Bifurcation of non-negative solutions for an elliptic system
bifurcation词源
分岔的词源英文版The word "bifurcation" is derived from the Latin word "bifurcus," which means "having two forks." It is a term that is used to describe the process of dividing into two branches or parts. The word "bifurcation" is often used in the context of mathematics, physics, and biology.In mathematics, bifurcation is a term that is used to describe the occurrence of two solutions to a problem where there was previously only one. This can happen when a parameter in the problem is changed. For example, in the case of the quadratic equation y = ax^2 + bx + c, the discriminant b^2 - 4ac determines the number of solutions to the equation. If the discriminant is positive, there are two real solutions. If the discriminant is zero, there is one real solution. And if the discriminant is negative, there are no real solutions.In physics, bifurcation is often used to describe the behavior of systems that undergo phase transitions. For example, when a fluid undergoes a phase transition from a liquid to a gas, the fluid can either condense into a single drop or it can split into two or more drops. This behavior is known as a bifurcation.In biology, bifurcation is often used to describe the branching of organisms. For example, the branching of trees is a type of bifurcation. Bifurcation is also used to describe the branching of nerves and blood vessels.The word "bifurcation" is a versatile term that can be used to describe a wide variety of phenomena. It is a term that is used in a variety of fields, including mathematics, physics, and biology.中文版分岔的词源“分岔”一词源自拉丁语“bifurcus”，意为“有两个叉子”。
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●●。
非靶向代谢组学方法英语
非靶向代谢组学方法英语Non-targeted Metabolomics Methods in EnglishIntroductionNon-targeted metabolomics is an innovative approach in the field of metabolomics that aims to identify and quantify as many metabolites as possible in a given biological sample without any prior knowledge or bias towards specific metabolites. This method provides comprehensive insights into the global biochemical changes occurring in a biological system, such as a cell, tissue, or organism. In recent years, non-targeted metabolomics has gained immense popularity due to its ability to unravel intricate metabolic pathways and discover novel biomarkers for various diseases.Sample Collection and PreparationThe first step in non-targeted metabolomics is the collection and preparation of the biological sample. The choice of sample depends on the research question and can range from blood, urine, tissues, or even fecal samples. It is crucial to handle the samples with extreme care to avoid any degradation or contamination of metabolites. Sample preparation involves various techniques such as extraction, filtration, and derivatization, to enhance the stability and visibility of metabolites during subsequent analysis.Mass Spectrometry-Based AnalysisMass spectrometry (MS) is the key analytical technique used in non-targeted metabolomics. It detects and quantifies metabolites based on their mass-to-charge ratio (m/z) and abundance. Liquid chromatography-massspectrometry (LC-MS) and gas chromatography-mass spectrometry (GC-MS) are commonly used platforms for metabolite analysis. LC-MS is suitable for hydrophilic compounds, while GC-MS is preferred for volatile and thermally stable metabolites.Data Acquisition and PreprocessingOnce the samples are analyzed using MS, the raw data obtained needs to be processed and converted into a format suitable for downstream analysis. This step involves data acquisition, which includes peak picking, alignment, and normalization. Peak picking identifies and quantifies metabolite peaks in the acquired spectra, while alignment corrects any potential retention time variations. Normalization ensures that all samples are comparably represented, eliminating any technical biases.Statistical Analysis and IdentificationStatistical analysis is a crucial step in non-targeted metabolomics, as it helps in identifying significant metabolites and detecting patterns within the dataset. Multivariate statistical techniques, such as principal component analysis (PCA) and partial least squares-discriminant analysis (PLS-DA), are commonly used to visualize and interpret the data. Additionally, metabolite identification is performed by matching the acquired mass spectra with metabolite databases, such as the Human Metabolome Database (HMDB) and the Kyoto Encyclopedia of Genes and Genomes (KEGG), using tools like MassBank, MetFrag, or Metlin.Metabolic Pathway AnalysisOne of the key strengths of non-targeted metabolomics is its ability to unravel complex metabolic pathways. Pathway analysis tools, such as MetaboAnalyst, MetaboMiner, and Ingenuity Pathway Analysis (IPA), are used to identify significantly altered pathways and discover potential biomarkers. These analyses provide crucial insights into the underlying biochemical mechanisms and aid in understanding the disease pathogenesis or physiological responses.Challenges and Future PerspectivesDespite its numerous advantages, non-targeted metabolomics faces several challenges. Metabolite identification remains a major bottleneck due to the limited coverage of metabolite databases and the lack of standardization in data reporting. Additionally, the high complexity and dynamic range of metabolomes make it difficult to detect low-abundance metabolites accurately. Nevertheless, advancements in analytical techniques, bioinformatics, and collaborative efforts are steadily overcoming these challenges and driving the field forward.In conclusion, non-targeted metabolomics plays a vital role in understanding the complex metabolic dynamics within biological systems. Through the use of advanced mass spectrometry techniques, data analysis tools, and metabolite identification strategies, this approach has the potential to uncover novel biomarkers and therapeutic targets for various diseases. With continued advancements, non-targeted metabolomics is poised to revolutionize personalized medicine and contribute significantly to the field of biomedical research.。
常微分方程中的英文单词和短语
常微分方程
Ordinary Differential Equtions
第一章 基本概念
Chapter 1 Basic Concepts
常微分方 程
阶 线性 非线性 偏微分方 程
通解 特解 初值条件 初值问题 柯西问题 几何解释 积分曲线 线素 线素场 方向场 等斜线 对称形式 奇异点 通积分
boundary finite closed interval open set finite semi-open interval
Lagrange formula
l ocal finite covering theorem
bar area monotone decreasing sectorial area comparability solution minimum solution maximum solution sl ope
topological dynamical system
differential dynamical system
stability of solution Liapunov stability
asymptotic stability asymptotic stability field attraction domain globe asymptotic stability
第六章 线 性 微 分 方 程 组
Chapter 6 Linear Differential Equations
存在和唯一 性定理 齐次线性微 分方程组 基本解组 朗斯基行列 式 刘维尔公 式 解矩阵 基解矩阵 常数矩阵 非齐次线性 微分方程 组 常数列向 量 常数变异公 式 计算公式 结构公式 常系数线性 微分方程 组 矩阵指数函 数 标准基解矩 阵 若尔当标准 型 矩阵函数 实值解 复值解 代数余子 式 特征方程 算子式 算子多项 式 拉普拉斯逆 变换
一类具有交叉扩散项的捕食-食饵模型的局部分歧
一类具有交叉扩散项的捕食-食饵模型的局部分歧容跃堂;董苗娜;何堤;王晓丽【摘要】研究一类带有交叉扩散项的捕食-食饵模型在齐次Dirichlet边界条件下分歧解的存在性.利用极大值原理和上下解法得到正解的先验估计,并借助Crandall-Rabinowitz分歧理论,得出局部分歧正解存在的充分条件.%The existence of bifurcation solutions for a predator-prey model with cross-diffusion under homogeneous Dirichlet boundary conditions is concerned.By the maximum principle,a priori estimate of positive solutions are obtained.Then by Crandall-Rabinowitz bifurcation theory,the sufficient conditions for the existence of positive solutions to a local bifurcation is proved.【期刊名称】《纺织高校基础科学学报》【年(卷),期】2016(029)004【总页数】7页(P443-449)【关键词】捕食-食饵;自扩散;交叉扩散;先验估计;局部分歧【作者】容跃堂;董苗娜;何堤;王晓丽【作者单位】西安工程大学理学院,陕西西安710048;西安工程大学理学院,陕西西安710048;西安工程大学理学院,陕西西安710048;西安工程大学理学院,陕西西安710048【正文语种】中文【中图分类】O175.26近年来,关于生物数学领域的捕食食饵模型的研究已经成为热点,尤其是对于种群扩散影响下的捕食模型,国内外学者均已取得了一些符合实际的研究成果.文献[1]研究了一类捕食模型的正常数平衡态解的稳定性及分歧;文献[2-3]利用极大值原理和分歧定理研究了一类捕食模型局部解的延拓;文献[4-7]利用分歧定理研究了模型在交叉扩散影响下的正解的存在性问题.在文献[8]中,作者提出了一类具有扩散项的捕食食饵模型,通过给出正解的先验估计及局部分歧解存在条件,进而得到该系统平衡态的全局分歧解及其走向;文献[9]则在上述基础上研究了该类模型在交叉扩散项影响下的分歧.在同时考虑交叉扩散和自扩散项时，本文将继续研究如下捕食-食饵模型在齐次Dirichlet边界条件下正解的存在性，即其中:Ω为RN中具有光滑边界∂Ω上的有界区域;u,v分别表示食饵和捕食者的种群密度;a,b,c,d,α,β都是正常数,m1,m3表示自扩散系数;m2,m4表示交叉扩散系数,反应函数是Bazykin研究捕食者的饱和不稳定性与食饵的稳定性时建立的功能反应函数,生物背景参见文献[10].本文将针对模型(1)的如下平衡态方程展开讨论.注:对于问题(2)的解(u,v),如果在Ω中,(u,v)中只有一个分量为0,则称其为半平凡解. 记}.定义中的范数为通常的Banach空间C1 (Ω)中的范数,令,则X是Banach空间. 首先,考虑特征值问题引理1[11] 假设为常数,则问题(3)的所有特征值满足λ1(p,q)<λ2(p,q)≤λ3(p,q)≤…→∞,相应的特征函数为φ1,φ2,….由文献[11]知λ1(p,q)是简单的且关于q(x)严格单调递增.为方便起见,简记λ1=λ1(0),相应的主特征函数φ1>0.再考虑边值问题引理2[11] (1) 如果a≤λ1,则u=0是问题(4)的唯一非负解;若a>λ1,则问题(4)的唯一正解为θa.(2) 如果c≤λ1,则v=0是问题(5)的唯一非负解;当c>λ1时,其存在唯一正解θc.因此,当a>λ1,问题(2)存在半平凡解(θa,0);当c>λ1,问题(2)存在半平凡解(0,θc).定义Z=(U,V),其中U=(1+m1u+m2v)u,V=(1+m3v+m4u)v,则即(u,v)≥0与(U,V)≥0之间存在一一对应的关系.现在,引入和问题(2)等价的半线性椭圆系统易知,当a,c>λ1时,问题(6)的两个半平凡解分别为,其中.引理3[12] 假设a>λ1,令,则L(a)的特征值均大于0.引理4 设c>λ1,则当cm4>d时,存在唯一的a=a*(c)∈(λ1,∞),满足,且a=a*(c)关于c严格单调递增.此外,∃ψ*≥0满足证明取.显然A(λ1,c)=λ1(-c)=λ1-c<0.由于当a→∞时θa→∞,故有.经计算得又因为与均严格单调递增,可知A(a,c)关于a严格单调递增.从而存在唯一的a=a*(c)>λ1,使得A(a*(c),c)=0.再对A(a*(c),c)=0两边关于c求导,得Aa(a*(c),c)·a*′(c)+Ac(a*(c),c)=0.由于Ac(a,c)<0,结合Aa(a,c)>0得知a*′(c)>0,即a=a*(c)关于c严格单调递增.类似可以证明以下引理.引理5 假设c>λ1,则当aβ>b时,就存在唯一的a=a*(c)∈(λ1,∞),满足,且a=a*(c)关于c严格单调递增.此外,∃φ*≥0满足现在,结合文献[12-13]中的方法给出系统(6)的正解存在的必要条件及先验估计.定理1 当a≤λ1,或者,则问题(6)没有正解.证明若问题(6)存在正解(U,V),由问题(6)中的第2个方程得两边同乘以V,分部积分得由Poincare不等式‖‖，可得,同理可证a>λ1.与已知条件矛盾,则定理1得证. 定理2 设且b-βa(1+αa)>0.若(U,V)是问题(6)的任意正解,则∀x∈Ω,有证明设∃x0∈Ω,使得(x).由于故有,则同理可得由(u,v)与(U,V)之间的关系知定理2成立.现在以a为分歧参数,参考文献[14-19],利用Crandall-Rabinowitz局部分歧定理,给出问题(6)发自半平凡解与的局部分歧正解的存在性.定理3 设且cm4>d,则为问题(6)的分歧点,且的领域内存在正解其中a*由唯一确定,ψ*>0满足证明令其中u,v均为(U,V)的函数.将问题(6)在(U,V)处Taylor展开为这里,偏导数为处的导数值,满足.同时对(U,V)求导,得令,则有显然T(a;0,0)=0.记关于在(a*;0,0)处的Frechlet导数是L(a*;0,0).经计算,L(a*;0,0)·(φ,ψ)=0等价于如果ψ≡0,那么由算子La*可逆知φ≡0,矛盾,所以ψ不恒为零.又,故有因此,算子L(a*;0,0)的核空间N(L(a*;0,0))=span{U0},U0=(φ*,ψ*)T,其中又令L*(a*;0,0)为L(a*;0,0)的自伴算子,类似可得由Fredholm选择公理知因此可得dimN(L(a*;0,0))=1,codimR(L(a*;0,0))=1.令,下面采用反证法证明假设∃(h,k)∈X,使得L1(a*;0,0)·(φ*,ψ*)=L(a*;0,0)·(h,k).经计算得那么有两边同时乘以ψ*,分部积分得由于cm4-d>0,且θa关于a严格单调递增,则上式左端大于0,矛盾.由Crandall-Rabinowitz局部分歧定理知,存在充分小的δ>0及C1连续曲线(a(s):Φ1(s),Ψ1(s)):(-δ,δ)→R×X满足a(0)=a*,Φ1(0)=0,Ψ1(0)=0,Φ1(s),Ψ1(s)∈Z 使得(a(s):(φ*+Φ1(s)),s(ψ*+Ψ1(s)))是T(a(s):的零点,其中X=Z⨁N(L(a*;0,0)),由于,因此可得到发自的局部分歧正解Γ*.同理可得到发自半平凡分支的局部分歧正解.定理4 设且aβ>b,则为问题(5)的分歧点,且的领域内存在正解a*由唯一确定,φ*>0满足充分小.这里(Φ2(s),Ψ2(s);a(s))是连续函数,满足a(0)=a*,Φ2(0)=0,Ψ2(0)=0,∫Ωψ2φ*dx=0,且RONG Yuetang,DONG Miaona,HE Di,et al.The local bifurcation for a kind of prey-predator model with cross-diffusion[J].Basic Sciences Journal of Textile Universities,2016,29(4):443-449.【相关文献】[1] 周冬梅,李艳玲.一类捕食模型正常数平衡态解的稳定性及分歧[J].科学技术与工程,2010,10 (23):5615-5619.ZHOU Dongmei,LI Yanling.Stability and bifurcation of positive constant steady-state solution for predator-prey model[J].Science Technology andEngineering,2010,10(23):5615-5619.[2] 李海侠,李艳玲.一类捕食模型正平衡解的整体分歧[J].西北师范大学学报:自然科学版,2006,42(2):8-12.LI Haixia,LI Yanling.Bifurcation of positive steady-state solutions for a king of predator-prey model[J].Journal of Northwest Normal University:Natural Science,2006,42(2):8-12. [3] 王妮娅,李艳玲.一类带收获率的的捕食模型的全局分歧和稳定性[J].安徽师范大学学报:自然科学版,2015,38(1):25-30.WANG Niya,LI Yanling.Global bifurcation and stability of a class of predator-prey models with prey harvesting [J].Journal of Anhui University:Natural Science Edition,2015,38(1):25-30.[4] KUTO K,YAMADA Y.Multiple coexistence states for a prey-predator system with cross-diffusion[J].J Differential Equations,2004,197(2):315-348.[5] 张晓晶,容跃堂,何堤,等.一类带有交叉扩散的捕食-食饵模型的分歧性[J].纺织高校基础科学学报,2014,27(3):322-326.ZHANG Xiaojing,RONG Yuetang,HE Di.Bifurcation for a prey-predator model with cross-diffusion[J].Basic Sciences Journal of Textile Universities,2014,27(3):322-326.[6] DUBEY B,DAS B,HASSAIN J.A prey-predator interaction model with self and cross-diffusion[J].Ecol Modelling,2002,141:67-76.[7] ZHANG Cunhua,YAN Xiangping.Positive solutions bifurcating from zero solution in a Lotka-Volterra competitive system with cross-diffusion effects[J].Appl Math J China Univ,2011,26(3):342-352.[8] 冯孝周,吴建华.具有饱和与竞争项的捕食系统的全局分歧及稳定性[J].系统科学与数学,2010,30(7):979-989.FENG Xiaozhou,WU Jianhua.Global bifurcation and stability for predator-prey model with predator saturation and competition[J].Journal of System Science and Mathematical Sciences,2010,30(7):979-989.[9] 何堤,容跃堂,张晓晶.一类具有交叉扩散的捕食-食饵模型的分歧[J].纺织高校基础科学学报,2015,28(4):426-430.HE Di,RONG Yuetang,ZHANG Xiaojing.Bifurcation for a prey-predator model with cross-diffusion[J].Basic Sciences Journal of Textile Universities,2015,28(4):426-430.[10] BAZYKIN A D.Nonlinear dynamics of interacting population[M].Singapore:World Scientific,1998.[11] 叶其孝,李正元,王明新.反应扩散方程引论[M].北京：科学出版社,2011:40-56.YE Qixiao,LI Zhengyuan,WANG Mingxin.Introduction of reaction-diffusionequations[M].Beijing:Science Press,2011:40-56.[12] 何堤,容跃堂,王晓丽,等.一类具有交叉扩散的捕食-食饵模型的局部分歧[J].西安工业大学学报,2015,35(11):872-876.HE Di,RONG Yuetang,WANG Xiaoli,et al.Local bifurcation for a prey-predator model with cross-diffusion[J].Journal of Xi′an Technological University,2015,35(11):872-876.[13] 容跃堂,何堤,张晓晶.带交叉扩散项的Holling Ⅳ捕食-食饵模型的全局分歧[J].纺织高校基础科学学报,2015,26(3):287-293.RONG Yuetang,HE Di,ZHANG Xiaojing.The global bifurcation for a prey-predator model with cross-diffusion and Holling Ⅳ [J].Basic Sciences Journal of TextileUniversities,2015,26(3):287-293.[14] 马晓丽,冯孝周.一类具有交叉扩散的捕食模型的正解的存在性[J].安徽大学学报:自然科学版,2011,35(5):26-31.MA Xiaoli,FENG Xiaozhou.The existence of positive solutions for a predator-prey model with cross-diffusion[J].Journal of Anhui University:Natural Science Edition,2011,35(5):26-31.[15] 马晓丽.一类具有交叉扩散的捕食模型的整体分歧[J].西安工业大学学报,2010,30(5):506-510. MA Xiaoli.Global bifurcation for a predator-prey model with cross-diffusion[J].Journal of Xi′an Technological University,2010,30(5):506-510.[16] 戴婉仪,付一平.一类交叉扩散系统定态解的分歧与稳定性[J].华南理工大学大学报:自然科学版,2005,33(2):99-102.DAI Wanyi,FU Yiping.Bifurcation and stability of the steady-state solutions to a system with cross-diffusion effect[J].Journal of South China University of Technology:Natural Science Edition,2005,33(2):99-102.[17] 柴俊平,李艳玲.带有交叉扩散项的捕食-食饵模型的全局分歧[J].纺织高校基础科学学报,2011,24(4):490-494.CHAI Junping,LI Yaning.Global bifurcation of a class of predator-prey models with cross-diffusion effect[J].Basic Sciences Journal of Textile Universities,2011,24(4):490-494. [18] WU J H.Global bifurcation of coexistence states for the competition model in the chemostat[J].Nonlinear Analysis,2000,39(7):817-835.[19] CRANDALL M G,RABINOWITZ P H.Bifurcation from simple eigenvalues[J].J Functional Analysis,1971,8(2):321-340.。
Bifurcation+Theory
Bifurcation Theory © John J. Tyson, 20091. Saddle-Node BifurcationSuppose we want to study the solutions of a pair of nonlinear ODEs as a particular parameter, p , is varied:d d (,;), (,;)d d (*,*;)0, (*,*;)0o o x y f x y p g x y p t tf x y pg x y p ====The steady state solution, at (x *, y *), depends on the chosen value of the parameter, p = p o . We want to know how the steady state changes as p is varied away from p o , so we introduce new variables that are deviations from the steady state:*, *, o x x y y p p ξηδ=−=−=−Then, the steady state equations can be approximated by:11121p 21222p 0 (neglecting 'higher-order' terms)0 (neglecting 'higher-order' terms)a a a a a a ξηδξηδ≈++≈++where a 11, a 12, a 21, a 22 are the elements of the Jacobian matrix, and a 1p , a 2p are partial derivatives of f (x,y;p ) and g (x,y;p ) with respect to p , evaluated at the steady state. These linear equations can be written in vector form1p 2p , where and a a ξδη⎛⎞⎛⎞===⎜⎟⎜⎟⎝⎠⎝⎠Jz b z b .This matrix equation has a non-trivial solution (Hint: Cramer’s Rule) as long as D =Det(J ) = a 11a 22–a 12a 21 ≠ 0. Hence, as long as D is non-zero, we can ‘continue’ the steady state by taking small steps, δ, in the parameter value. Whenever D becomes small, we simply rearrange the linear equations as follows,121p 11222p 21a a a a a a ηδξηδξ+=+=and now follow η and δ as we take small steps, ξ, in the x-direction. This trick allows us (i.e., AUTO) to ‘turn the corner’ at a saddle-node bifurcation point, where D = 0.As AUTO tracks a steady state solution for changing values of a parameter, p , it monitors D (p ), the determinant of the Jacobian matrix, and also Re{λ(p )} for all complex-conjugate pairs of eigenvalues. Whenever D (p c ) = 0 for some ‘critical’ value of p , AUTOrecords p c as a ‘limit point’ (LP), also known as a saddle-node (SN) bifurcation point. Whenever Re{λ(p c )} = 0, then AUTO records p c as a ‘Hopf bifurcation’ point (HB).A ‘one-parameter’ bifurcation diagram is a plot of the steady state value of a variable, say x *(p 1), as a function of the changing value of a parameter p . A typical bifurcation diagram might look like this:AUTO leaves little ‘labels’ (the circles in the diagram above) at bifurcation points, allowing the user to ‘grab’ a bifurcation point and follow it in a ‘two-parameter’bifurcation diagram. For example, we might grab one of the SN bifurcation points and follow it as two parameters, call them p and q , are varied. In this case, AUTO must solve 3 nonlinear algebraic equations in four unknowns (x *, y *, p and q ):(*,*;,)0, (*,*;,)0, (*,*;,)0f x y p q g x y p q D x y p q ===where D = Determinant of the Jacobian matrix at the steady state. We know that these three equations are satisfied simultaneously at a SN bifurcation point, say at p o and q o . To follow the SN as p and q are varied, AUTO linearizes the algebraic equations around the known solution11121p 1q 21222p 2q 12p q 0 0 0a a a a a a a a D D D D ξηδγξηδγξηδγ=+++=+++=+++where 1/, ..., /q D D x D D q =∂∂=∂∂. We can solve these equations, as before, for ξ, η and γ as functions of δ, and then plot o q q γ=+ as a function of o p p δ=+. A typical two-parameter bifurcation diagram for SN bifurcations looks like thisThe ‘cusp point’ is known as a ‘codimension-2’ bifurcation point because we must choose precise values of two different parameters, p and q , in order to reach the cusp.2. Hopf BifurcationIn 1942, E. Hopf proved a remarkable and powerful theorem about limit cycleoscillations in the vicinity of a critical point, p = p c , where Re{λ(p c )} = 0. (Let us assume that Re{λ(p )} < 0 for p < p c , and Re{λ(p )} > 0 for p > p c . If the opposite is true, then replace p by q = −p , and you have the assumed case.) In informal language, Hopf proved that for p in the vicinity of p c , there exists a one-parameter family of small-amplitudelimit cycle solutions for one of the following three cases: (i) p > p c . (ii) p < p c . (iii) p = p c . For cases (i) and (ii) the limit cycles are parameterized by their distance from thebifurcation point, i.e., | p − p c|. The amplitude and period of each limit cycle scales likec 2 A T b p p πω==+−where a and b are constants. (Case (iii) is rarely encountered, and we will not bother with it.)When the amplitude of the bifurcating limit cycles are plotted on a one-parameter bifurcation diagram, we get the following typical figures:Case (i) is called a super-critical Hopf bifurcation (creating stable limit cycles around an unstable steady state), and case (ii) is called a sub-critical Hopf bifurcation (creating unstable limit cycles around a stable steady state).Hopf bifurcation points can be ‘followed’ in a two-parameter bifurcation diagram, using exactly the same strategy outlined for following a SN bifurcation. In this case, the typical 1case (i), diagrammed above. For q = q 2, the one-parameter bifurcation diagram looks so:bifurcation point, called a ‘degenerate Hopf bifurcation’, where a locus of cyclic folds meets the locus of HB’s tangentially.If one follows a locus of HB’s on a two-parameter bifurcation diagram, then typically the line may (i) ‘run off to infinity’, or (ii) close in a loop, or (iii) end at a codimension-2‘Takens-Bogdanov’ bifurcation. A TB bifurcation point has a complex structure3. Homoclinic BifurcationsThe final sort of bifurcations that we must learn concern the ‘birth’ of limit cycles fromtrajectories that begin and end on saddle-type steady states.3a. Saddle-Loop (SL) BifurcationFirst, consider the sequence of changes to a vector field as a parameter p is changed:As p increases through p c, a limit cycle is born with large amplitude and infinite period.(At p = p c, it takes an infinite amount of time to go around the saddle loop.) Notice thequalitative difference between limit cycle creation by a Hopf bifurcation and by a saddle-loop bifurcation:Period AmplitudeFrequencyHopf Bifurcation infinitesimal finite finiteSaddle-Loop Bif’n finite infinitesimal InfiniteLike HB’s, SL bifurcations may give rise to either stable or unstable limit cycles.SL bifurcations are often found in conjunction with Hopf bifurcations, as in the exampleof a Takens-Bogdanov bifurcation above. If we were to compute a one-parameterbifurcation diagram for q = q1 (a little above the TB point), it would look like this:cnode’ point, and the unique trajectory that proceeds out of the saddle-node loops around to the other side and comes back into the saddle-node, forming an ‘invariant circle’. It takes an infinite amount of time to go around the invariant circle. For p > p c, the saddle-node point disappears and the invariant circle becomes a stable limit cycle, with finite amplitude and very long period, because there is a ‘slow’ part of the vector field in the region where the steady states used to be. As before, SNIC bifurcations can generate either stable or unstable limit cycles.SNIC bifurcations are created at a codimension-2 bifurcation point, when an SL locus merges tangentially into an SN locus. You will find such a bifurcation point in the two-parameter bifurcation diagram I used to illustrate the Takens-Bogdanov bifurcation.As I have said, SL and SNIC bifurcations can create either stable or unstable limit cycles. In the same way that supercritical HB changes to subcritical HB at a degenerate HB point, the stability of SL and SNIC limit cycles may change at a codimension-2 degenerate bifurcation point by throwing off a locus of cyclic fold (CF) bifurcations.4. SummaryAt a bifurcation point the solutions of a set on nonlinear ODEs undergo a dramatic, qualitative change in their characteristics: steady states appear or disappear, and limit cycles appear or disappear. Because the stable attractors of the ODEs determine, in large measure, the temporal behavior of the dynamical system, the bifurcations exhibited by a dynamical system determine, in large measure, the signal-response characteristics of the molecular control system. Because there are only a limited number of ‘generic’ bifurcations exhibited by nonlinear ODEs, there is only a small selection of bifurcations from which nature must build all the complex information processing capabilities of macromolecular reaction networks. We are now familiar with all the fundamental bifurcations of codimension-1 and -2. They are:Codimension-1Saddle-Node SN Coalescence of saddle and node to annihilate bothsteady statesHopf HB Steady state loses stability and gives rise to a small-amplitude limit cycle of finite periodCyclic Fold CF Coalescence of stable and unstable limit cycles toannihilate both oscillatory solutionsSaddle Loop SL Trajectory that loops from a saddle point back to itself,creating a large-amplitude limit cycle of infinite periodSaddle-Node Invariant-Circle SNIC Trajectory that loops from a saddle-node point back to itself, creating a large-amplitude limit cycle of infiniteperiodCodimension-2Cusp C Tangential coalescence of two SN loci; the dynamicalsystem is bistable inside the cuspDegenerateHopf DH Tangentialcoalescence of HB and CF loci, whichchanges the HB from supercritical to subcriticalTakens-Bogdanov TB Tangentialcoalescence of SN, HB and SL loci. Limitcycles exist in the small wedge between HB and SL. Neutral Saddle Loop NSL Tangential coalescence of SL and CF loci, whichchanges the stability of the bifurcating limit cycles Saddle-Node Loop SNL Tangential coalescence of SL and SN loci to create alocus of SNIC bifurcations。
神经外科5本杂志目录 英文-中文 2022年10月
神外杂志英-中文目录(2022年10月) Neurosurgery1.Assessment of Spinal Metastases Surgery Risk Stratification Tools in BreastCancer by Molecular Subtype按照分子亚型评估乳腺癌脊柱转移手术风险分层工具2.Microsurgery versus Microsurgery With Preoperative Embolization for BrainArteriovenous Malformation Treatment: A Systematic Review and Meta-analysis 显微手术与显微手术联合术前栓塞治疗脑动静脉畸形的系统评价和荟萃分析mentary: Silk Vista Baby for the Treatment of Complex Posterior InferiorCerebellar Artery Aneurysms点评: Silk Vista Baby用于治疗复杂的小脑下后动脉动脉瘤4.Targeted Public Health Training for Neurosurgeons: An Essential Task for thePrioritization of Neurosurgery in the Evolving Global Health Landscape针对神经外科医生的有针对性的公共卫生培训:在不断变化的全球卫生格局中确定神经外科手术优先顺序的一项重要任务5.Chronic Encapsulated Expanding Hematomas After Stereotactic Radiosurgery forIntracranial Arteriovenous Malformations: An International Multicenter Case Series立体定向放射外科治疗颅内动静脉畸形后的慢性包裹性扩张血肿:国际多中心病例系列6.Trends in Reimbursement and Approach Selection for Lumbar Arthrodesis腰椎融合术的费用报销和入路选择趋势7.Diffusion Basis Spectrum Imaging Provides Insights Into Cervical SpondyloticMyelopathy Pathology扩散基础频谱成像提供了脊髓型颈椎病病理学的见解8.Association Between Neighborhood-Level Socioeconomic Disadvantage andPatient-Reported Outcomes in Lumbar Spine Surgery邻域水平的社会经济劣势与腰椎手术患者报告结果之间的关系mentary: Prognostic Models for Traumatic Brain Injury Have GoodDiscrimination But Poor Overall Model Performance for Predicting Mortality and Unfavorable Outcomes评论:创伤性脑损伤的预后模型在预测death率和不良结局方面具有良好的区分性，但总体模型性能较差mentary: Serum Levels of Myo-inositol Predicts Clinical Outcome 1 YearAfter Aneurysmal Subarachnoid Hemorrhage评论:血清肌醇水平预测动脉瘤性蛛网膜下腔出血1年后的临床结局mentary: Laser Interstitial Thermal Therapy for First-Line Treatment ofSurgically Accessible Recurrent Glioblastoma: Outcomes Compared With a Surgical Cohort评论:激光间质热疗用于手术可及复发性胶质母细胞瘤的一线治疗:与手术队列的结果比较12.Functional Reorganization of the Mesial Frontal Premotor Cortex in Patients WithSupplementary Motor Area Seizures辅助性运动区癫痫患者中额内侧运动前皮质的功能重组13.Concurrent Administration of Immune Checkpoint Inhibitors and StereotacticRadiosurgery Is Well-Tolerated in Patients With Melanoma Brain Metastases: An International Multicenter Study of 203 Patients免疫检查点抑制剂联合立体定向放射外科治疗对黑色素瘤脑转移患者的耐受性良好:一项针对203例患者的国际多中心研究14.Prognosis of Rotational Angiography-Based Stereotactic Radiosurgery for DuralArteriovenous Fistulas: A Retrospective Analysis基于旋转血管造影术的立体定向放射外科治疗硬脑膜动静脉瘘的预后:回顾性分析15.Letter: Development and Internal Validation of the ARISE Prediction Models forRebleeding After Aneurysmal Subarachnoid Hemorrhage信件:动脉瘤性蛛网膜下腔出血后再出血的ARISE预测模型的开发和内部验证16.Development of Risk Stratification Predictive Models for Cervical DeformitySurgery颈椎畸形手术风险分层预测模型的建立17.First-Pass Effect Predicts Clinical Outcome and Infarct Growth AfterThrombectomy for Distal Medium Vessel Occlusions首过效应预测远端中血管闭塞血栓切除术后的临床结局和梗死生长mentary: Risk for Hemorrhage the First 2 Years After Gamma Knife Surgeryfor Arteriovenous Malformations: An Update评论:动静脉畸形伽玛刀手术后前2年出血风险:更新19.A Systematic Review of Neuropsychological Outcomes After Treatment ofIntracranial Aneurysms颅内动脉瘤治疗后神经心理结局的系统评价20.Does a Screening Trial for Spinal Cord Stimulation in Patients With Chronic Painof Neuropathic Origin Have Clinical Utility (TRIAL-STIM)? 36-Month Results From a Randomized Controlled Trial神经性慢性疼痛患者脊髓刺激筛选试验是否具有临床实用性(TRIAL-STIM)？一项随机对照试验的36个月结果21.Letter: Transcriptomic Profiling Revealed Lnc-GOLGA6A-1 as a NovelPrognostic Biomarker of Meningioma Recurrence信件:转录组分析显示Lnc-GOLGA6A-1是脑膜瘤复发的一种新的预后生物标志物mentary: The Impact of Frailty on Traumatic Brain Injury Outcomes: AnAnalysis of 691 821 Nationwide Cases评论:虚弱对创伤性脑损伤结局的影响:全国691821例病例分析23.Optimal Cost-Effective Screening Strategy for Unruptured Intracranial Aneurysmsin Female Smokers女性吸烟者中未破裂颅内动脉瘤的最佳成本效益筛查策略24.Letter: Pressure to Publish—A Precarious Precedent Among Medical Students信件:出版压力——医学研究者中一个不稳定的先例25.Letter: Protocol for a Multicenter, Prospective, Observational Pilot Study on theImplementation of Resource-Stratified Algorithms for the Treatment of SevereTraumatic Brain Injury Across Four Treatment Phases: Prehospital, Emergency Department, Neurosurgery, and Intensive Care Unit信件:一项跨四个治疗阶段(院前、急诊科、神经外科和重症监护室)实施资源分层算法的多中心、前瞻性、观察性试点研究的协议26.Risk for Hemorrhage the First 2 Years After Gamma Knife Surgery forArteriovenous Malformations: An Update动静脉畸形伽玛刀手术后前2年出血风险:更新Journal of Neurosurgery27.Association of homotopic areas in the right hemisphere with language deficits inthe short term after tumor resection肿瘤切除术后短期内右半球同话题区与语言缺陷的关系28.Association of preoperative glucose concentration with mortality in patientsundergoing craniotomy for brain tumor脑肿瘤开颅手术患者术前血糖浓度与death率的关系29.Deep brain stimulation for movement disorders after stroke: a systematic review ofthe literature脑深部电刺激治疗脑卒中后运动障碍的系统评价30.Effectiveness of immune checkpoint inhibitors in combination with stereotacticradiosurgery for patients with brain metastases from renal cell carcinoma: inverse probability of treatment weighting using propensity scores免疫检查点抑制剂联合立体定向放射外科治疗肾细胞癌脑转移患者的有效性:使用倾向评分进行治疗加权的反向概率31.Endovascular treatment of brain arteriovenous malformations: clinical outcomesof patients included in the registry of a pragmatic randomized trial脑动静脉畸形的血管内治疗:纳入实用随机试验登记处的患者的临床结果32.Feasibility of bevacizumab-IRDye800CW as a tracer for fluorescence-guidedmeningioma surgery贝伐单抗- IRDye800CW作为荧光导向脑膜瘤手术示踪剂的可行性33.Precuneal gliomas promote behaviorally relevant remodeling of the functionalconnectome前神经胶质瘤促进功能性连接体的行为相关重塑34.Pursuing perfect 2D and 3D photography in neuroanatomy: a new paradigm forstaying up to date with digital technology在神经解剖学中追求完美的2D和三维摄影:跟上数字技术的新范式35.Recurrent insular low-grade gliomas: factors guiding the decision to reoperate复发性岛叶低级别胶质瘤:决定再次手术的指导因素36.Relationship between phenotypic features in Loeys-Dietz syndrome and thepresence of intracranial aneurysmsLoeys-Dietz综合征的表型特征与颅内动脉瘤存在的关系37.Continued underrepresentation of historically excluded groups in the neurosurgerypipeline: an analysis of racial and ethnic trends across stages of medical training from 2012 to 2020神经外科管道中历史上被排除群体的代表性持续不足:2012年至2020年不同医学培训阶段的种族和族裔趋势分析38.Management strategies in clival and craniovertebral junction chordomas: a 29-yearexperience斜坡和颅椎交界脊索瘤的治疗策略:29年经验39.A national stratification of the global macroeconomic burden of central nervoussystem cancer中枢神经系统癌症全球宏观经济负担的国家分层40.Phase II trial of icotinib in adult patients with neurofibromatosis type 2 andprogressive vestibular schwannoma在患有2型神经纤维瘤病和进行性前庭神经鞘瘤的成人患者中进行的盐酸埃克替尼II期试验41.Predicting leptomeningeal disease spread after resection of brain metastases usingmachine learning用机器学习预测脑转移瘤切除术后软脑膜疾病的扩散42.Short- and long-term outcomes of moyamoya patients post-revascularization烟雾病患者血运重建后的短期和长期结局43.Alteration of default mode network: association with executive dysfunction infrontal glioma patients默认模式网络的改变:与额叶胶质瘤患者执行功能障碍的相关性44.Correlation between tumor volume and serum prolactin and its effect on surgicaloutcomes in a cohort of 219 prolactinoma patients219例泌乳素瘤患者的肿瘤体积与血清催乳素的相关性及其对手术结果的影响45.Is intracranial electroencephalography mandatory for MRI-negative neocorticalepilepsy surgery?对于MRI阴性的新皮质癫痫手术，是否必须进行颅内脑电图检查？46.Neurosurgeons as complete stroke doctors: the time is now神经外科医生作为完全中风的医生:时间是现在47.Seizure outcome after resection of insular glioma: a systematic review, meta-analysis, and institutional experience岛叶胶质瘤切除术后癫痫发作结局:一项系统综述、荟萃分析和机构经验48.Surgery for glioblastomas in the elderly: an Association des Neuro-oncologuesd’Expression Française (ANOCEF) trial老年人成胶质细胞瘤的手术治疗:法国神经肿瘤学与表达协会(ANOCEF)试验49.Surgical instruments and catheter damage during ventriculoperitoneal shuntassembly脑室腹腔分流术装配过程中的手术器械和导管损坏50.Cost-effectiveness analysis on small (< 5 mm) unruptured intracranial aneurysmfollow-up strategies较小(< 5 mm)未破裂颅内动脉瘤随访策略的成本-效果分析51.Evaluating syntactic comprehension during awake intraoperative corticalstimulation mapping清醒术中皮质刺激标测时句法理解能力的评估52.Factors associated with radiation toxicity and long-term tumor control more than10 years after Gamma Knife surgery for non–skull base, nonperioptic benignsupratentorial meningiomas非颅底、非周期性良性幕上脑膜瘤伽玛刀术后10年以上与放射毒性和长期肿瘤控制相关的因素53.Multidisciplinary management of patients with non–small cell lung cancer withleptomeningeal metastasis in the tyrosine kinase inhibitor era酪氨酸激酶抑制剂时代有软脑膜转移的非小细胞肺癌患者的多学科管理54.Predicting the growth of middle cerebral artery bifurcation aneurysms usingdifferences in the bifurcation angle and inflow coefficient利用分叉角和流入系数的差异预测大脑中动脉分叉动脉瘤的生长55.Predictors of surgical site infection in glioblastoma patients undergoing craniotomyfor tumor resection胶质母细胞瘤患者行开颅手术切除肿瘤时手术部位感染的预测因素56.Stereotactic radiosurgery for orbital cavernous hemangiomas立体定向放射外科治疗眼眶海绵状血管瘤57.Surgical management of large cerebellopontine angle meningiomas: long-termresults of a less aggressive resection strategy大型桥小脑角脑膜瘤的手术治疗:较小侵袭性切除策略的长期结果Journal of Neurosurgery: Case Lessons58.5-ALA fluorescence–guided resection of a recurrent anaplastic pleomorphicxanthoastrocytoma: illustrative case5-ALA荧光引导下切除复发性间变性多形性黄色星形细胞瘤:说明性病例59.Flossing technique for endovascular repair of a penetrating cerebrovascular injury:illustrative case牙线技术用于血管内修复穿透性脑血管损伤:例证性病例60.Nerve transfers in a patient with asymmetrical neurological deficit followingtraumatic cervical spinal cord injury: simultaneous bilateral restoration of pinch grip and elbow extension. Illustrative case创伤性颈脊髓损伤后不对称神经功能缺损患者的神经转移:同时双侧恢复捏手和肘关节伸展。
_一类非线性奇性薛定谔方程的精确解
2.1 动力系统.................................................................................................................. 4
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