Degenerate Hopf bifurcations control the extended Sprott E system with only one stable equilibrium.
具時滞的单种群模型和SIS模型的稳定性和分支分析
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
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
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
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Zhouchao WEI1, ∗, Irene MOROZ2 , Anping LIU1 School of Mathematics and Physics, China University of Geosciences, Wuhan, P.R. China 2 Mathematical Institute, Oxford University, Oxford, UK
1. Introduction Since chaotic attractors were found by Lorenz in 1963 [10], many chaotic systems have been constructed, such as the R¨ ossler [16], the Chen [4], and the L¨ u [11] systems. Because of potential applications in engineering, the study of chaotic systems has attracted the interest of more and more researchers. By exhaustive computer searching, Sprott [21–23] found about 20 simple chaotic systems with no more than 3 equilibria. These systems have either 5 terms and 2 nonlinearities or 6 terms and 1 nonlinearity. Later, many 3-dimensional (3-D) Lorenz-like or Lorenz-based chaotic systems were proposed and investigated [1,3,5,9,12,13,14,24,25,27,29,32]. Methods for generating multiscroll attractors have commonly used analytical criteria for generating and proving chaos in autonomous systems, based on the fundamental work of Sil’nikov [17,18] and its subsequent embellishment and extension [19]. Chaos in the Sil’nikov type of 3-D autonomous quadratic dynamical systems may be classified into 4 subclasses [34]: (1) chaos of homoclinic-orbit type; (2) chaos of heteroclinic-orbit type; (3) chaos of the hybrid type with both homoclinic and heteroclinic orbits; (4) chaos of other types. Therefore, Sil’nikov’s criteria are sufficient but certainly not necessary for the emergence of chaos. Creating a chaotic system with a more complicated topological structure such as chaotic attractors with only stable equilibria, therefore, becomes a desirable task and sometimes a key issue for many engineering applications.
• Accepted: 01.02.2014 • Published Online: 25.04.2014 • Printed: 23.05.2014
1
Received: 28.05.2013
Abstract: In this paper, we introduce an extended Sprott E system by a general quadratic control scheme with 3 arbitrary parameters for the new system. The resulting system can exhibit codimension-one Hopf bifurcations as parameters vary. The control strategy used can be applied to create degenerate Hopf bifurcations at desired locations with preferred stability. A complex chaotic attractor with only one stable equilibrium is derived in the sense of having a positive largest Lyapunov exponent. The chaotic attractor with only one stable equilibrium can be generated via a period-doubling bifurcation. To further suppress chaos in the extended Sprott E system coexisting with only one stable equilibrium, adaptive control laws are designed to stabilize the extended Sprott E system based on adaptive control theory and Lyapunov stability theory. Numerical simulations are shown to validate and demonstrate the effectiveness of the proposed adaptive control. Key words: Chaotic attractor, stable equilibrium, Sil’nikov’s theorem, degenerate Hopf bifurcations, hidden attractor
Turkish Journal of Mathematics
.tr/math/
Turk J Math (2014) 38: 672 – 687 ¨ ITAK ˙ c TUB ⃝ doi:10.3906/mat-1305-64
Research Article
∗Correspondence:
weizhouchao@ 2010 AMS Mathematics Subject Classification: 34C23, 34C28.
672
WEI et al./Turk J Math
To further the investigation of chaos theory and its applications, it is very important to generate new chaotic systems or to enhance the complex dynamics and topological structure based on the existing chaotic attractors. In this endeavor, Yang et al. [33] studied an unusual 3-D autonomous quadratic Lorenz-like chaotic system with only 2 stable node-foci. Moreover, a new 3-D chaotic system with 6 terms including only 1 nonlinear term in the form of an exponential function was proposed and studied in [30]. This system has double-scroll chaotic attractors in a very wide region of parameter space with only 2 stable equilibria. Wei and Yang [31] analyzed the generalized Sprott C system with only 2 stable equilibria. They computed some basic dynamical properties: Lyapunov exponent spectra, fractal dimensions, bifurcations, and routes to chaos. Wang and Chen [25] obtained chaotic attractors with only one stable node-focus by adding a simple constant control parameter to Sprott’s E system. Recently, a chaotic system with no equilibria was proposed by Wei [28], which showed a peri