary conjugate pair.The conditions that we are looking for so that a Hopf bifurcation occurs (see theorem above) for a parametric continuous dynamical system are given by this last proposition.[edit ] Example Let us consider the classical Van der Pol oscillator written with ordinary differential equations:The Jacobian matrix associated to this system follows:The characteristic polynomial (in λ) of the linearization at (0,0) is equal to:P (λ) = λ2 " μλ + 1.The coefficients are: a 0 = 1,a 1 = " μ,a 2 = 1 The associated Sturm series is:The Sturm polynomials can be written as (here i = 0,1):The above proposition 2 tells that one must have:c 0,0 = 1 > 0,c 1,0 = " μ = 0,c 0,1 = " 1 < 0.Because 1 > 0 and 1 < 0 are obvious, one can conclude that a Hopf bifurcation may occur for Van der Pol oscillator if μ = 0.[edit ] References1. ^ a b Strogatz, Steven H. (1994). Nonlinear Dynamics and Chaos . Addison Wesley publishing company.2. ^ Kuznetsov, Yuri A. (2004). Elements of Applied Bifurcation Theory . New York: Springer-Verlag. ISBN 0-387-21906-4.3. ^ a b Hale, J.; Ko ak, H. (1991). Dynamics and Bifurcations . Texts in Applied Mathematics. 3. New York: Springer-Verlag.4. ^ Guckenheimer, J.; Myers, M.; Sturmfels, B. (1997). "Computing Hopf Bifurcations I". SIAM Journal on Numerical Analysis .5. ^ a b Hairer, E.; Norsett, S. P.; Wanner, G. (1993). Solving ordinary differential equations I: nonstiff problems (Second ed.). New York: Springer-Verlag.6. ^ Wilhelm, T.; Heinrich, R. (1995). "Smallest chemical reaction system with Hopf bifurcation". Journal of Mathematical Chemistry 17 (1): 1–14.doi :10.1007/BF01165134. http://www.fli-leibniz.de/~wilhelm/JMC1995.pdf .7. ^ Kirk, P. D. W.; Toni, T.; Stumpf, MP (2008). "Parameter inference for biochemical systems that undergo a Hopf bifurcation". Biophysical Journal 95 (2):540–549. doi :10.1529/biophysj.107.126086. PMC 2440454. PMID 18456830. /biophysj/pdf/PIIS0006349508702315.pdf .8. ^ a bKahoui, M. E.; Weber, A. (2000). "Deciding Hopf bifurcations by quantifier elimination in a software component architecture". Journal of SymbolicComputation 30 (2): 161–179. doi:10.1006/jsco.1999.0353. [edit] External links● Reaction-diffusion systems● The Hopf Bifurcation● Andronov–Hopf bifurcation page at ScholarpediaCategories: Bifurcation theoryPersonal tools● Log in / create accountNamespaces● Article● DiscussionVariantsViews● Read● Edit● View historyActionsSearchInteractionToolboxPrint/exportLanguages● This page was last modified on 25 May 2011 at 02:56.● Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. See Terms of use for details. Wikipedia is a registered trademark of the W ikimedia Foundation, Inc., a non-profit organization.● Contact us● Privacy policy● About Wikipedia● Disclaimers●●。

β甘油磷酸和壳聚糖水凝胶降解时间

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分岔与混沌

稳定性。对于一般动力系统，控制参数的变
化会引起特征值的变化，当控制参数达到分
岔参数值时，系统稳定性发生质的变化，它
可以表现为 ( ) 在复平面的运动。由此也可以
定义三种分岔类型:
2022/1/10
机械系统与振动国家重点实验室
9
叉型分岔
霍普分岔
特征值为
实数，沿复
平面的实轴由
负变正穿过虚
轴。
平衡点
x 0 和 x＝ ,
x 0,
而对应特征值则为
0
0
x2
对于图3，当 c时，平衡态的一个分支是稳定的；然而当 c时，这
一支就变得不稳定了；一旦当 c 有新的平衡分支解 x 又变成稳
定的了，这种情况被称为超临界分岔。反过来，若新的平衡分支解x
2022/1/10
机械系统与振动国家重点实验室
22
庞加莱-伯克霍夫(PB)规范形方法
考虑微分方程
x=f（x），x∈Rn
（1）
设f（x）足够光滑，且f（0）=0。
现在研究对于某个给定正整数r≥2，通过坐标的多项式变换，
使得在f的泰勒展开式中直到r次的项都有比较简单的形式。
庞加莱伯克霍夫范式定理设f（x）是Cr向量场（r≥2），f
23
庞加莱-伯克霍夫(PB)规范形方法
需要注意：
1.对于给定的r来说，r阶PB范式的取法一般不是唯一的。
2.在平衡点附近，截断规范形系统与原来的系统的拓扑结
构往往有密切的关系，但并不一定相同。一般来说，对于
给定的r，r阶PB范式到底能在多大程度上反映原系统的定
性性态仍然是一个未完全解决的问题。
3.尽管如此，在大量研究中发现，阶数不太高的PB范式通

生物化学名词解释集锦

生物化学名词解释集锦第一章蛋白质1．两性离子（dipolarion）2．必需氨基酸（essential amino acid）3．等电点(isoelectric point,pI)4．稀有氨基酸(rare amino acid)5．非蛋白质氨基酸(nonprotein amino acid) 6．构型(configuration)7．蛋白质的一级结构(protein primary structure)8．构象(conformation)9．蛋白质的二级结构(protein secondary structure)10．结构域(domain)11．蛋白质的三级结构(protein tertiary structure)12．氢键(hydrogen bond)13．蛋白质的四级结构(protein quaternary structure)14．离子键(ionic bond)15．超二级结构(super-secondary structure) 16．疏水键(hydrophobic bond)17．范德华力(van der Waals force) 18．盐析(salting out)19．盐溶(salting in)20．蛋白质的变性(denaturation)21．蛋白质的复性(renaturation)22．蛋白质的沉淀作用(precipitation) 23．凝胶电泳（gel electrophoresis）24．层析（chromatography）第二章核酸1．单核苷酸(mononucleotide)2．磷酸二酯键（phosphodiester bonds）3．不对称比率（dissymmetry ratio）4．碱基互补规律(complementary base pairing)5．反密码子（anticodon）6．顺反子（cistron）7．核酸的变性与复性（denaturation、renaturation）8．退火（annealing）9．增色效应（hyper chromic effect）10．减色效应（hypo chromic effect）11．噬菌体（phage）12．发夹结构（hairpin structure）13．DNA的熔解温度（melting temperature T m）14．分子杂交（molecular hybridization）15．环化核苷酸(cyclic nucleotide)第三章酶与辅酶1．米氏常数（K m值）2．底物专一性（substrate specificity）3．辅基（prosthetic group）4．单体酶（monomeric enzyme）5．寡聚酶（oligomeric enzyme）6．多酶体系（multienzyme system）7．激活剂（activator）8．抑制剂（inhibitor inhibiton）9．变构酶（allosteric enzyme）10．同工酶（isozyme）11．诱导酶（induced enzyme）12．酶原（zymogen）13．酶的比活力（enzymatic compare energy）14．活性中心（active center）第四章生物氧化与氧化磷酸化1．生物氧化（biological oxidation）2．呼吸链（respiratory chain）3．氧化磷酸化（oxidative phosphorylation）4．磷氧比P/O（P/O）5．底物水平磷酸化（substrate level phosphorylation）6．能荷（energy charg第五章糖代谢1．糖异生(glycogenolysis)2．Q酶(Q-enzyme)3．乳酸循环(lactate cycle)4．发酵(fermentation)5．变构调节(allosteric regulation)6．糖酵解途径(glycolytic pathway)7．糖的有氧氧化(aerobic oxidation)8．肝糖原分解(glycogenolysis)9．磷酸戊糖途径(pentose phosphate pathway) 10．D-酶（D-enzyme）11．糖核苷酸（sugar-nucleotide）第六章脂类代谢1．必需脂肪酸（essential fatty acid）2．脂肪酸的α-氧化(α-oxidation)3．脂肪酸的β-氧化(β-oxidation)4．脂肪酸的ω-氧化(ω-oxidation)5．乙醛酸循环（glyoxylate cycle）6．柠檬酸穿梭(citriate shuttle)7．乙酰CoA羧化酶系（acetyl-CoA carnoxylase）8．脂肪酸合成酶系统（fatty acid synthase system）第八章含氮化合物代谢1．蛋白酶（Proteinase）2．肽酶（Peptidase）3．氮平衡（Nitrogen balance）4．生物固氮（Biological nitrogen fixation）5．硝酸还原作用（Nitrate reduction）6．氨的同化（Incorporation of ammonium ions into organic molecules）7．转氨作用（Transamination）8．尿素循环（Urea cycle）9．生糖氨基酸（Glucogenic amino acid）10．生酮氨基酸（Ketogenic amino acid）11．核酸酶（Nuclease）12．限制性核酸内切酶（Restriction endonuclease）13．氨基蝶呤（Aminopterin）14．一碳单位（One carbon unit）第九章核酸的生物合成1．半保留复制（semiconservative replication）2．不对称转录（asymmetric trancription）3．逆转录（reverse transcription）4．冈崎片段（Okazaki fragment）5．复制叉（replication fork）6．领头链（leading strand）7．随后链（lagging strand）8．有意义链（sense strand）9．光复活（photoreactivation）10．重组修复（recombination repair）11．内含子（intron）12．外显子（exon）13．基因载体（genonic vector）14．质粒（plasmid）第十一章代谢调节1．诱导酶（Inducible enzyme）2．标兵酶（Pacemaker enzyme）3．操纵子（Operon）4．衰减子（Attenuator）5．阻遏物（Repressor）6．辅阻遏物（Corepressor）7．降解物基因活化蛋白（Catabolic gene activator protein）8．腺苷酸环化酶（Adenylate cyclase）9．共价修饰（Covalent modification）10．级联系统（Cascade system）11．反馈抑制（Feedback inhibition）12．交叉调节（Cross regulation）13．前馈激活（Feedforward activation）14．钙调蛋白（Calmodulin）第十二章蛋白质的生物合成1．密码子(codon)2．反义密码子(synonymous codon) 3．反密码子(anticodon)4．变偶假说(wobble hypothesis)5．移码突变(frameshift mutant)6．氨基酸同功受体(isoacceptor)7．反义RNA(antisense RNA)8．信号肽(signal peptide)9．简并密码(degenerate code)10．核糖体(ribosome)11．多核糖体(poly some)12．氨酰基部位(aminoacyl site) 13．肽酰基部位(peptidy site)14．肽基转移酶(peptidyl transferase) 15．氨酰-tRNA合成酶(amino acy-tRNA synthetase)16．蛋白质折叠(protein folding) 17．核蛋白体循环(polyribosome) 18．锌指(zine finger)19．亮氨酸拉链(leucine zipper) 20．顺式作用元件(cis-acting element) 21．反式作用因子(trans-acting factor) 22．螺旋-环-螺旋(helix-loop-helix)第一章蛋白质1．两性离子：指在同一氨基酸分子上含有等量的正负两种电荷，又称兼性离子或偶极离子。

基于多尺度方法的1∶3共振双Hopf分岔分析

基于多尺度方法的1∶3共振双Hopf分岔分析王万永;陈丽娟;郭静【摘要】利用改进的多尺度方法对一个电路振子模型1∶3共振附近的动力学行为进行了研究。

应用该方法得到了系统的复振幅方程，进而得到一个振幅与相位解耦的三维实振幅系统，通过分析实振幅方程的平衡点个数及其稳定性，将系统共振点附近的动力学行为进行分类，发现了双稳态等动力学现象，数值模拟验证了理论结果的正确性。

%The dynamical behavior near a 1∶3 resonance of an electric oscillator was investigated. By using the method of multiple scale, the complex amplitude equations of the system were obtained. Then a three dimension real amplitude system in which the amplitudes decouple from the phases was given. Ana-lyzing the number of equilibrium and its stability of the real amplitude equation, the dynamical behavior around the resonant point was classified. Some interesting dynamical phenomenon were found, for exam-ple,the bistability. Numerical simulations for justifying the theoretical analysis were also provided.【期刊名称】《郑州大学学报（理学版）》【年(卷),期】2016(048)003【总页数】5页(P23-27)【关键词】电路振子;1∶3共振;多尺度方法;分岔【作者】王万永;陈丽娟;郭静【作者单位】河南工程学院理学院河南郑州451191;河南工程学院理学院河南郑州451191;郑州铁路职业技术学院公共教学部河南郑州450052【正文语种】中文【中图分类】O175.1在非线性动力学的研究中，内共振由于能够反应系统线性模态之间的相互作用，有着非常重要的研究价值.文献[1]通过研究一个两端固支屈曲梁模型的内共振，构建了该模型在1∶1和1∶3内共振情形下的非线性模态.文献[2]研究了一个悬索模型的1∶2内共振，并讨论了三次非线性和高阶修正项对系统解的影响.文献[3]研究了一个极限环振子系统发生的1∶3共振双Hopf分岔，并研究了非线性对共振附近动力学行为的影响.文献[4]通过利用3∶1内共振的性质设计了一个非线性振动吸振器.文献[5]研究了内共振条件下风力发电机风轮叶片的空气动力学行为.在内共振和双Hopf分岔的研究中，常用的方法有中心流形和规范型方法、多尺度方法、摄动增量法、Liapunov-Schmidt约化和奇异摄动法.这些方法都存在一些问题，例如中心流形方法计算过程复杂，奇异性理论更加数学化，晦涩难懂，而多尺度方法得到的强共振的实振幅方程中，平衡点是非孤立的平衡点[6]，因而使稳定性分析和分岔分析无法进行.在本文的研究中，将应用一种改进的多尺度方法，把1∶3共振的规范型化为一个三维的实振幅系统，进而可以研究系统在共振点附近的动力学行为.本文以一个电路振子模型为例，利用改进的多尺度方法研究其1∶3共振点附近的动力学行为.其电路示意图如图1所示[7].其数学模型为[7]：其中：x1=v1，x2=i1，x3=v2，x4=i2是状态变量；η1=1/C1，η2=R，η3=1/L1，ρ1=1/C2，ρ2=1/L2是参数；α1、α2、α3是辅助参数.非线性电路模型的动力学行为是非线性动力学研究的重要内容之一.目前已有不少的文献从实验和理论方面对其进行了研究[8-12]，并发现了次谐波振荡、周期解、概周期解、分岔以及混沌等大量的非线性现象[11].本文将应用改进的多尺度方法对该电路系统的1∶3共振进行研究，计算其振幅方程并分析共振点附近的动力学行为.系统(1)在其唯一平衡点(0，0，0，0)处的线性化系统为,其特征方程为λ4+(-α1η1+η2ρ2)λ3+(η1η3+η1ρ2-α1η1η2ρ2+ρ1ρ2)λ3+(η1η2η3ρ2-α1η1ρ1ρ2)λ+η1η3ρ1ρ2=0.为了研究该系统1∶3共振点附近的动力学行为，设其特征方程有两对纯虚根λ1,3=±iω1和λ2,4=±iω2，其中ω1∶ω2=1∶3.可以求得当,时，特征方程(2)有两对纯虚根和.为了得到1∶3共振的规范型方程,将应用改进的多尺度方法对系统(1)进行分析.首先按照如下形式摄动参数设，则系统(1)可写为其多尺度形式的解具有如下形式将式(3)、(5)带入式(4),并对式(4)的右端进行Taylor展开,令两端ε的各次幂的系数相等,可得方程(6)的解具有如下形式其中：Aj(j=1,2)是复振幅，为时间尺度T2的函数；p1和p2是相应于特征值iω1和iω2的右特征向量；c.c. 表示前面各项的复共轭.将式(9)代入式(7),可求得式(7)的解为其中zij是复系数.将式(9)、(10)代入式(8),令长期项的系数为零,可得到A1和A2关于时间尺度T2导数的两个方程.应用左特征向量消去D2A1和D2A2的系数并吸收参数ε[13],可得Cijk和Ciμ με是复系数.在式(11)中，A1和A2为复振幅，为了将式(11)转化为实数振幅方程，通常将A1和A2设为极坐标形式.但是，在强共振条件下，如果将A1和A2设为极坐标形式，将会得到一个实振幅与相位变量耦合的三维系统，其平衡点将是非孤立的平衡点，平衡点的稳定性将无法研究.为了避免这种情况，将复振幅A1和A2设为一种混合形式(极坐标-笛卡尔形式)[13]，将式(12)代入式(11),分离其实部和虚部,可得到一个振幅与相位解耦的三维实振幅方程，如下:0.210 018uv2-0.532 248v3+0.080 357 1uη1ε-0.139 382vη1ε-0.21967uη2ε+0.168 86vη2ε+ 0.258 519 u η3ε+1.345 23vη3ε,0.210 018u2v+0.532 248uv2-0.210 018v3+0.139 382uη1ε+0.080 3571vη1ε-0.168 86uη2ε-0.219 67vη2ε-1.345 23uη3ε+0.258 519vη3ε.若设,则相应于原系统的状态变量x的Hopf分岔是振幅变量a1、a2的静态分岔. 由前面的分析可知1∶3共振的振幅方程是由3个变量组成的三维系统，并且含有3个分岔参数.为了分析共振点(η1c，η2c，η3c)附近的动力学行为，可以固定其中一个分岔参数，分析系统在二维参数平面上共振点附近的动力学行为.为此，固定参数η3，在η1-η2平面内对系统的动力学行为进行分类.根据实振幅方程的平衡点个数及每个平衡点稳定性的不同, 将平面η1-η2分为6个不同的区域，如图2所示.在Ⅰ区中，其平凡平衡点E0(0,0)是稳定的平衡点，对应于原系统的原点.当参数进入Ⅱ区，一个稳定的单模态平衡点E1(a10,0)出现，而平凡平衡点E0(0,0)变为不稳定的平衡点.当参数进入Ⅲ区，一个不稳定的平衡点E2(0,a20)出现，而平衡点E1(a10,0)保持其稳定性，平衡点E0(0,0)仍然是不稳定的.在Ⅳ区，一个新的不稳定的双模态平衡点E3(a12,a22)产生，而平衡点E1(a10,0)和E2(0,a20)是稳定的平衡点.在Ⅴ区，双模态平衡点E3(a12,a22)消失，平衡点E1(a10,0)失稳，平衡点E2(0,a20)仍然是稳定的.在Ⅵ区，平衡点E2(0,a20)保持稳定性，平衡点E1(a10,0)消失.其中单模态平衡点E1(a10,0)和E2(0,a20)分别相应于原系统频率为ω1和ω2的周期解，双模态平衡点E3(a12,a22)则相应于原系统的一个概周期解.为了验证理论分析的正确性，对原系统进行数值模拟，模拟的结果如图3～图8所示.可以发现，当参数在共振点附近变化时，系统出现两个不同频率的周期解，其频率比值接近1∶3.同时在分类图的Ⅳ区，两个不同频率的周期解同时出现，系统出现双稳态现象.本文研究了一个电路振子模型中发生的1∶3共振双Hopf分岔，通过应用改进的多尺度方法得到了该1∶3共振的规范型方程，进而分析其共振点附近的动力学行为，发现了周期解、双稳态等动力学现象，并通过数值模拟验证了结果的正确性.本文在揭示电路振子系统动力学现象的同时，应用了一种研究1∶3共振的新方法，该方法通过应用多尺度方法的过程，并将1∶3共振的复振幅设为一种混合形式，可以得到1∶3共振实振幅系统，从而能够研究共振点附近的动力学行为.【相关文献】[1] LACARBONARA W，REGA G，NAYFEH A H．Resonant non-linear normal modes．Part I：analytical treatment for structural one-dimensional systems [J]．Int JNon-linear Mech,2003，38(6)：851-872．[2] LEE C L， PERKINS N C．Nonlinear oscillations of suspended cables containing atwo-to-one internal resonance [J]．Nonlinear Dyn，1992，3(6)：465-490.[3] 王万永，陈丽娟．非线性时滞反馈对共振附近动力学行为的影响 [J]．信阳师范学院学报(自然科学版)，2014，27(1)：15-18．[4] JI J C， ZHANG N．Design of a nonlinear vibration absorber using three-to-one internal resonances [J]．Mech Syst Signal Processing，2014，42(1/2)： 236-246．[5] LI L,LI Y H,LIU Q K,et al. Flap wise non-linear dynamics of wind turbine blades with both external and internal resonances[J]．Int J Non-Linear Mech，2014，61(1)：1-14．[6] LUONGO A，DI EGIDIO A，PAOLONE A．On the proper form of the amplitude modulation equations for resonant systems [J]．Nonlinear Dyn，2002，27(3)：237-254．[7] REVEL G，ALONSO D M，MOIOLA J L．Numerical semi-global analysis of a 1∶2 resonant Hopf-Hopf bifurcation [J]．Physica D-nonlinear phenomena，2012，247(1)：40-53．[8] 徐兴磊，李红．压缩真空态的激发态下介观串并联RLC电路的量子涨落 [J]．郑州大学学报(理学版)，2007，39(1)：67-70．[9] 方天申，董学义．LC串联电路非共振固有振荡与谐波共振的区别 [J]．信阳师范学院学报(自然科学版)，2007，20(4)：429-431．[10] CHUA L O, WU C W, HUANG A, et al．A universal circuit for studying and generating chaos-Ⅱ：Strange attractors[J]. IEEE T Circuits Sys I, 1993, 40(10): 745-761. [11] 张晓芳，陈章耀，毕勤胜．非线性电路系统动力学的研究进展及展望[J]．电路与系统学报，2012，17(5)：124-129．[12] 苏利捷，魏兆博，杨广德．单相逆变器共模电磁干扰特性研究 [J]．郑州大学学报(理学版)，2014，46(4)：57-62．[13] LUONGO A，PAOLONE A，DI EGIDIO A．Multiple timescales analysis for 1∶2 and 1∶3 resonant Hopf bifurcations [J]．Nonlinear dynamics，2003，34(3/4)：269-291．。

非线性动力系统的两类分岔控制与混沌控制研究

湖南大学硕士学位论文非线性动力系统的两类分岔控制与混沌控制研究姓名：欧阳克俭申请学位级别：硕士专业：固体力学指导教师：唐驾时 20070428
硕士学位论文
摘
要
分岔控制作为非线性科学中的前沿研究课题，极具挑战性。分岔控制的目的是对给定的非线性动力系统设计一个控制器，用来改变系统的分岔特性，从而去掉系统中有害的动力学行为，使之产生人们需要的动力学行为。本文在全面分析和总结非线性动力系统分岔控制研究现状的基础上，基于非线性动力学、非线性控制理论、分岔理论等非线性科学的现代分析方法，对倍周期分岔、Hopf 分岔等进行控制，工作具有较大的理论意义和应用价值。研究内容如下：第一章对非线性控制理论、分岔控制的研究方法、现状和进展进行综述，介绍本文的研究目的、研究内容和创新点。第二章介绍动力学研究的一些基本概念，简述发生鞍结分岔、跨临界分岔、叉形分岔的充分必要条件，以及这三种静态分岔相互转换的条件；介绍分岔控制器设计及分析的主要方法。第三章设计了线性和非线性的状态反馈控制器，对 Logistic 模型的倍周期分岔进行了控制，得到了系统在控制前和控制后的分岔图 , 通过设计不同的参数控制器，改变了动力系统的分岔特性。根据实际应用目的，设计了不同的控制器改变了存在的分岔点的参数值，并且调整了分岔链的形状。通过优化控制器可以使 Logistic 模型的分岔行为满足一定的要求。第四章设计了状态反馈控制器和 washout filter 控制器对 van der Pol-Duffing 系统的 Hopf 分岔的极限环幅值进行了控制。通过对控制方程的分析，了解了控制参数和极限环幅值的影响情况，进而提出控制策略，设计了状态反馈控制器对系统的 Hopf 分岔进行了控制。第五章设计了线性反馈控制器对 Lorenz 系统的平衡点和周期轨道进行了控制，首先利用 Routh-Hurwitz 准则对受控系统进行了稳定性分析，严格证明了达到控制目标反馈系数的选择原则，最后通过数值计算证明了该方法能够有效地控制混沌系统到稳定的平衡点同时也能使系统控制到 1P 周期轨道，并且得到了控制到稳定的 1P 周期轨道的控制参数的选取范围。本文的主要创新点在于将分岔控制理论应用于非线性振动系统的研究，丰富了非线性控制理论研究的内容，加深了分岔理论研究的深度。具体表现在：对 Logistic 模型的倍周期分岔进行了反馈控制；首次将 washout filter 技术应用于二维 van der Pol-Duffing 系统的 Hopf 分岔控制；应用线性反馈控制成功实现了对 Lorenz 系统平衡点的混沌控制和 1P 周期轨道控制。关键词：分岔控制；非线性动力系统；状态反馈控制；多尺度法； Hopf 分岔

静止无功补偿器在电力系统Hopf分岔控制中的应用

PE 电力电子年第期6静止无功补偿器在电力系统Hopf 分岔控制中的应用夏加宽1刘康珍2米欣1（1.沈阳工业大学，沈阳110178；2.山东省冶金建设开发公司，济南250101）摘要分析研究了电力系统中的静止无功补偿器（SVC ）在抑制功率/频率振荡问题中的应用。

这些振荡现象都与Hopf 分岔有关，因此可以根据分岔理论分析该现象并设计相应的控制器。

通过一个实际系统，研究了SVC 对分岔临界特征值的影响，进而研究了其对功率/频率振荡的影响。

关键词：功率振荡控制；Hopf 分岔；静止无功补偿器Hopf Bifurcation Control in Power Systems Using Static Var CompensatorsX ia Jiakuan 1Liu Kangzhen 2Mi Xin 1(1.Shenyang University of Technology,Shenyang 110178;2.Shandong Province Metallurgical Construction Company,Ji ’nan 250101)Abstr act This paper prescribed the use of Static V ar Compensators (SVC)to damp power/frequency oscillations in power systems.These oscillations are first shown to be associated with Hopf bifurcations.Then,bifurcation theory is used to analyze the phenomena and design corrective measures.The effect on the bifurcation critical eigenvalues,and hence on the power/frequency oscillations,of the gains of the SVC controls is also studied on a 16-bus test system.Key words ：power oscillation control ；Hopf bifurcations ；SVC1引言电力系统中的功率/频率振荡问题通常采用特征值计算的方法进行分析，并通过对发电机组添加电力系统稳定器（PSS ）进行解决[1]。

发光调控英语

发光调控英语Title: The Complexity and Intricacies of Luminescence RegulationLuminescence regulation, a field at the intersection of physics, chemistry, and biology, holds immense potential in various applications ranging from displays and lighting to biomedical imaging and sensing. It involves the precise control of the emission of light from a material, either spontaneously or in response to an external stimulus. This article delves into the complexities and intricacies of luminescence regulation, exploring its principles, techniques, and evolving applications.Firstly, it's crucial to understand the fundamental mechanisms of luminescence. Luminescence occurs when a material absorbs energy, either in the form of light, electricity, or heat, and subsequently emits light. This process is typically characterized by the excitation of electrons within the material, followed by their relaxation and emission of photons. The color and intensity of the emitted light depend on the material's chemical composition, structure, and the nature of the excitation.Luminescence regulation involves manipulating these mechanisms to achieve desired emission properties. One approach is through the use of dopants or activators, which introduce additional energy states within the material. These dopants can enhance or modify the emission spectrum, enabling the tuning of color and intensity. Another method involves manipulating the material's physical structure, such as through nanostructuring or the use of porous materials, to alter the path and efficiency of light emission.Moreover, the field of luminescence regulation has benefited significantly from the advancement of synthetic techniques and material science. The ability to synthesize materials with precise compositional and structural control has opened new avenues for precise luminescence tuning. For instance, the development of colloidal quantum dots and perovskite nanocrystals has enabled the creation of luminescent materials with tunable emission wavelengths and high brightness.In terms of applications, luminescence regulation finds widespread use in various fields. In displays and lighting,luminescent materials are used to generate vibrant colors and efficient light emission. The precise control of emission properties enables the creation of displays with high color accuracy and contrast, as well as lighting systems with optimized energy efficiency.In the biomedical field, luminescent materials have revolutionized imaging and sensing techniques. Fluorescence microscopy, for instance, relies on the ability to label specific molecules or cells with luminescent probes, enabling their visualization with high spatial and temporal resolution. Luminescent probes are also used in biosensing applications, where they can detect and quantify biological analytes with high sensitivity and specificity.Furthermore, the emergence of photoluminescence-based solar cells has highlighted the potential of luminescence regulation in renewable energy applications. By engineering the luminescent properties of photovoltaic materials, researchers aim to improve the efficiency and stability of solar cells, addressing key challenges in solar energy conversion.However, the field of luminescence regulation remains challenging and evolving. The complexity of luminescent mechanisms, coupled with the diverse range of materials and applications, poses significant challenges in achieving precise and reliable luminescence control. Ongoing research efforts are focused on developing novel materials and techniques that can further enhance the performance and versatility of luminescent systems.In conclusion, luminescence regulation represents a vibrant and dynamic field with vast potential for innovation and applications. As the understanding of luminescent mechanisms deepens and synthetic techniques improve, the capabilities of luminescent materials will continue to expand, opening new doors in various fields from displays and lighting to biomedicine and renewable energy.。

几类时滞反应-扩散神经网络的Hopf分岔研究

目录摘要 (I)Abstract (III)第一章引言 (1)1.1神经网络的定义 (1)1.2神经网络分岔行为的研究现状 (2)1.3神经网络的扩散现象 (4)1.4本文主要研究内容 (4)第二章预备知识 (7)2.1符号说明 (7)2.2概念解释 (7)2.3数学理论阐述 (8)2.3.1关于雅可比(Jacobi)矩阵 (8)2.3.2特征方程的根的判别式 (9)第三章时滞反应-扩散中立型神经元模型的Hopf分岔分析 (11)3.1引言 (11)3.2系统局部稳定性和分岔 (13)3.3Hopf分岔与周期解稳定性 (20)3.4数值仿真 (25)第四章带有时滞的反应-扩散神经网络的图灵不稳定性和Hopf分岔分析 (29)4.1引言 (29)4.2没有延迟时的图灵不稳定性 (31)4.3Hopf分岔分析 (33)4.4Hopf分岔的规范性 (37)4.5仿真 (42)4.6小结 (43)第五章总结与展望 (49)5.1本文的工作总结 (49)5.2后续展望 (50)参考文献 (53)致谢 (57)攻读硕士期间已发表的学术论文 (59)几类时滞反应-扩散神经网络的Hopf分岔研究学科专业：信号与信息处理研究方向：神经网络分岔研究指导教师：董滔作者：夏林茂摘要关于对神经网络分岔行为的研究一直以来都是十分热门的话题，也是在神经网络动力学行为研究中的一大重点和难点。

而时滞反应-扩散神经网络作为普通神经网络的扩展，由于其更符合现实生物神经网络的特点、存在更加丰富的动力学行为、更加适用于工业发展与应用而逐渐成为学者们的重点研究领域。

本文分别研究了时滞中立型反应-扩散神经元模型的Hopf分岔和二维反应-扩散神经网络的Hopf分岔及图灵不稳定性，本文的主要内容和创新点如下：（1）带有延时的一维中立型反应-扩散神经网络的Hopf分岔分析本文提出了一类具有时滞的反应-扩散中立型神经元模型，并研究了在扩散的影响下，该系统的局部稳定性和Hopf分岔。

温敏水凝胶的英语

温敏水凝胶的英语The English Composition on Thermo-Sensitive HydrogelsThermo-sensitive hydrogels have gained significant attention in the field of biomedicine due to their unique properties and potential applications. These intelligent materials possess the ability to undergo reversible phase transitions in response to changes in temperature, making them particularly useful in various biomedical applications.Hydrogels are a class of hydrophilic polymeric networks that can absorb and retain large amounts of water or biological fluids within their three-dimensional structure. Thermo-sensitive hydrogels, specifically, exhibit a temperature-dependent phase transition, which means they can undergo a sol-gel transition as the temperature changes. This property is often referred to as the lower critical solution temperature (LCST) or upper critical solution temperature (UCST), depending on the specific polymer system.One of the most well-known thermo-sensitive hydrogels is poly(N-isopropylacrylamide) (PNIPAAm), wh ich has an LCST around 32°C, close to the human body temperature. Below the LCST, PNIPAAmhydrogels are in a swollen, hydrophilic state, allowing for the incorporation and release of various therapeutic agents. However, as the temperature increases above the LCST, the polymer chains undergo a conformational change, leading to the collapse of the hydrogel structure and the expulsion of water. This temperature-induced phase transition makes PNIPAAm-based hydrogels particularly useful for controlled drug delivery applications.The mechanism behind the temperature-responsive behavior of thermo-sensitive hydrogels, such as PNIPAAm, is related to the delicate balance between hydrophobic and hydrophilic interactions within the polymer network. At temperatures below the LCST, the polymer chains are hydrated, and the hydrogen bonding between water molecules and the polymer's amide groups dominates, leading to a swollen, hydrophilic state. As the temperature increases above the LCST, the hydrogen bonding between water and the polymer becomes weaker, and the hydrophobic interactions between the isopropyl groups of the polymer become more prominent. This results in the collapse of the polymer chains, causing the expulsion of water and the formation of a more compact, hydrophobic structure.The unique temperature-responsive behavior of thermo-sensitive hydrogels has led to their widespread application in various biomedical fields. One of the primary applications is in controlleddrug delivery systems. Thermo-sensitive hydrogels can be used as carriers for therapeutic agents, such as small-molecule drugs, proteins, or even cells. These hydrogels can be designed to release the encapsulated drugs in a controlled manner by responding to the temperature changes in the body. For example, a PNIPAAm-based hydrogel loaded with a drug can be administered in a liquid state at room temperature and then undergo a phase transition to a gel state upon reaching body temperature, effectively trapping the drug within the hydrogel matrix. As the temperature increases further, the hydrogel can undergo a volume phase transition, leading to the release of the drug in a controlled manner.Another important application of thermo-sensitive hydrogels is in tissue engineering and regenerative medicine. These hydrogels can be used as scaffolds for cell growth and tissue regeneration. The temperature-responsive nature of the hydrogels allows for easy administration and in situ gelation, which can facilitate the encapsulation of cells or the delivery of growth factors directly to the site of injury or disease. The hydrogel scaffold can then provide a suitable microenvironment for cell proliferation, differentiation, and tissue formation.Thermo-sensitive hydrogels have also found applications in wound healing and burn treatment. The ability of these hydrogels to undergo a sol-gel transition in response to temperature changes canbe exploited to create wound dressings that can be easily applied in a liquid form and then transition to a gel state upon contact with the body. This can help maintain a moist environment, promote wound healing, and prevent infection.Furthermore, thermo-sensitive hydrogels have been investigated for use in various diagnostic and sensing applications. For instance, they can be designed to incorporate responsive elements, such as enzyme-substrate pairs or antibody-antigen interactions, which can trigger a detectable change in the hydrogel's physical properties in response to the presence of specific analytes or biomarkers.The development of thermo-sensitive hydrogels has also led to advancements in the field of injectable biomaterials. These hydrogels can be designed to be injected in a liquid form and then undergo in situ gelation at the target site, allowing for minimally invasive procedures and the delivery of therapeutic agents or cells directly to the site of interest.Despite the numerous promising applications of thermo-sensitive hydrogels, there are still several challenges that need to be addressed. One of the key challenges is the optimization of the LCST or UCST to match the specific requirements of the target application. Researchers are exploring ways to fine-tune the polymer composition and structure to achieve the desired temperature-responsive behavior. Additionally, the long-term biocompatibility and biodegradability of these hydrogels need to be thoroughly investigated to ensure their safe and effective use in biomedical applications.In conclusion, thermo-sensitive hydrogels have emerged as a versatile class of biomaterials with tremendous potential in the field of biomedical engineering. Their temperature-responsive behavior, coupled with their ability to encapsulate and deliver therapeutic agents, make them a promising platform for a wide range of applications, from controlled drug delivery to tissue engineering and regenerative medicine. As research in this field continues to advance, we can expect to see even more innovative and impactful applications of thermo-sensitive hydrogels in the years to come.。

Hopf bifurcation in a three-species system with delays

JAMC J Appl Math Comput (2011)35:635–661DOI 10.1007/s12190-010-0383-x Hopf bifurcation in a three-species system with delays Xin-You Meng ·Hai-Feng Huo ·Hong XiangReceived:18August 2009/Published online:11February 2010©Korean Society for Computational and Applied Mathematics 2010Abstract 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 nor-mal 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 bifurcat-ing from Hopf bifurcations are established by using a global Hopf bifurcation result.Numerical simulation results are also given to support our theoretical predictions.Keywords Stability ·Hopf bifurcation ·Delays ·Functional responseMathematics Subject Classiﬁcation (2000)92D251IntroductionRecently,Ruan et al.[1]considered the following two-competitor/one-prey model with a Holling II functional response˙u(t)=ru(t) 1−u(t)K −au(t)v(t)1+bu(t)−Au(t)w(t)1+Bu(t),This work was partially supported by the NNSF of China (10961018),the Key Project of Chinese Ministry of Education (209131),the Project Sponsored by the Scientiﬁc Research Foundation for the Returned Overseas Chinese Scholars,State Education Ministry,the NSF of Bureau of Education of Gansu Province of China (0803-01)and the Development Program for Outstanding Young Teachers in Lanzhou University of Technology (Q200703)and the Doctor’s Foundation of LanzhouUniversity of Technology.X.-Y .Meng ·H.-F.Huo ( )·H.XiangInstitute of Applied Mathematics,Lanzhou University of Technology,Lanzhou,Gansu 730050,Chinae-mail:hfhuo@636X.-Y .Meng et al.˙v(t)=v(t)−d +eu(t)1+bu(t) ,(1.1)˙w(t)=w(t)−D −Gw(t)+Eu(t)1+Bu(t),where u(t)is the density of the prey and v(t),w(t)are the density of the two preda-tors at the time t .r,a,A,b,B,d,D,e,E,K,G are positive constants.By construct-ing a suitable Lyapunov functional,they obtained the sufﬁcient conditions for the global stability of the positive equilibrium of system (1.1).It is well-known that time delays of one type or another have been incorporated into mathematical models of population dynamics due to maturation time,capturing time or other reasons.The effect of the past history on the stability of system is an im-portant problem in population biology.In general,delay differential equations exhibit much more complicated dynamics than ordinary differential equations since a time delay could cause a stable equilibrium to become unstable and cause the population to ﬂuctuate.A great deal of research has been devoted to the delay model,see [2–10]and references therein.Yang [2]introduced the delay in system (1.1)and this delay is referred to as the gestation period˙u(t)=ru(t) 1−u(t)K −au(t)v(t)1+bu(t)−Au(t)w(t)1+Bu(t),˙v(t)=v(t −d +eu(t −τ)1+bu(t −τ) ,(1.2)˙w(t)=w(t) −D −Gw(t)+Eu(t −τ)1+Bu(t −τ),and considered the stability of equilibrium and the existence of Hopf bifurcation.In this paper,as a generality,we will not only introduce the delay τ1due to gestating but also the delay τ2due to maturating to the system (1.1),and consider the following system˙x(t)=rx(t) 1−x(t)K −a 1x(t)y 1(t)1+b 1x(t)−a 2x(t)y 2(t)1+b 2x(t),˙y 1(t)=y 1(t) −d 1+e 1x(t −τ1)1+b 1x(t −τ1) ,(1.3)˙y 2(t)=y 2(t) −d 2−Gy 2(t −τ2)+e 2x(t −τ1)1+b 2x(t −τ1),where x(t)is the density of the prey and y 1(t),y 2(t)are the density of the two predators at the time t .We assume that both predators feed upon the prey accord-ing to the Holling II functional response,and they compete for the common prey.r,a 1,a 2,b 1,b 2,d 1,d 2,e 1,e 2,K,G are positive constants.τ1>0is a lag due to ges-tating,τ2>0is a lag due to the growth to maturity of the second predator.The rest of this paper is organized as follows.In next section,by analyzing the character equation of linearized system of system (1.3)at positive equilibrium,someHopf bifurcation in a three-species system with delays637 sufﬁcient conditions ensuring the local stability of the positive equilibrium and the existence of Hopf bifurcation are obtained.Some explicit formulas are given to de-termine the direction and stability of periodic solutions bifurcating from Hopf bi-furcations by applying the normal form method and center manifold theory due to Hassard et al.[11]in Sect.3.In Sect.4,the existence of periodic solutions forτfar away from Hopf bifurcation values is established by using a global Hopf bifurcation result of Wu[12].To support our theoretical predictions,some numerical simulations are also included in Sect.5.A brief discussion is also given in last section.2Local stability and Hopf bifurcationIt is obvious that system(1.3)has a unique positive equilibrium E∗=(x∗,y∗1,y∗2) deﬁned byx∗=d1e11d1,y∗1=1+b1x∗a1r−rx∗K−a2y∗21+b2x∗,(2.1)y∗2=1G−d2+e2d1e1+b2d1−b1d1provided(H1)r−rx∗K −a2y∗21+b2x∗>0and(H2)e1b1<d1,e2d1e1+b2d1−b1d1>d2.Note that two inequalities in(H2)simply mean that the death rates of the two com-petitors must be smaller than the corresponding growth rate,otherwise the competing species cannot survive and the positive steady state does not exist.Let u1(t)=x(t)−x∗,u2(t)=y1(t)−y∗1,u3(t)=y2(t)−y∗2,we can rewrite system(1.3)as the following system˙u1(t)=a11u1(t)+a12u2(t)+a13u3(t)+i+j+k≥2f ijk1u i1u j2u k3,˙u2(t)=a21u1(t−τ1)+i+j≥2f ij2u i1(t−τ1)u j2,(2.2)˙u3(t)=a31u1(t−τ1)+a32u3(t−τ2)+i+j≥2f ij3u i1(t−τ1)u j3(t−τ2),wherea11=−rx∗K+a1b1x∗y∗1(1+b1x∗)2+a2b2x∗y∗2(1+b2x∗)2,638X.-Y .Meng et al.a 12=−a 1x ∗1+b 1x ∗,a 13=−a 2x ∗1+b 2x ∗,a 21=e 1y ∗1(1+b 1x ∗),a 31=e 2y ∗2(1+b 2x ∗),a 32=−Gy ∗2,f ijk 1=1i !j !k !∂i +j +k f 1∂u i 1∂u 2∂u k 3|(x ∗,y ∗1,y ∗2),f ij 2=1i !j !∂i +j f 2∂u 1(t −τ1)i ∂u j 2|(x ∗,y ∗1,y ∗2),f ij 3=1i !j !∂i +j f 3∂u 1(t −τ1)i ∂u 3(t −τ2)j|(x ∗,y ∗1,y ∗2),f 1=ru 1(t) 1−u 1(t)K −a 1u 1(t)u 2(t)1+b 1u 1(t)−a 2u 1(t)u 3(t)1+b 2u 1(t),f 2=u 2(t) −d 1+e 1u 1(t −τ1)1+b 1u 1(t −τ1) ,f 3=u 3(t) −d 2−Gu 3(t −τ2)+e 2u 1(t −τ1)1+b 2u 1(t −τ1).To study the stability of the equilibrium point E ∗=(x ∗,y ∗1,y ∗2),it is sufﬁcient to study the stability of the origin for system (2.2).Consider the linearized system ofsystem (2.2)at (0,0,0)˙u 1(t)=a 11u 1(t)+a 12u 2(t)+a 13u 3(t),˙u 2(t)=a 21u 1(t −τ1),(2.3)˙u 3(t)=a 31u 1(t −τ1)+a 32u 3(t −τ2).2.1τ1=τ2=τ>0The associated characteristic equation of system (2.3)isλ3+Aλ2+(Bλ2+Cλ)e −λτ+De −2λτ=0,(2.4)where A =−a 11,B =−a 32,C =a 11a 32−a 12a 21−a 13a 31,D =a 12a 21a 32.Multi-plying e λτon both sides of (2.4),it is obvious to obtainBλ2+Cλ+(λ3+Aλ2)e λτ+De −λτ=0.(2.5)When τ=0,(2.5)becomesλ3+(A +B)λ2+Cλ+D =0.(2.6)Suppose(H3)A +B >0and (A +B)C >D .Hopf bifurcation in a three-species system with delays 639The Routh-Hurwitz criterion implies the equilibrium point is locally asymptoti-cally stable if (H3)holds.We want to determine if the real part of some root increases to reach zero and eventually becomes positive as τvaries.Now for τ=0,if λ=iω(ω>0)is a root of (2.5),then we have−Bω2+Cωi +(−ω3i −Aω2)e ωτi +De −ωτi =0.Separating the real and imaginary parts,we haveω3sin ωτ+(D −Aω2)cos ωτ=Bω2,ω3cos ωτ+(D +Aω2)sin ωτ=Cω.(2.7)It follows from (2.7)thatsin ωτ=Bω5+ACω3−CDωω6+A 2ω4−D 2,cos ωτ=(C −AB)ω4−BDω2ω6+A 2ω4−D 2.(2.8)As is known to all,sin 2ωτ+cos 2ωτ=1.So we haveω12+e 5ω10+e 4ω8+e 3ω6+e 2ω4+e 2ω4+e 1ω2+e 0=0,(2.9)where e 5=2A 2−B 2,e 4=A 4−A 2B 2−C 2,e 3=4BCD −2ADB 2−2D 2−A 2C 2,e 2=2ADC 2−2A 2D 2−B 2D 2,e 1=−C 2D 2,e 0=D 4.Denote v =ω2,then (2.9)becomesv 6+e 5v 5+e 4v 4+e 3v 3+e 2v 2+e 1v +e 0=0.(2.10)A delay system is absolutely stable if the real parts of all eigenvalues will remain negative for all values of the delay when the coefﬁcients of the exponential polyno-mial satisfy certain assumption (see [13]and [14]).A general result in Hale et al.[14]says that a delay system is absolutely stable if and only if the corresponding ODE system is asymptotically stable and the characteristic equation has no purely imaginary roots.Since (2.10)is a equation of the sixth degree,it is difﬁcult to study the distribution of the roots of (2.10)by Ruan’method [13].In addition,since e 0>0,the methods of [15,16]and [17]is invalid.If all the parameters of system (1.3)are given,it is easy to calculate the roots of (2.10)by using computer (see,for example,(6.2)).Thus,we give the following assumption.(H4)Equation (2.10)has at least one positive real root.Suppose that (H4)holds.Without loss of generality,we assume that it has six real positive roots,which are deﬁned by v 1,v 2,v 3,v 4,v 5and v 6,respectively.Then (2.9)has six positive roots ωk =√v k ,k =1,2, (6)Thus,denotingτ(j)k =1ωk arcos Cω4k −ABω4k −BDω2k ωk +A 2ωk−D 2 +1ωk 2jπ,k =1,2,3,4,5,6,j =0,1,2,...,(2.11)640X.-Y.Meng et al.then±iωk are a pair of purely imaginary root of(2.5)withτ=τ(j)k.Deﬁneτ0=τ0k=mink∈{1,2,···6}{τ0k},ω0=ωk.(2.12)Letλ(τ)=ξ(τ)+iω(τ)(2.13) be the root of(2.5)nearτ=τ0satisfyingξ(τ0)=0,ω(τ0)=ω0.(2.14) Taking the derivative ofλwith respect toτin(2.5),we obtain(2Bλ+C)dλdτ+(3λ2+2Aλ)eλτdλdτ+(λ3+Aλ2)eλτλ+τdλdτ+De−λτ−λ−τdλdτ=0,(2.15)it follows thatdλdτ=λ(De−λτ−(λ3+Aλ2)eλτ)2Bλ+C+(3λ)e−λτ.(2.16)Thendλdτ −1=2Bλ+C+(3λ2+2Aλ)eλτλDe−λτ)e−τλ.(2.17)Let=[(Dω0−Aω30)sinω0τ0−ω40cosω0τ0]2+[(Aω30−Dω0)cosω0τ0−ω40sinω0τ0]2.(2.18) Substituteλ=ω0i(ω0>0)into(2.17),we haved Reλ(τ0)dτ −1=Re2Bλ+C+(3λ2+2Aλ)eλτλDe−λτ)eλ=iω0=1(Dω0−Aω30)sinω0τ0−ω40cosω0τ0×C−3ω30cosω0τ0−2Aω0sinω0τ0×(Aω30−Dω0)cosω0τ0−ω40sinω0τ0×2Bω0−3ω20sinω0τ0+2Aω0cosω0τ0.Noting thatsignd Reλ(τ0)dτ=signd Reλ(τ0)dτ−1.(2.19)Hopf bifurcation in a three-species system with delays641 In order to give the main results,it is necessary to make the following assumption: (H5)d Reλ(τ0)dτ=0.By Corollary2.4in[10](see also[6]),we have the following theorem.Theorem2.1Suppose that(H3),(H4)and(H5)hold,the following results are true.(i)The positive equilibrium E∗of system(1.3)is asymptotically stable forτ∈[0,τ0).(ii)If >0,then the positive equilibrium E∗of system(1.3)undergoes a Hopf bifurcation whenτ=τ0.That is,system(1.3)has a branch of periodic solution bi-furcation from the zero solution nearτ=τ0.2.2τ1=τ2,τ1>0andτ2>0The associated characteristic equation of system(2.3)isλ3+(a12a21−a13a31)λe−λτ1+(−a32λ2+a11a32λ)e−λτ2−a12a21a32e−λ(τ1+τ2)=0.(2.20)We consider(2.20)withτ1in its stable interval of Yang[2].Regardingτ2as a parameter.Letωi(ω>0)be a root of(2.20),then we can obtainc1(ω)+2c2(ω)sinωτ1+2c3(ω)cosωτ1=0,(2.21) where c1(ω)=ω6+(a211−a232)ω4+((a12a21−a13a31)2−a211a232)ω2+(a12a21a32)2,c2(ω)=(a11a12a21−a11a13a31)ω2−a11a12a21a232ω,c3(ω)=(a13a31−a12a21)ω4−a12a21a232ω2.Suppose thatF(ω)=c1(ω)+2c2(ω)sinωτ1+2c3(ω)cosωτ1.Since F(0)=−(a12a21a32)2<0,and F(+∞)=+∞,then(2.21)has at least one positive root.Without loss of generality,the roots of(2.21)deﬁned byω1,ω2,...,ωk.For everyﬁxedωi(i=1,2,...,k),there exists a sequence{τ(j)2i|j=1,2,...},suchthat(2.21)holds.Letτ2∗={minτ(j)2i |i=1,2,...,k,j=1,2,...}.Whenτ2=τ2∗,(2.21)has a pari of purely imaginary roots±iω∗forτ1∈[0,τ10)(hereτ10=τ0, which was obtained in Yang[2]).In the following,we assume that(H6)[dReλ(τ2)dτ2]τ2=τ(j)2i=0.Therefore,by the general Hopf bifurcation theorem for FDEs in Hale[14],we have the following results on stability and bifurcation in system(1.3).Theorem2.2For system(1.3),suppose that(H6)holds andτ1∈[0,τ10).Then the positive equilibrium E∗of system(1.3)is asymptotically stable forτ2∈[0,τ2∗),and system(1.3)undergoes a Hopf bifurcation whenτ2=τ2∗.That is,system(1.3)has a branch of periodic solution bifurcation from the zero solution nearτ2=τ2∗.642X.-Y.Meng et al. 3Direction and stability of the Hopf bifurcationIn this section,we shall study the direction of the Hopf bifurcation and the stability of bifurcating periodic solution of system(1.3)atτ1=τ2=τ=τ0.The approach employed here is the normal form method and center manifold theorem introduced by Hassard et al.[11].More precisely,we will compute the reduced system on the center manifold with the pair of conjugate complex,purely imaginary solution of the characteristic equation(2.5).By this reduction we can determine the Hopf bi-furcation direction,i.e.,to answer the question of whether the bifurcation branch of periodic solution exists locally for supercritical bifurcation or subcritical bifurcation. The Taylor expansion of system(1.3)about the equilibrium point is˙u1(t)=a11u1(t)+a12u2(t)+a13u3(t)+a14u21(t)+a15u1(t)u2(t)+a16u1(t)u3(t),˙u2(t)=a21u1(t−τ)+a22u21(t−τ)+a23u1(t−τ)u2(t),(3.1)˙u3(t)=a31u1(t−τ)+a32u3(t−τ)+a33u21(t−τ)+a34u1(t−τ)u3(t) +a35u3(t−τ)u3(t),wherea11=−rx∗K+a1b1x∗y∗1(1+b1x∗)+a1b2x∗y∗2(1+b2x∗),a12=−a1x∗1+b1x∗,a13=−a2x∗1+b2x∗,a14=−2rK+a1b1y∗1(2+b1x∗)(1+b1x∗)3+a1b2y∗2(2+b2x∗)(1+b2x∗)3,a15=−a1(1+b1x∗)2,a16=−a2(1+b2x∗)2,a21=e1y∗1(1+b1x∗),a22=−2e1b1y∗1(1+b1x∗),a23=e1(1+b1x∗)2,a31=e2y∗2(1+b2x∗)2,a33=−2e2b2y∗2(1+b2x∗)3,a34=e2(1+b2x∗)2,a32=−Gy∗2,a35=G.Letτ=τ0+μ,u(t)=(u1(t),u2(t),u3(t))T,and u t(θ)=u(t+θ)forθ∈[−τ,0]. Denote C k[−τ,0]={φ|φ:[−τ,0]→R3,φhas k-order continuous derivative}.We can rewrite system(3.1)as˙u(t)=Lμ+F(u t,μ)(3.2)Hopf bifurcation in a three-species system with delays643 withLμ(φ)=B1φ(0)+B2φ(−τ)(3.3) andF(φ,μ)=⎛⎝a14φ21(0)+a15φ1(0)φ2(0)+a16φ1(0)φ3(0)a22φ21(−τ)+a23φ1(−τ)φ2(0)a33φ21(−τ)+a34φ1(−τ)φ3(0)+a35φ3(−τ)φ3(0)⎞⎠,(3.4)whereB1= a11a12a13000000,B2=000a2100a310a32.(3.5)Then Lμis one parameter family of bounded linear operator in C[−τ,0].By the Reisz representation theorem,there exists3×3matrix-valued functionη(·,μ):[−τ,0]→R3×3(3.6) forφ∈C[−τ,0]such thatLμ= 0−τdη(θ,μ)φ(θ).(3.7)In fact,we can chooseη(θ,μ)=B1δ(θ)+B2δ(θ+τ),(3.8) whereδ(θ)is a Dirac delta function.Next,forφ∈C [−τ,0],we deﬁneA(μ)φ(θ)=dφdθ,θ∈[−τ,0),−τdη(θ,μ)φ(θ),θ=0,(3.9)andR(μ)φ(θ)=0,θ∈[−τ,0),F(μ,φ),θ=0.(3.10)Since du tdθ=du tdt,(3.2)can be rewritten as˙u(t)=A(μ)u t+R(μ)u t,(3.11)which is an equation of the form we desired.Forθ∈[−τ,0),(3.11)is just the trivialequation du tdθ=du tdt;forθ=0,it is(3.2).Forθ∈[−τ,0),the adjoint A∗of A is deﬁned asA∗(μ)ψ(θ)=−dψdθ,θ∈(0,τ],−τψ(−θ)dη(θ,μ),θ=0.(3.12)644X.-Y.Meng et al. Forφ∈[−τ,0)andφ∈[−τ,0),we deﬁne a bilinear formψ,φ =¯ψT(0)φ(0)− 0θ=−τθξ=0¯ψT(ξ−θ)dη(θ)φ(ξ)dξ,(3.13)whereη(θ)=η(θ,0).From the above analysis,we obtain that±iω0are the eigen-values of A(0).Let q(θ)be eigenvector of A(0)corresponding to iω0,then we haveA(0)q(θ)=iω0q(θ).(3.14) Since±iω0are the eigenvalues of A(0),and other eigenvalues have strictly negative real parts,∓iω0are the eigenvalues of A∗(0).Then we have the following lemma.Lemma3.1Let q(θ)=V e iω0θbe eigenvector of A associated with iω0,and q∗(θ)=DV∗e iω0θbe eigenvector of A∗associated with−iω0.Thenq∗,q =1, q∗,¯q =0,(3.15) whereV=(1,ρ1,ρ2)T,V∗=(1,ρ∗1,ρ∗2)T,ρ1=−e1iω0(1+b1x∗)2,ρ2=−e2(G−iω0)(ω2+G2)(1+b2x∗)2,ρ∗1=−a11x∗iω0(1+b1x∗)2,ρ∗2=−a2x∗(G+iω0)(ω2+G2)(1+b2x∗)2,¯D=[1+¯ρ∗1ρ1+¯ρ∗2ρ2+τ0e−iω0τ0(a21¯ρ∗1+a31¯ρ∗2+a32¯ρ∗2ρ∗1)]−1.Proof Since q(θ)is eigenvector of A(0)corresponding to iω0,then we haveA(0)q(θ)=iω0q(θ).(3.16) For(3.9),we can rewrite(3.16)asdq(θ)dθ=iω0q(θ),θ∈[−τ,0),L(0)q(0)=iω0q(0),θ=0.(3.17)Therefore,we can obtainq(θ)=V e iω0θ,θ∈[−τ,0],(3.18) where V=(v1,v2,v3)T∈C3is a constant vector.Based on(3.3)and(3.17),we have[iω0I−(B1+B2e−iω0τ0)]V=0,(3.19) where I is identity matrix,that isiω0−a11−a12−a13−a21e−iω0τ0iω00−a31e−iω0τ00iω0−a32e−iω0τ0V=0.(3.20)So,we can chooseV= V1V2V3=⎛⎜⎜⎝1−e1i01∗2−e2(G−iω0)(ω2+G2)(1+b2x∗)2⎞⎟⎟⎠=1ρ1ρ2.(3.21)From(3.12),we can getA∗ψ= 0−τdηT(t,0)φ(−t)=B T1φ(0)+B T2φ(τ).(3.22)Letq∗(θ)=DV∗e iω0θ,θ∈[0,τ],(3.23) where q∗=(V∗1,V∗2,V∗3)T∈C3is a constant vector.Similar to the proof of(3.16) to(3.19),we can obtainV∗= V∗1V∗2V∗3=⎛⎜⎜⎝1−a11x∗iω0(1+b1x∗)2−a2x∗(G+iω0)(ω2+G2)(1+b2x∗)2⎞⎟⎟⎠=1ρ∗1ρ∗2.(3.24)Now,we can calculate q∗,q as follows:q∗,q =¯q∗T(0)q(0)− 0θ=−τ0θξ=0¯q T(ξ−θ)dη(θ)q(ξ)dξ=¯D¯V∗T V−θ=−τ0θξ=0¯V∗T e−iω0(ξ−θ)dη(θ)V e iω0ξdξ=¯D¯V∗T V−θ=−τ0¯V∗T dη(θ)e iω0θV=¯D[¯V∗T V+τ0e−iω0τ0¯V∗T B2V]=¯D[1+¯ρ∗1ρ1+¯ρ∗2ρ2+τ0e−iω0τ0(a21¯ρ∗1+a31¯ρ∗2+a32¯ρ∗2ρ∗1)].(3.25) So,when¯D=[1+¯ρ∗1ρ1+¯ρ∗2ρ2+τ0e−iω0τ0(a21¯ρ∗1+a31¯ρ∗2+a32¯ρ∗2ρ∗1)]−1,we can get q∗,q =1.On the other hand,since ψ,Aφ = A∗ψ,φ ,we have iω q∗,¯q = q∗,A¯q = A∗q∗,¯q = −iω0q∗,¯q =iω0 q∗,¯q .(3.26) Therefore, q∗,¯q =0.This completes the proof.In the remainder of this section,by using the same notations as in Hassard et al.[11],weﬁrst compute the coordinates to describe the center manifold center C0at μ=0,which is locally invariant,attracting three-dimensional manifold in C0.Let u tbe solution of (3.11)when μ=0.Deﬁnez(t)= q ∗,u t ,W (t,θ)=u t −zq −¯z ¯q =u t −2Re z(t)q(θ).(3.27)On the center manifold C 0,we haveW (z,¯z ,θ)=W 20(θ)z 22+W 11(θ)z ¯z+W 02(θ)¯z 22+···,(3.28)z and ¯z are local coordinates of center manifold C 0in the direction of q ∗and ¯q ∗.Note the W is real if u t is real.We only consider real solutions.From (3.27),we getq ∗,W = q ∗,u t −zq −¯z ¯q = q ∗,u t − q ∗,q − q ∗,¯q =0.(3.29)For a solution u t ∈C 0of (3.11),from (3.7),(3.13)and μ=0,we have˙z (t)= q ∗,˙u t = q ∗,A(0)u t +R(0)u t = A ∗(0)q ∗,u t +¯q ∗(0)F (u t ,0):=iω0z(t)+¯q ∗T(0)f 0(z,¯z).(3.30)We rewrite in abbreviated form as˙z (t)=iω0z(t)+g(z,¯z ),(3.31)whereg(z,¯z )=g 20z 22+g 11z ¯z +g 02¯z 22+g 21z 2¯z 2+···.(3.32)By (3.11)and (3.31),we have˙W =˙u t −˙z q −˙¯z ¯q =A(0)u t +R(0)u t −(iω0z +g)q −(−iω0¯z +¯g)q =A(0)u t +R(0)u t −2Re (gq)=A(0)W −2Re (q ∗(0)f 0q(θ)),θ∈[−τ,0),A(0)W −2Re (¯q ∗(0)f 0q(0))+f 0,θ=0,:=A(0)W +H (z,¯z ,θ),(3.33)whereH (z,¯z ,θ)=H 20(θ)z 22+H 11(θ)z ¯z +H 02(θ)¯z 22+···.(3.34)On the other hand,On C 0˙W =W z ˙z +W ¯z ˙¯z.(3.35)Substituting (3.13)and (3.31)into (3.35),and comparing the coefﬁcients of the above equation with those of (3.33),we get(A(0)−2iω0)W 20(θ)=−H 20(θ),A(0)W 11(θ)=−H 11(θ),(A(0)+2iω0)W 11(θ)=−H 02(θ)(3.36)Since u t=y(t+θ)=W(z,z,θ)+zq+¯z¯q,we haveu t= y1(t+θ)y2(t+θ)y3(t+θ)=W(1)(t+θ)W(2)(t+θ)W(2)(t+θ)+z1ρ1ρ2e iω0θ+z1ρ1ρ2e−iω0θ.(3.37)Therefore,we can obtain⎧⎪⎨⎪⎩y1(t+θ)=ze iω0θ+¯z e−iω0θ+W(1)20(θ)z22+W(1)11(θ)z¯z+W(1)02(θ)¯z22+···,y2(t+θ)=zρ1e iω0θ+¯z¯ρ1e−iω0θ+W(2)20(θ)z22+W(2)11(θ)z¯z+W(2)02(θ)¯z22+···, y3(t+θ)=zρ2e iω0θ+¯z¯ρ2e−iω0θ+W(3)20(θ)z22+W(3)11(θ)z¯z+W(3)02(θ)¯z22+···.(3.38)It is obvious that⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ϕ1(0)=y1(t+θ)|θ=0=z+¯z+W(1)20(0)z22+W(1)11(0)z¯z+W(1)02(0)¯z22+···,ϕ2(0)=y2(t+θ)|θ=0=zρ1+¯z¯ρ1+W(2)20(0)z22+W(2)11(0)z¯z+W(2)02(0)¯z22+···,ϕ3(0)=y2(t+θ)|θ=0=zρ2+¯z¯ρ2+W(3)20(0)z22+W(3)11(0)z¯z+W(3)02(0)¯z22+···,ϕ1(−τ0)=ze−iω0τ0+¯z e iω0τ0+W(1)20(0)z22+W(1)11(0)z¯z+W(1)02(0)¯z22+···,ϕ2(−τ0)=zρ1e−iω0τ0+¯z¯ρ1e iω0τ0+W(2)20(0)z22+W(2)11(0)z¯z+W(2)02(0)¯z22+···,ϕ3(−τ0)=zρ2e−iω0τ0+¯z¯ρ2e iω0τ0+W(3)20(0)z22+W(3)11(0)z¯z+W(3)02(0)¯z22+···,ϕ21(0)=z2+2z¯z+¯z2+[W(1)20(0)+2W(1)11(0)]z2¯z+···,ϕ1(0)ϕ2(0)=ρ1z2+(ρ1+¯ρ1)z¯z+¯ρ1¯z2+[W(2)11(0)+W(2)20(0)2+W(2)20(0)2¯ρ1+W(2)11(0)ρ1]z2¯z+···,ϕ1(0)ϕ3(0)=ρ2z2+(ρ2+¯ρ2)z¯z+¯ρ2¯z2+[W(3)11(0)+W(3)20(0)2+W(3)20(0)2¯ρ2+W(3)11(0)ρ2]z2¯z+···,ϕ1(−τ0)ϕ2(0)=e−iω0τ0ρ1z2+(e iω0τ0ρ1+e−iω0τ0¯ρ1)z¯z+e iω0τ0¯ρ1¯z2+[e−iω0τ0W(2)11(0)+e iω0τ0W(2)20(0)2+W(1)20(−τ0)2¯ρ1+W(1)11(−τ0)ρ1]z2¯z+···,ϕ1(−τ0)ϕ3(0)=e−iω0τ0ρ2z2+(e iω0τ0ρ2+e−iω0τ0¯ρ2)z¯z+e iω0τ0¯ρ2¯z2+[e−iω0τ0W(3)11(0)+e iω0τ0W(3)20(0)2+W(1)20(−τ0)2¯ρ2+W(1)11(−τ0)ρ2]z2¯z+···,ϕ21(−τ0)=e−2iω0τ0z2+2zz+e2iω0τ0z2+[2e−iω0τ0W(2)11(−τ0)+e iω0τ0W(1)20(−τ0)+···,ϕ2(−τ0)ϕ3(0)=e−iω0τ0ρ22z2+(e iω0τ0+e−iω0τ0)ρ2¯ρ2z¯z+e iω0τ0¯ρ22z2+[ρ2W(3)11(−τ0)+e iω0τ0W(3)20(−τ0)2+W(3)20(0)2¯ρ2+W(3)11(0)ρ2e−iω0τ0]z2¯z+···.(3.39)It follows thatf0(z,z)= K11z2+K12z¯z+K13¯z2+K14z2¯zK21z2+K22z¯z+K23¯z2+K24z2¯zK31z2+K32z¯z+K33¯z2+K34z2¯z+···,(3.40)where⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩K 11=a 14+a 15ρ1+a 16ρ2,K 12=2a 14+a 15(ρ1+¯ρ1)+a 16(ρ2+¯ρ2),K 13=a 14+a 15¯ρ1+a 16¯ρ2,K 14=a 14[2W (1)11(0)+W (1)20(0)]+a 15[W (2)11(0)+W (2)20(0)2+W (1)20(0)2¯ρ1+W (1)11(0)ρ1]+a 16[W (3)11(0)+W (3)20(0)2+W (1)20(0)2¯ρ2+W (1)11(0)ρ2],K 21=a 22e −2iω0τ0+a 23e −iω0τ0,K 22=2a 22+a 23(e −iω0τ0+e iω0τ0),K 23=a 22e 2iω0τ0+a 23e iω0τ0,K 24=a 22[2e −iω0τ0W (1)11(−τ0)+e iω0τ0W (1)20(−τ0)]+a 23[e −iω0τ0W (1)11(0)+e iω0τ0W (2)20(0)2+W (1)20(−τ0)2+W (1)11(−τ0)]K 31=a 33e −2iω0τ0+a 34e −iω0τ0ρ2+a 35e −iω0τ0ρ22,K 32=2a 33+a 34[−¯ρ2e −iω0τ0+ρ2e iω0τ0+ρ2¯ρ2(e iω0τ0+e −iω0τ0)],K 33=a 33e 2iω0τ0+a 34e iω0τ0¯ρ2+a 35e iω0τ0¯ρ22,K 34=a 33[2e −iω0τ0W (1)11(−τ0)+e iω0τ0W (1)20(−τ0)]+a 34[e −iω0τ0W (3)11(0)+e iω0τ0W (3)20(0)2+W (1)20(−τ0)2¯ρ2+W (1)11(−τ0)ρ2]+a 35[e −iω0τ0W (3)11(−τ0)+¯ρ2W (3)20(−τ0)2+e iω0τ0W (3)20(0)2¯ρ2+W (1)11(0)ρ2e −iω0τ0].(3.41)Since q ∗T (0)=¯D(1,¯ρ∗1,¯ρ∗2),we have g(z,¯z )=¯q ∗T (0)f 0(z,¯z )=¯D(1¯ρ∗1¯ρ∗2) K 11z 2+K 12z ¯z +K 13¯z 2+K 14z 2¯z K 21z 2+K 22z ¯z +K 23¯z 2+K 24z 2¯z K 31z 2+K 32z ¯z+K 33¯z 2+K 34z 2¯z +···=¯D {(K 11+¯ρ∗1K 21+¯ρ∗2K 31)z 2+(K 12+¯ρ∗1K 22+¯ρ∗2K 32)zz +(K 13+¯ρ∗1K 23+¯ρ∗2K 33)z 2+(K 14+¯ρ∗1K 24+¯ρ∗2K 34)z 2z }+···.Comparing the coefﬁcients of the above equation with (3.32),we haveg 20=2¯D(K 11+¯ρ∗1K 21+¯ρ∗2K 31),g 11=¯D(K 12+¯ρ∗1K 22+¯ρ∗2K 32),g 02=2¯D(K 13+¯ρ∗1K 23+¯ρ∗2K 33),g 21=2¯D(K 14+¯ρ∗1K 24+¯ρ∗2K 34).(3.42)We still need to W 20(θ)and W 11(θ)for θ∈[−τ0,0)for the expression of g 21.Indeed,we haveH (z,¯z ,θ)=−2Re (¯q ∗(0)G 0q(θ))=−2Re (g(z,¯z )q(θ))=−g(z,¯z )q(θ)−¯g(z,¯z )¯q(θ)=− g 20z 22+g 11z ¯z +g 02¯z 22+g 21z 2¯z 2q(θ)− ¯g 20¯z 22+¯g 11z ¯z +¯g 02z 22+¯g21¯z 2z2¯q(θ)···.Comparing the coefﬁcients of the above equation with those in (3.34),it is obviousthatH 20(θ)=−g 20q(θ)−¯g 02¯q(θ),H 11(θ)=−g 11q(θ)−¯g 11¯q(θ).(3.43)It follows from (3.9)and (3.36)that˙W20(θ)=AW 20(θ)=2iω0W 20(θ)−H 20(θ)=2iω0W 20(θ)+g 20q(θ)+g 02¯q(θ)=2iω0W 20(θ)+g 20q(0)e iω0θ+¯g 02q(0)e −iω0θ.Solving for W 20(θ),we can obtainW 20(θ)=ig 20q(0)ω0e iω0θ+i ¯g 02¯q(0)3ω0e −iω0θ+E 1e 2iω0θ.(3.44)By a similar way,we getW 11(θ)=−ig 11q(0)ω0e iω0θ+i ¯g 11¯q(0)ω0e −iω0θ+E 2.(3.45)Where E 1=(E (1)1,E (2)1,E (3)1)T and E 2=(E (1)2,E (2)2,E (3)2)T are both three dimen-sional vectors,and can be determined by setting θ=0in H (z,¯z ,θ).In fact,we haveH (z,¯z ,0)=−2Re (¯q ∗T(0)f 0q(0))+f 0(z,¯z)=−g 20z 22+g 11z ¯z +g 02¯z 22+g 21z 2¯z 2 q(0)− ¯g 20¯z 22+¯g 11z ¯z +¯g 02z 22+¯g21¯z 2z2¯q(0)+ K 11z 2+K 12z ¯z +K 13¯z 2+K 14z 2¯z K 21z 2+K 22z ¯z +K 23¯z 2+K 24z 2¯z K 31z 2+K 32z ¯z+K 33¯z 2+K 34z 2¯z +···.Comparing the coefﬁcients of the above equation with those in (3.34),it follows thatH 20(0)=−g 20q(0)−¯g02¯q(0)+2(K 11,K 21,K 31)T .(3.46)H11(0)=−g11q(0)−¯g11¯q(0)+(K12,K22,K32)T.(3.47) By the deﬁnition of A and(3.9),(3.36),we can get−1dη(0,θ)W20(θ)=2iω0W20(0)−H20(0).(3.48)−1dη(0,θ)W11(θ)=−H11(0).(3.49) Notice thatiω0I− 0−1dη(0,θ)e iω0θq(0)=0,−iω0I− 0−1dη(0,θ)e−iω0θ¯q(0)=0.Substituting(3.44)and(3.48)into(3.46),we obtain2iω0I− 0−1dη(0,θ)e2iω0θE1=2K11K21K31,(3.50)which leads to2iω0−a11−a12−a13−a21e−2iω0τ02iω00−a31e−2iω0τ002iω0−a32e−2iω0τ0E1=2K11K21K31.(3.51)It follows thatE1=2 2iω0−a11−a12−a13−a21e−2iω0τ02iω00−a31e−2iω0τ002iω0−a32e−2iω0τ0−1 K11K21K31.(3.52)Similarly,substituting(3.45)and(3.49)into(3.47),we obtainE2=− a11a12a13a2100a310a32−1 K12K22K32.(3.53)Thus,we can determine W20(θ)and W11(θ)from(3.2)and(3.45).Furthermore,we can see that each g ij in(3.42)is determined by parameters and delays in system(1.3). Thus,we can compute the following quantities.⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩c1(0)=i2ω0(g20g11−2|g11|2−|g02|23)+g212,μ2=−Re{c1(0)}Re{λ (τ0)},β2=2Re{c1(0)},T2=−Im{c1(0)}+μ2Im{λ(τ0)}ω0,(3.54)which determine the quantities of bifurcation periodic solutions in the center mani-fold at the critical value τ0.By the results of Hassard et al.[11],we have the following results.Theorem 3.1In (3.54),the following results hold :(i)The sign of μ2determines the direction of the Hopf bifurcation :if μ2>0(μ2<0),then the Hopf bifurcation is supercritical (subcritical )and the bifur-cation periodic solutions exist for τ>τ0(τ<τ0).(ii)The sign of β2determines the stability of the bifurcating periodic solution :thebifurcation periodic solutions are stable (unstable )if β2<0(β2>0).(iii)The sign of T 2determines the period of the bifurcating periodic solutions :theperiod increase (decrease )if T 2>0(T 2<0).4Global existence of periodic solutionsIn this section,we shall investigate the global existence of periodic solutions bifurcat-ing from the point (E ∗,τ(j)k ),j =0,1,2,...by applying a global Hopf bifurcation result due to Wu [12].Let R 3+={(x,y 1,y 2)∈R 3,x >0,y 1>0,y 2>0},X =C([−τ,0],R 3+)and z t =(x t ,y 1t ,t 2t )T ∈X be deﬁned as z t (θ)=z(t +θ)for t ≥0and θ∈[−τ,0].Since x(t),y 1(t)and y 2(t)denote the densities of the prey,the ﬁrst predator and the second predator,respectively,the positive solution of system (1.3)is of interest and its periodic solutions only arise in the ﬁrst quadrant.Thus,throughout this section,we consider system (1.3)only in the domain R 3+.It is easy to see that system (1.3)can be rewritten as the following functional differential equation˙z (t)=F (z t ,τ,p),(4.1)parameterized by two nonnegative real parameters (τ,p)∈R +×R +,where R +=[0,+∞).It is obvious that (4.1)has only an equilibria z ∗=E ∗in R 3+under the assumption (H1),(H2)and easily to see that the mapping F :X ×R +×R +is completely continuous.If we restrict F to the subspace of constant function Z ,then F is identiﬁed with R 3+and thus we obtain a mapping F =F |R 3+×R +×R +:R 3+×R +×R +→R 3+.Let ¯z 0∈X be the constant mapping with value z 0∈R 3+.The point (¯z 0,τ0,p 0)iscalled a stationary solution of system (4.1)if F (¯z 0,τ0,p 0)=0.From system (1.3)we know easily that the following condition regarding the mapping Fholds.(A1) F∈C 2(R 3+×R +×R +,R 3+).It follows from system (1.3)that D z F(z,τ,p)=⎛⎜⎝r −2rx K−a 1y 1(1+b 1x)2−a 2y 2(1+b 2x)2−a 1x 1+b 1x −a 2x1+b 2xe 1y 1(1+b 1x)2−d +e 1x 1+b 1xe 2y 2(1+b 2x)2−d 2−2Gy 2+e 2x 1+b 2x⎞⎟⎠.(4.2)。

液相色谱词汇中英文对照

液相色谱词汇中英文对照液相色谱词汇中英文对照高效毛细管电泳high—performance capillary electrophoresis归一化法normalization method毛细管等电聚焦capillary isoelectric focusing毛细管等速电泳isotachophoresis毛细管电色谱capillary electrochromatography毛细管电泳capillary electrophoresis毛细管电泳电喷雾质谱联用capillary electrophoresis – electr芯片电泳microchip electrophoresis色谱法chromatography色谱峰chromatographic peak色谱峰区域宽度peak width色谱富集过样samt injection of chromatography色谱工作站chromatographic working station色谱图chromatogram色谱仪chromatograph色谱柱chromatographic column色谱柱column色谱柱切换技术switching column technique毛细管超临界流体色谱法capillary supercritical fluid chromat…毛细管电泳基质辅助激光解吸电离质谱离线检测off—line capillar…毛细管电泳离子分析capillary ion analysis毛细管电泳免疫分析immunity analysis of capillary electropho…毛细管胶束电动色谱micellar electrokinetic chromatography毛细管凝胶电泳capillary gel electrophoresis毛细管凝胶柱capillary gel column毛细管亲和电泳affinity capillary electrophoresis毛细管区带电泳capillary zone electrophoresis毛细管有效长度the effective length of capillary electrophor…间接检测indirect detection间接荧光检测indirect fluorescence detection间接紫外检测indirect ultraviolet detection检测器detector检测器检测限detector detectability检测器灵敏度detector sensitivity检测器线性范围detector linear range阴离子交换剂anion exchanger阴离子交换色谱法anion exchange chromatography, AEC高速逆流色谱法high speed counter—current chromatography高温凝胶色谱法high temperature gel chromatography高效液相色谱-付里叶变换红外分析法high performance liquid ch…高效液相色谱法high performance liquid chromatography高效柱high performance column高压流通池技术high pressure flow cell technique高压输液泵high pressure pump高压梯度high-pressure gradient高压液相色谱法high pressure liquid chromatography阴离子交换树脂anion exchange resin荧光薄层板fluorescent thin layer plate荧光检测器fluorescence detector荧光色谱法fluorescence chromatography迎头色谱法frontal chromatography迎头色谱法frontal method硬（质)凝胶hard gel有机改进剂organic modifier有机相生物传感器Organic biosensor有效峰数effective peak number EPN有效理论塔板数number of effective theoretical plates有效塔板高度effective plate height有效淌度effective mobility淤浆填充法slurry packing method予柱pre-column在线电堆集on-line electrical stacking在柱电导率检测on—column electrical conductivity detection噪声noise噪信比noise –signal ratio增强紫外-可见吸收检测技术UV—visible absorption enhanced det…窄粒度分布narrow particle size distribution折射率检测器refractive index detector，RID真空脱气装置vacuum degasser阵列毛细管电泳capillary array electrophoresis蒸发光散射检测器evaporative light—scattering detector, ELSD整体性质检测器integral property detector正相高效液相色谱法normal phase high performance liquid chro…正相离子对色谱法normal phase ion-pair chromatography正相毛细管电色谱positive capillary electrokinetic chromatog…直接化学离子化direct chemical ionization GC-MS直接激光在柱吸收检测on-column direct laser detection纸色谱法paper chromatography置换色谱法displacement chromatography制备色谱preparative chromatography制备色谱仪preparative chromatograph制备柱preparation column智能色谱chromatography with artificial intelligence质量色谱mass chromatography质量型检测器mass detector质量型检测器mass flow rate sensitive detector中压液相色谱middle—pressure liquid chromatography重建色谱图reconstructive chromatogram重均分子量weight mean molecular weight轴向扩散longitudinal diffusion轴向吸收池absorption pool of axial direction轴向压缩柱axial compression column柱端电导率检测out—let end detection of electrical conductiv…柱负载能力column loadability柱后衍生化post-column derivatization柱老化condition （aging) of column柱流出物（column) effluent柱流失column bleeding柱内径column internal diameter柱前衍生化pro-column derivatization柱切换技术column switching technique柱清洗column cleaning柱容量column capacity柱入口压力column inlet pressure柱色谱法column chromatography柱上检测on—line detection柱渗透性column permeability柱寿命column life柱头进样column head sampling柱外效应extra—column effect柱温箱column oven柱效column efficiency柱压column pressure柱再生column regeneration柱中衍生化on-column derivatization注射泵syringe pump转化定量法trans-quantitative method紫外-可见光检测器ultraviolet visible detector，UV-Vis紫外吸收检测器ultraviolet absorption detector自动进样器automatic sampler自由溶液毛细管电泳free solution capillary electrophoresis总分离效能指标over-all resolution efficiency总交换容量total exchange capacity总渗透体积total osmotic volume纵向扩散longitudinal diffusion组合式仪器系统building block instrument最佳流速optimum flow rate最佳实际流速optimum practical flow rate最小检测量minimum detectable quantity最小检测浓度minimum detectable concentration萃取色谱法extraction chromatography脱气装置degasser外标法external standard method外梯度outside gradient网状结构reticular structure往复泵reciprocating pump往复式隔膜泵reciprocating diaphragm pump微分型检测器differential detector微孔树脂micro—reticular resin微库仑检测器micro coulometric detector微量进样针micro-syringe微量色谱法micro-chromatography微乳液电动色谱microemulsion electrokinetic chromatography微生物传感器Microbial sensor微生物显影bioautography微填充柱micro-packed column微吸附检测器micro adsorption detector微型柱micro-column涡流扩散eddy diffusion无机离子交换剂inorganic ion exchanger无胶筛分毛细管电泳non-gel capillary electrophoresis无孔单分散填料non-porous monodisperse packing无脉动色谱泵pulse-free chromatographic pump物理钝化法physical deactivation吸附等温线adsorption isotherm吸附剂adsorbing material吸附剂活性adsorbent activity吸附平衡常数adsorption equilibrium constant吸附溶剂强度参数adsorption solvent strength parameter吸附色谱法adsorption chromatography吸附型PLOT柱adsorption type porous—layer open tubular colum…吸附柱adsorption column吸光度比值法absorbance ratio method洗脱强度eluting power显色器color—developing sprayer限制扩散理论theory of restricted diffusion线速度linear velocity线性梯度linear gradient相比率phase ratio相对保留值relative retention value相对比移值relative Rf value相对挥发度relative volatility相对灵敏度relative sensitivity相对碳（重量)响应因子relative carbon response factor相对响应值relative response相对校正因子relative correction factor相交束激光诱导的热透镜测量heat lens detection of intersect …相似相溶原则rule of similarity响应时间response time响应值response小角激光散射光度计low—angle laser light scattering photomet…小内径毛细管柱Microbore column校正保留体积corrected retention volume校正曲线法calibration curve method校正因子correction factor旋转薄层法rotating thin layer chromatography旋转小室逆流色谱rotational little-chamber counter—current c…选择性检测器selective detector循环色谱法recycling chromatography压电晶体piezoelectric crystal压电免疫传感器Piezoelectric Immunosensor压电转换器piezoelectric transducer压力保护pressure protect压力上限pressure high limit压力梯度校正因子pressure gradient correction factor压力下限pressure low limit衍生化法derivatization method衍生化试剂derivatization reagent阳离子交换剂cation exchanger阳离子交换色谱法cation exchange chromatography, CEC氧化铝色谱法alumina chromatography样品环sample loop样品预处理sample pretreatment液-液分配色谱法liquid—liquid partition chromatography液—液色谱法liquid—liquid chromatography液滴逆流色谱drop counter-current chromatography液固色谱liquid-solid chromatography液晶固定相liquid crystal stationary phase液态离子交换剂liquid ion exchanger液相传质阻力resistance of liquid mass transfer液相色谱—傅里叶变换红外光谱联用liquid chromatography—FTIR 液相色谱—质谱分析法liquid chromatography-mass spectrometry 液相色谱—质谱仪liquid chromatography-mass spectrometer液相色谱法liquid chromatography液相载荷量liquid phase loading溶剂效率solvent efficiency溶解度参数solubility parameter溶液性能检测器solution property detector溶胀swelling溶质性质检测器solute property detector容量因子capacity factor渗透极限分子量permeation limit molecular weight生物色谱biological chromatography生物特异性柱biospecific column生物自显影法bioautography升温速率temperature rate湿法柱填充wet column packing十八烷基键合硅胶octadecyl silane石墨化碳黑graphitized carbon black示差折光检测器differential refraction detector试剂显色法reagent color—developing method手动进样器manual injector手性氨基酸衍生物GC固定相chiral amino aci d derivatives stat…手性拆分试剂chiral selectors手性固定相chiral stationary phase手性固定相拆分法chiral solid phase separation手性环糊精衍生物GC固定相chiral cyclodextrin der GC手性金属络合物GC固定相chirametal stationary phase in GC 手性流动相chiral mobile phase手性流动相拆分法chiral mobile phase separation手性色谱chiral chromatography手性试剂chiral reagent手性衍生化法chiral derivation method疏溶剂理论solvophobic theory疏溶剂色谱法solvophobic chromatography疏溶剂作用理论solvophobic interaction principle疏水作用色谱hydrophobic interaction chromatography树脂交换容量exchange capacity of resin数均分子量number mean molecular weight双保留机理dual reservation mechanism双活塞往复泵two-piston reciprocating pump双束差分检测器detector of dual-beam difference双柱色谱法dual column chromatography水凝胶hydragel水系凝胶色谱柱aqua—system gel column死区域dead zone死体积dead volume塔板理论方程plate theory equation碳分子筛carbon molecular sieve特殊选择固定液selective stationary phase梯度洗脱gradient elution梯度洗脱装置gradient elution device梯度液相色谱gradient liquid chromatography体积排斥理论size exclusion theory体积排斥色谱size exclusion chromatography体积色谱法volumetric chromatography填充柱packed column填料packing material停流进样stop—flow injection通用型检测器common detector涂层毛细管coated capillary拖尾峰tailing peak拖尾因子tailing factor流动分离理论separation by flow流动相mobile phase流动相梯度eluent gradient流体动力学进样hydrostatic pressure injection流体力学体积hydrodynamic volume流型扩散dispersion due to flow profile脉冲阻尼器pulse damper酶传感器Enzyme sensor酶联免疫传感器Enzyme linked immunosensor酶免疫分析enzyme immnunoassay内标法internal standard method内标物internal standard内梯度inside gradient逆流色谱法counter-current chromatography逆流色谱仪counter current chromatograph凝胶过滤色谱gel filtration chromatography凝胶内体积gel inner volume凝胶色谱法gel chromatography凝胶色谱仪gel chromatograph凝胶渗透色谱gel permeation chromatography凝胶外体积gel interstitial volume凝胶柱gel column浓度梯度成像检测器concentration gradient imaging detector 浓度型检测器concentration detector排斥极限分子量exclusion limit molecular weight排斥体积exclusion volume排阻薄层色谱法exclusion TLC漂移drift迁移时间migration time迁移时间窗口the window of migration time前延峰leading peak前沿色谱法frontal chromatography强碱性阴离子交换剂strong-base anion exchanger强酸性阳离子交换剂strongly acidic cation exchanger切换时间switching time去离子水deionized water全多孔硅胶macro-reticular silica gel全多孔型填料macro-reticular packing material全二维色谱Comprehensive two-dim ensional gas chromatography…全硅烷化去活complete silylanization deactivation溶剂强度solvent strength激光解吸质谱法laser desorption MS，LDMS激光色谱laser chromatography激光诱导光束干涉检测detection of laser—induced light beam I…激光诱导毛细管振动测量laser—reduced capillary vibration det…激光诱导荧光检测器laser—induced fluorescence detector记忆峰memory peak记忆效应memory effect夹层槽sandwich chamber假峰ghost peak间断洗脱色谱法interrupted—elution chromatography间接光度（检测)离子色谱法indirect photometric ion chromato…间接光度（检测）色谱法indirect photometric chromatography减压液相色谱vacuum liquid chromatography键合固定相bonded stationary phase键合型离子交换剂bonded ion exchanger焦耳热joule heating胶束薄层色谱法micellar thin layer chromatography胶束液相色谱法micellar liquid chromatography交联度crosslinking degree阶梯梯度stagewise gradient进样阀injection valve进样量sample size进样器injector聚苯乙烯PSDVB聚硅氧烷高温裂解去活high—temperature pyrolysis deactivation…聚合物基质离子交换剂polymer substrate ion exchanger绝对检测器absolute detector可见光检测器visible light detector可交换离子exchangable ion空间性谱带加宽band broadening in space空穴色谱法vacancy chromatography孔结构pore structure孔径pore diameter孔径分布pore size distribution控制单元control unit快速色谱法high—speed chromatography理论塔板高度height equivalent to a theoretical plate(HETP）理论塔板数number of theoretical plates峰面积peak area峰面积测量法measurement of peak area峰面积校正calibration of peak area峰容量peak capacity固定相stationary phase固定液stationary liquid固定液的相对极性relative polarity of stationary liquid固定液极性stationary liquid polarity固相扩散solid diffusion固相荧光免疫分析solid phase fluorescence immunoassay固有粘度intrinsic viscosity光散射检测器light scattering detector硅胶silica gel硅烷化法silanization硅烷化法silanizing硅烷化载体silanized support过压液相色谱法over pressured liquid chromatography,OPLC恒流泵constant flow pump恒温操作constant temperature method恒压泵constant pressure pump红色载体red support红外检测器infrared detector红外总吸光度重建色谱图total infrared absorbance reconstruct…化合物形成色谱compound-formation chromatography化学发光检测器chemiluminescence detector化学发光检测器Chemiluminescence detector，SCD化学键合固定相bonded stationary phase化学键合相色谱bonded phase chromatography化学色谱法chemi—chromatography环糊精电动色谱cyclodextrin electrokinetic chromatography环形展开比移值circular development Rf value环形展开法circular development缓冲溶液添加剂buffer additives辉光放电检测器glow discharge detector混合床离子交换固定相mixed-bed ion exchange stationary phase 混合床柱mixed bed column活塞泵piston pump活性activation活性硅胶activated silica gel活性氧化铝activated aluminium oxide基流background current or base current基线baseline基线宽度baseline width基质substrate materials基质隔离技术matrix isolation technique电歧视效应the effect of electrical discrimination电迁移进样electrophoretic injection电渗流electroendosmotic flow电渗流标记物electroendosmotic flow marker电渗流淌度electroendosmotic mobility电泳淌度electrophoretic mobility调整保留时间adjusted retention time调整保留体积adjusted retention volume叠加内标法added internal standard method二极管阵列检测器diode-array detector，DAD二维色谱法two-dimensional chromatography二元溶剂体系dual solvent system反冲洗back wash反吹技术back flushing technique反峰negative peak反离子counter ion反相高效液相色谱法reversed phas e high performance liquid ch…反相离子对色谱reversed phase ion pair chromatography反相离子对色谱法reversed phase ion—pair chromatography反相毛细管电色谱reverse capillary electrokinetic chromatogr…反相柱reversed phase column反应色谱reaction chromatography反圆心式展开anti-circular development反转电渗流reverse electroendosmotic flow范第姆特方程式van Deemter equation仿生传感器Biomimic electrode放射性检测器radioactivity detector放射自显影autoradiography非极性固定相non—polar stationary phase非极性键合相non—polar bonded phase非水系凝胶色谱柱non-aqua—system gel column非水相色谱nonaqueous phase chromatography非吸附性载体non-adsorptive support非线性分流non-linearity split stream非线性色谱non—linear chromatography非线性吸附等温线non-linear adsorption isotherm酚醛离子交换树脂phenolic ion exchange resin分离－反应－分离展开SRS development分离数separation number分离因子separation factor分离柱separation column分配等温线distribution isotherm分配色谱partition chromatography分配系数partition coefficient分析型色谱仪analytical type chromatograph分子扩散molecular diffusion封尾endcapping峰高peak heightpH梯度动态分离dynamic separation of the pH gradient pH值梯度洗脱pH gradient elutionZata电势Zata potentialZ形池Z-form pool氨基键合相amino-bonded phase氨基酸分析仪amino acid analyzer安培检测器ampere detector白色载体white support半微柱semimicro-column半制备柱semi-preparation column包覆型离子交换剂coated ion exchanger包覆型填料coated packing material保护柱guard column保留间隙retention gap保留时间retention time保留体积retention volume保留温度retention temperature保留值定性法retention qualitative method保留值沸点规律boiling point rule of retention保留值碳数规律carbon number rule of retention保留指数retention index保留指数定性法retention index qualitative method背景电导background conductance苯酚磺酸树脂phenol sulfonic acid resin苯乙烯styrene比保留体积specific retention volume比例阀proportional valve比渗透率specific permeability比移值Rf value便携式色谱仪portable chromatograph标准偏差standard deviation表观电泳淌度apparent electrophoretic mobility表观交换容量apparent exchange capacity表面电位检测器surface potential detector表面多孔硅胶superficially porous silica gel表面多孔填料superficially porous packing material表面多孔型离子交换剂superficially porous ion—exchanger玻璃球载体glass beads support不分流进样splitless sampling参比柱reference column场放大进样electrical field magnified injection场流分离field-flow fractionation场流分离仪field-flow fractionation场效应生物传感器Field effect transistor based Biosensor常压液相色谱法common-pressure liquid chromatography超声波脱气ultrasonic degas程序变流色谱法programmed flow (gas）chromatography程序升温进样programmed temperature sampling程序升温色谱法programmed temperature （gas) chromatography 程序升温蒸发器programmed temperature vaporizer ，PTV程序升压programmed pressure大孔树脂macro-reticular resin大孔填料macro-reticular packing material大内径毛细管柱Megaobore column单活塞往复泵single piston reciprocating pump单相色谱仪single phase chromatograph单向阀one—way valve单柱离子色谱法single column ion chromatography等度洗脱isocratic elution等离子体色谱法plasma chromatography等途电泳—毛细管区带电泳耦合进样isotachophoresis injection—c…低负荷柱low load column低容量柱low capacity column低压梯度low—pressure gradient低压液相色谱low—pressure liquid chromatography电导池conductance cell电导检测法conductance detection电荷转移分光光度法charge transfer spectrophotometry电化学检测器electrochemical detector电解抑制器electrolyze suppressor。