Application of Chaos Theory and Fractal Analysis for EEGsignal Processing in Patients with
混沌建筑
l裎。弦热夺眦。。。。。。
混沌建筑
Choos of Architecture
李世芬 孔宇航
Lj Sh汗en Konq Yuhang
中吲分类号Tu一80
夏射标识码A
文章蝙号1 o。3 739×(2002)05—0064—03
摘要混沌理佗揭示了物质世 界更为真实的形态和结构,使人 类思维从机械的宇宙论转向有机 主义,茬沌学的引入为建鬣复杂 空司的探索注入,活力,非线忤 思维与分形理论拓展了人们的观 念和手法.自然q9类比是建筑创 1乍的丰富琼果。 美毽词 混沌学自然结稠非 线性分形自然的娄l:
1 978年.埃森蛙在住宅,n(图4)-f-借 用分形的比例缩放的慨念.以立体“r彤的 复杂旋转构成了建筑的怍量。此外.埃森曼“罗 蜜欧与朱丽ur中也运ni r多磋缓的分形构 成往cannareg Lo城的广场J_,埃森曼叉布置 ,一系列不㈦R度的住宅,b的相似形,最小 的一个与』、同高;最大的一个远远大于房了; 而房子人小的物体里却充满了无限的自相似 彤使其不能作为房了使用不㈣尺度的物体 相互嵌套.成为一系列自我相似、自我参照的 建筑‘生成兀”。埃森曼的分彤构成肘建筑与 止常R发(人的身体)的关系,对人的透视慨 念提出了质疑,同时他试图证明:昕有尺度部 哇续.所有尺度闸的间隔代表了物体的自然形 态,而且通过它们的稃JJⅡ产牛新的物体 结 果.没有仟何东四是稳定的,没有江何来西是 可以预知的;它是个复杂的人工物.通过其 形成过程的痕迹被标,下出来.这正是埃森星在 他混沌的斛构建筑中所刻意追求的
Chaos and its Application2010 1&2
工科博士生讲座
Complex Network Example: Telecomm Networks (Stephen G. Eick)
工科博士生讲座
Metabolic Network 新陈代谢网络
Nodes: chemicals (substrates)
Links: bio-chemical reactions
x(0) 0 dx sin t dt x(1) 0
的解
x 1
1
(cos t cos )
它写不成如下形式
x 1
(cos (t 1) 1)
工科博士生讲座
稳定性问题:设原系统
x f (t , x)
x(t0 ) x0
如果初始受一微小的扰动 x(t0 ) x1,而 x1 x0 很小, 问:解是否发生很大的变化? 0 定义: 0 (对于一般非自治系统, 除了与 有关,还与 t 0 有关), 当 x1 x0 ,对于 t t 0 有 (t, x1 ) (t, x0 ) 则称原系统 x (t , x0 )在 x0 是稳定的,否则是 不稳定的
差之毫厘,谬之千里 《中国成语》
工科博士生讲座
钉子缺,蹄铁卸; 蹄铁卸,战马蹶; 战马蹶,骑士绝; 骑士绝,战事折; 战事折,国家灭。 ---美国学者、控制论的创立者维纳编著的 民谣。说的是:古代某国,飞马传书, 因马掌缺了一颗钉子,造成马掌脱落, 战马仆倒,战士摔死、信未送到,战争 失败,最终导致了国家灭亡。
工科博士生讲座
例
dx a 2 x dt x(0) x0
的解为
x(t ) x0e
x(0) x1
THEORY AND APPLICATIONS OF FRACTIONAL DIFFERENTIAL EQUATIONS
1 6 10 18 24 27 32 40 45 49 54 58 67
2 FRACTIONAL INTEGRALS A N D FRACTIONAL DERIVATIVES 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 Riemann-Liouville Fractional Integrals and Fractional Derivatives Liouville Fractional Integrals and Fractional Derivatives on the HalfAxis Liouville Fractional Integrals and Fractional Derivatives on the Real Axis Caputo Fractional Derivatives Fractional Integrals and Fractional Derivatives of a Function with Respect to Another Function Erdelyi-Kober Type Fractional Integrals and Fractional Derivatives Hadamard Type Fractional Integrals and Fractional Derivatives . . Griinwald-Letnikov Fractional Derivatives Partial and Mixed Fractional Integrals and Fractional Derivatives Riesz Fractional Integro-Differentiation Comments and Observations
chaotic
780
Z. Odibat / Nonlinear Analysis: Real World Applications 13 (2012) 779–789
generalizations of many well-known systems, can also behave chaotically, such as the fractional Duffing system [13], the fractional Chua system [12,14], the fractional Rössler system [15], the fractional Chen system [16,17], the fractional Lorenz system [18], fractional Arneodo’s system [19] and the fractional Lü system [20]. In [14–16] it has been shown that some fractional order systems can produce chaotic attractors with order less than 3. Moreover, recent studies show that chaotic fractional order systems can also be synchronized [21–32]. In many literatures, synchronization among fractional order systems is only investigated through numerical simulations that are based on the stability criteria of linear fractional order systems, such as the work presented in [24–26], or based on Laplace transform theory, such as the work presented in [27–29]. Very recently some researchers have shown the existence of phase synchronization in chaotic fractional differential equations [33,34]. The main aim of this paper is to study the phase synchronization of coupled chaotic Caputo based fractional order systems. We have employed the stability results of linear fractional order systems in our analysis to achieve phase and complete synchronization. An active control scheme, which consists of a nonlinear dynamic feedback controller, has been proposed. The effectiveness of the proposed scheme is demonstrated via its application to the phase synchronization of the fractional models of Lorenz, Lü and Rössler systems. 1.1. Basic concepts There are several definitions of a fractional derivative of order α > 0 [35–39]. The two most commonly used are the Riemann–Liouville and Caputo definitions. Each definition uses Riemann–Liouville fractional integration and derivatives of whole order. The difference between the two definitions is in the order of evaluation. The Riemann–Liouville fractional integral operator of order α ≥ 0 of the function f (t ) is defined as, J f (t ) =
Synchronizing chaotic systems using backstepping design
Synchronizing chaotic systems using backstepping designXiaohui Tana,*,Jiye Zhang b ,Yiren YangaaDepartment of Applied Mechanics and Engineering,Southwest Jiaotong University,Chengdu 610031,PR ChinabNational Power Traction Laboratory,Southwest Jiaotong University,Chengdu 610031,PR ChinaAccepted 20May 2002AbstractBackstepping design is a recursive procedure that combines the choice of a Lyapunov function with the design of a controller.In this paper it is proposed for synchronizing chaotic systems.There are several advantages in this method for synchronizing chaotic systems:(a)it presents a systematic procedure for selecting a proper controller in chaos synchronization;(b)it can be applied to a variety of chaotic systems whether they contain external excitation or not;(c)it needs only one controller to realize synchronization between chaotic systems;(d)there is no derivatives in controller,so it is easy to be complemented.Examples of Lorenz system,Chua Õs circuit and Duffing system are presented.Ó2002Elsevier Science Ltd.All rights reserved.1.IntroductionSynchronizing chaotic systems and circuits has received great interest in recent years since the seminal paper by Ott–Grebogi–Yorke [1].Generally the two chaotic systems in synchronization are called drive system and response system respectively.The idea of synchronization is to use the output of the drive system to control the response system,so that the output of the response system follows the output of the drive system asymptotically.Some attempts to solve the problem have been made recently [2–7].However,although they derive some methods for solving the problem,some of them can be only applied to chaotic systems with two dimensions like Duffing system [7]and some of them need several controllers to realize synchronization [4].Some focus on the condition with unknown parameters and disturbances [4,6].This paper focuses on increasing the effectiveness of the method to a wider variety of chaotic systems by using only one controller.So it has not studied the condition with unknown parameters and disturbances.The method here to use is backstepping design [8].It consists in a recursive procedure that links the choice of a Lyapunov function with the design of a controller.It has been successfully used to stabilize and track chaotic systems by Saverio Mascolo and Giuseppe Grassi [9].The paper is organized as follows:In Section 2the class of chaotic systems considered in this work and the problem formulation are presented.In Section 3the tool is utilized for several systems such as Lorenz system,Chua Õs circuit and Duffing system.Numerical simulations are carried out to confirm the validity of the proposed theoretical approach.In Section 4conclusion is presented.2.Problem formulationIn general,typical dynamics of chaotic systems such as Lorenz system,R €ossler system,Chua Õs circuit and Duffing system all belong to the system as following:*Corresponding author.E-mail address:xhtan@ (X.Tan).0960-0779/03/$-see front matter Ó2002Elsevier Science Ltd.All rights reserved.PII:S 0960-0779(02)00153-4/locate/chaos_x1¼f1ðx1;x2Þ_x2¼f2ðx1;x2;x3Þ..._x n¼f nðx1;x2;...;x nÞþf nþ1ðtÞð1Þwhere f1is a linear function and f iði¼2;3;...;nþ1Þare nonlinear functions and when it comes to Lorenz system, R€o ssler system and ChuaÕs circuit,f nþlðtÞ¼0.Assume that drive system is expressed as Eq.(1).Then response system which is coupled with system(1)by u is as following:_y1¼f1ðy1;y2Þ_y2¼f2ðy1;y2;y3Þ..._y n¼f nðy1;y2;...;y nÞþf0nþ1ðtÞþuð2Þwhere f0nþ1ðtÞhas similar characteristics as f nþ1ðtÞ.By properly choosing u,synchronization between drive system and response system can be achieved.Let us define the state errors between the response system and the drive system ase1¼y1Àx1e2¼y2Àx2ÁÁÁe n¼y nÀx nð3Þnamely,y1¼e1þx1y2¼e2þx2ÁÁÁy n¼e nþx nð4ÞSubtract(1)from(2).Notice Eqs.(3)and(4),finally error system can be derived as_e1¼g1ðe1;e2Þ_e2¼g2ðx1;x2;x3;e1;e2;e3Þ..._e n¼g nðx1;x2;...;x n;e1;e2;...;e nÞþf0nþ1ðtÞÀf nþ1ðtÞþuð5Þwhere g1is a linear function and g iði¼2;3;...;nÞare nonlinear functions with inputðx1;x2;...;x nÞfrom system(1). Apparently g iði¼2;3;...;nÞdepends not only on state variables but also on time t.The problem to realize the synchronization between two chaotic systems now transforms to another problem on how to choose a control law u to make e iði¼1;2;...;nÞgenerally converge to zero with time increasing.Here backstepping design is used to achieve the objective.3.Adaptive synchronization via a backstepping designIn this section,Lorenz system,ChuaÕs circuit and Duffing system are presented for synchronizing by backstepping design.3.1.Lorenz systemIn Lorenz system,external excitation does not exit and drive Lorenz system and response Lorenz system can be described respectively as(6)and(7)_x1¼rðy1Àx1Þ_y1¼q x1Ày1Àx1z1 _z1¼Àb z1þx1y1ð6Þ38X.Tan et al./Chaos,Solitons and Fractals16(2003)37–45_x2¼rðy2Àx2Þ_y2¼q x2Ày2Àx2z2_z2¼Àb z2þx2y2þuð7Þwhere r;q;b>0.Lete x¼x2Àx1e y¼y2Ày1e z¼z2Àz1ð8Þnamely,x2¼e xþx1y2¼e yþy1z2¼e zþz1ð9ÞSubtract Eq.(6)from Eq.(7)and consider Eqs.(8)and(9)_e x¼rðe yÀe xÞ_e y¼q e xÀe yÀe x e zÀe x z1Àx1e z_e z¼Àb e zþe x e yþe x y1þx1e yþuð10Þvariables x1,y1,z1in error system(10)can be considered as input signals from system(6).If system(10)did not have u, it would have an equilibrium(0,0,0).If we choose a u which would not change the equilibrium(0,0,0),problem of synchronization between drive and response system can be transformed into a problem on how to realize the as-ymptotical stabilization of system(10).Now the objective is tofind a control law u for stabilizing the error variables of system(10)at the origin.First we consider the stability of system(11)_e x¼rðe yÀe xÞð11Þwhere e y is regarded as a controller.Choose Lyapunov function V1as following:V1ðe xÞ¼12e2xð12ÞThe derivative of V1is as following:_V 1¼Àr e2xþr e x e yð13ÞAssume controller e y¼a1ðe xÞ,Eq.(13)can be rewritten as_V 1¼Àr e2xþr e x a1ðe xÞð14Þif a1ðe xÞ¼0(controller must be as simple as possible),then_V 1¼Àr e2x<0ð15Þmakes system(11)asymptotically stable.Function a1ðe xÞis an estimative function when e y is considered as a controller. The error between e y and a1ðe xÞisw2¼e yÀa1ðe xÞð16ÞStudyðe x;w2Þsystem(17)_e x¼rðw2Àe xÞ_w2¼q e xÀw2Àe x e zÀe x z1Àx1e zð17ÞConsider e z as a controller in system(17).Assume when it is equal to a2ðe x;w2Þ,it makes system(17)asymptoticallystable.Choose Lyapunov function V2ðe x;w2Þ¼V1ðe xÞþð1=2Þw22.The derivative of V2is_V 2¼Àr e2xÀw22þw2ðr e xþq e xÀz1e xÀe x e zÀx1e zÞð18ÞIf a2ðe x;w2Þ¼ðrþqÀz1Þe x=ðe xþx1Þ(according to paper[10]we have already known that x1and z1have boundaries respectively),_V2is_V 2¼Àr e2xÀw22<0ð19ÞX.Tan et al./Chaos,Solitons and Fractals16(2003)37–4539negative definite.Define the error variable w3asw3¼e zÀa2ðe x;w2Þð20ÞStudy full dimensionðe x;w2;w3Þsystem_e x¼rðw2Àe xÞ_w2¼q e xÀw2Àe xðw3þa2ÞÀe x z1Àx1ðw3þa2Þ_w3¼Àbðw3þa2Þþe x w2þe x y1þx1w2þuÀd a2d tð21Þwhered a2 d t ¼rðrþqÀz1Þðw2x1Àe x y1Þþðb z1Àx1y1Þðe2xþe x x1Þðe xþx1Þ2Choose Lyapunov function V3ðe x;w2;w3Þ¼V2ðe x;w2Þþð1=2Þw23.The derivative of V3is_V 3¼Àr e2xÀw22þw2½r e xþq e xÀz1e xÀðx1þe xÞa2 þw3Àbðw3þa2Þþe x y1þuÀd a2d tð22ÞConsidering r e xþq e xÀz1e xÀðx1þe xÞa2¼0,Eq.(22)can be rewritten as_V 3¼Àr e2xÀw22þw3Àbðw3þa2Þþe x y1þuÀd a2d tð23ÞLet u¼ba2Àe x y1þðd a2=d tÞ,and_V3can be described as_V 3¼Àr e2xÀw22Àb w23<0negative definite.Substituteðba2Àe x y1þd a2=d tÞfor u in Eq.(10),equilibrium is still(0,0,0)and has not been changed.So following above procedure we can conclude that equilibrium(0,0,0)of system(10)is asymptotically stable.As for an arbitrary initial error between systems(6)and(7),after afinite period of time,the initial error will converge to zero and syn-chronization between two Lorenz systems will be achieved.By taking r¼10,b¼8=3,q¼28and giving initial condition(x1ð0Þ¼20,y1ð0Þ¼5,z1ð0Þ¼2,x2ð0Þ¼24, y2ð0Þ¼20,z2ð0Þ¼28)and the numerical resolves are reported in Fig.1.Fig.1shows the time waveforms of x,y and z without u and Fig.2shows the waveforms of x,y and z with u.3.2.Chua’s circuitIn order to further test the effectiveness of the method ChuaÕs circuit,which was thefirst physical dynamical system capable of generating chaotic phenomena in the laboratory,is proposed for synchronizing.The circuit considered here contains a cubic nonlinearity and the drive system(24)and response system(25)are described by the following set of differential equations:_x1¼aðy1Àx31Àcx1Þ_y1¼x1Ày1þz1_z1¼Àb y1ð24Þ_x2¼aðy2Àx32Àcx2Þþu_y2¼x2Ày2þz2_z2¼Àb y2ð25Þwhere a,c and b are the circuit parameters.Subtract(24)from(25)and rearrange the order of the equations,the error system can be written as _e z¼Àb e y_e y¼e xÀe yþe z_e x¼a e yÀa e xðe2x þ3x1e xþ3x21ÞÀa ce xþuð26Þwhere e x¼x2Àx1;e y¼y2Ày1;e z¼z2Àz1.The objective is tofind a control law u so that system(26)is stabilized at the origin.Starting from thefirst equation of system(26),an estimative stabilizing function a1ðe zÞhas to be designedfor the virtual control e y in order to make the derivative of V1ðe zÞ¼ð1=2Þe2z ,namely_V1¼Àb e z a1ðe zÞ,negative definitewhen a1ðe zÞ¼e z.Define the error variable w2asw2¼e yÀa1ðe zÞð27ÞStudyðe z;w2Þsystem(28)_e z¼Àbðw2þe zÞ_w2¼e xÀw2þbðw2þe zÞð28ÞConsider e x as a controller in system(28).Assume when it is equal to a2ðe z;w2Þ,it makes system(28)asymptoticallystable.Choose Lyapunov function V2ðe z;w2Þ¼V1ðe zÞþð1=2Þw22.The derivative of V2is_V 2¼Àb e2zÀw22þw2ðe xþb w2Þð29ÞIf a2ðe z;w2Þ¼Àb w2,_V2is_V 2¼Àb e2zÀw22<0ð30Þnegative definite.Define the error variable w3asw3¼e xÀa2ðe z;w2Þð31ÞStudy full dimensionðe z;w2;w3Þsystem_e z¼Àbðw2þe zÞ_w2¼w3Àw2þb e z_w3¼aðw2þe zÞþbðw3Àw2þb e zÞÀa e xðe2x þ3e x x1þ3x21ÞÀa cðw3Àb w2Þþuð32ÞChoose Lyapunov function V3ðe z;w2;w3Þ¼V2ðe z;w2Þþð1=2Þw23.The derivative of V3is_V 3¼Àb e2zÀw22þw3b w2þaðw2þe zÞþbðw3Àw2þb e zÞÀa e xðe2xþ3e x x1þ3x21ÞÀa cðw3Àb w2Þþu c:ð33ÞLet u¼Àw2Àw3Àaðw2þe zÞÀbðw3Àw2þb e zÞþa e xðe2x þ3x1e xþ3x21Þþa cðw3Àb w2Þ,and_V3can be describedas_V 3¼Àb e2zÀw22Àw23<0negative definite.Following above procedure we have chosen a control law u.As for an initial error between systems(24)and(25), after afinite period of time,the initial error will converge to zero and synchronization between two ChuaÕs circuits will be achieved(Fig3).By taking a¼10,b¼16,c¼À0:143and giving initial conditionðx1ð0Þ¼1,y1ð0Þ¼2,z1ð0Þ¼1,x2ð0Þ¼10, y2ð0Þ¼5,z2ð0Þ¼5),the numerical resolves such as time waveforms of x,y and z are reported in Fig.1.The control law u is switched on at t¼5.42X.Tan et al./Chaos,Solitons and Fractals16(2003)37–453.3.Duffing systemLorenz system and ChuaÕs circuit discussed above can generate chaotic phenomena under no external excitation condition while Duffing system can generate chaotic phenomena only under external excitation.So here we classify Duffing system to another chaotic system and make Duffing system an example to illustrate how to use this method to synchronize chaotic systems with external excitation.The following set of differential equations formulates two Duffing systems.Thefirst is drive system and the second response system_x1¼y1_y1¼ax1þby1Àx31þc cosð0:4tÞð34Þ_x2¼y2_y2¼ax2þby2Àx32þc cos tþuð35Þwhere a,b and c are known parameters.Subtract(34)from(35),the error system can be written as _e x¼e y_e y¼ae xþbe yÀe xðe2x þ3x1e xþ3x21Þþc½cos tÀcosð0:4tÞ þuð36Þwhere e x¼x2Àx1;e y¼y2Ày1;e z¼z2Àz1.The objective is tofind a control law u so that system(36)is stabilized at the origin.Starting from thefirst equation of system(36),an estimative stabilizing function a1ðe xÞhas to be designed forthe virtual control e y in order to make the derivative of V1ðe xÞ¼ð1=2Þe2x ,namely_V1¼e x a1ðe xÞ,negative definite whena1ðe zÞ¼Àe x.Define the error variable w2asw2¼e yÀa1ðe xÞð37ÞStudyðe x;w2Þsystem(38):_e x¼w2Àe x_w2¼ðbþ1Þw2þðaÀbÀ1Þe xÀe xðe2x þ3x1e xþ3x21Þþc½cos tÀcosð0:4tÞ þuð38ÞChoose Lyapunov function V2ðe x;w2Þ¼V1ðe xÞþð1=2Þw22.The derivative of V2is_V 2¼Àe2xþw2bðbþ1Þw2þðaÀbÞe xÀe xðe2xþ3e x x1þ3x21Þþc½cos tÀcosð0:4tÞ þu cð39ÞLet u¼Àðbþ2Þw2ÀðaÀbÞe xþe xðe2x þ3e x x1þ3x21ÞÀc½cos tÀcosð0:4tÞ ,and_V2can be described as_V 2¼Àe2xÀw22<0negative definite.Following above procedure we have chosen a control law u.As for an initial error between systems(34)and(35), after afinite period of time,the initial error will converge to zero and synchronization between two Duffing systems will be achieved.By taking a¼1:8,b¼À0:1,c¼À1:1and giving initial conditionðx1ð0Þ¼1,y1ð0Þ¼1,x2ð0Þ¼2,y2ð0Þ¼2),the numerical resolves such as time waveforms of x and y are reported in Fig.4.The control law u is switched on at t¼5.4.ConclusionIn this paper,backstepping design has been used to synchronize chaotic systems.The advantages of this method can be summarized as follows:(a)it is a systematic procedure for synchronizing chaotic systems;(b)it can be applied to a variety of chaotic systems no matter whether it contains external excitation or not;(c)it needs only one controller to realize synchronization no matter how much dimensions the chaotic system contains;(d)there is no derivatives in controller,so it is easy to be complemented.The technique has been successfully applied to the Lorenz system,ChuaÕs circuit and Duffing system.Numerical simulations have verified the effectiveness of the method. AcknowledgementThe work is supported by the Natural Science Foundation of Southwest Jiaotong University(No.2001B09). References[1]Ott E,Grebogi C,Yorke JA.Controlling chaos.Phys Rev Lett1990;64:1196–9.[2]Carroll TL,Pecora LM.Synchronizing chaotic circuits.IEEE Trans Circ Syst I1991;38:453–6.X.Tan et al./Chaos,Solitons and Fractals16(2003)37–4545[3]Bai EW,Lonngran EE.Synchronization of two Lorenz systems using active control.Chaos,Solitons&Fractals1997;8:51–8.[4]Liao TL.Adaptive synchronization of two Lorenz systems.Chaos,Solitons&Fractals1998;9:1555–61.[5]Cuomo KM,Oppenheim AV,Strogatz SH.Synchronization of Lorenz-based chaotic circuits with applications to communi-cations.IEEE Trans Circ Syst I1993;40:626–33.[6]Liao T-L,Tsai S-H.Adaptive synchronization of chaotic systems and its application to secure communications.Chaos,Solitons&Fractals2000;11:1387–96.[7]Bai E-W,Lonngren KE.Synchronization and control of chaotic systems.Chaos,Solitons&Fractals1999;10:1571–5.[8]Krstic M,Kanellakopoulos I,Kokotovic P.Nonlinear and adaptive control design.New York:John Wiley;1995.[9]Mascolo S,Grassi G.Controlling chaotic dynamics using backstepping design with application to the Lorenz system and ChuaÕscircuit.Int J Bifur Chaos1999;9:1425–34.[10]Rodrigues HM,Alberto LFC,Bretas NG.On the invariance principle:generalizations and applications to synchronization.IEEETrans Circ Syst––I:Fundamental Theory Appl2000;47:730–9.。
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关于频率分裂原理的研究,通常涉及到物理学中的振动理论、波动理论以及相关的数学方法。
这个概念在机械工程、物理学和电子工程等领域中都有应用。
以下是一些可能会涉及到频率分裂原理或相关内容的书籍:1. 《振动理论》(Mechanical Vibrations)- S. Rao这本书详细介绍了振动理论的基础知识,包括单自由度系统、多自由度系统、连续系统的振动分析等内容,可能会涉及到频率分裂的概念。
2. 《波动物理》(The Physics of Waves)- Howard Georgi波动物理是频率分裂现象的理论基础,本书将帮助你理解波动的基本概念,以及频率如何在不同的物理环境中发生变化。
3. 《应用谐振理论》(Applied Resonance Theory)- 不同作者这类书籍通常会讨论共振现象,其中可能包含频率分裂原理的内容,尤其是在探讨系统的自然频率和受迫频率时。
4. 《固体物理学》(Solid State Physics)- Ashcroft & Mermin对于材料科学和凝聚态物理感兴趣的读者,这本经典教材提供了晶体结构、电子性质、晶格振动等方面的深入讨论,其中晶格振动部分可能会讨论到频率分裂现象。
5. 《非线性振动》(Nonlinear Vibrations)- 不同作者当系统的振动行为偏离线性时,可能会出现复杂的频率分裂现象,这类书籍将讨论非线性系统的动力学行为。
6. 《电磁波理论与应用》(Electromagnetic Waves & Applications)- 不同作者在电磁学领域,频率分裂可能与波导、腔体和天线的设计相关,这类书籍可能会涉及到相关的理论和应用。
7. 《量子力学》(Quantum Mechanics)- 不同作者在量子力学中,能级分裂是一个重要概念,它与频率分裂有着密切联系。
这类书籍会讨论量子系统的性质和测量结果,其中可能包括频率分裂的量子版本。
请注意,以上推荐的书籍可能需要一定的物理学和数学背景才能充分理解。
Chaos_1
Deterministic Systems Concept
System dynamics display three types of behaviours in their solutions:
– to approach constant solutions, – to converge toward periodic solutions, or – to converge toward quasi-periodic solutions.
– low-dimensional dynamical systems are capable of complex and unpredictable behaviour.
One of the characteristics of chaotic systems is sensitivity to initial conditions;
– trajectories from two relatively close initial values will diverge as the systems evolve, Or: a very small difference in the starting values of the function will, after the function has been iterated many times, lead to a great difference in the produced behaviours.
RMIT University
Slide 8
1.3 Chaos and Bifurcation in Nonlinear Systems
chaos theory
Exploration of fractal and chaos logics based on universal logicÃExploration of fractal and chaos logics based on universal logicCHEN Zhic-heng, HE Hua-can, MAO Ming-yi(Department of Computer Science and Engineering, Northwestern Polytechnical University, Xi’an, 710072, China)Abstract: Opinions that people think about the nature are basically changing, that is to say, turning to multimode, temporality and complexity. Fractal and chaos are sunrise sciences, which research nonlinear natural system. This paper mainly analyzes the basic characteristics of fractal phenomenon and chaos system, expatiates on the correlation of fractal, chaos and universal logic, fully discusses the necessary and probability of setting up fractal logic and chaos logic, and gives the compendium of fractal logic and the way of the implementing of chaos logic. The paper raises:a). Along with the fact that people search after the order of nature more and more thoroughly, the naissance of chaos logic and fractal logic is not only the inevitable production, but also the objective requirement for continuing to do that. b). The principles of universal logic offer the formal frame and the producing rules for establishing the architecture of a concrete logic system, and this makes it possible to set up fractal logic and chaos logic. c). Fractal and chaos logics will play an important and positive role in researching complicated system, and gradually consummate themselves in the applications. Key words: universal logic; fractal logic; chaos logic; nonlinear sciences1 BackgroundLogic is the science about forms and rules of right thought. The target of logic contains formal logic and dialectical logic, the purpose and task about right thinking, and the significance of logic for science and practice activities [1,2]. Such is that, and it is important to know the purpose of logic. It can help us to understand the question “Why do set up fractal logic and chaos logic” in the following content.1.1 The characteristics of classical logicsHere the classical logics include two-valued logic, multi-valued logic and mathematical logic. In the opinion of universal logic [3], classical logics are “rigid logics” that have the following main features:①.The truth-value of proposition is integerIn two-valued logic, the truth-value of proposition only is one of numeral zero and numeral one, Along with the request of the development of other subjects, especially cybernetics, multi-valued logic appears. It allows that the truth-value of proposition can be one of more than two numerals. But they are still “rigid logics”, because their proposition’s values are integer.②.Independence of the correlation of propositionsIn classical logic inferential systems, there is latent hypothesis that every proposition is independent from others. Otherwise the inferential march is difficult. In other words, generally don’t consider the correlation of different propositions.③. The uniqueness of the operating model of connectivesIn classical logics, even in the fuzzy logic[4,5], the operating rules of connectives are defined and cannot be changed in the process of reasoning, such as: ~P=1-P,P∧Q=min(P,Q),P∨Q=max(P,Q),P®Q =min(1,1-P+Q),P«Q =1-|P-Q|. Document [3] has fully discussed the rationality of this operating model of fuzzy prepositional connectives. Because of the complexity of the real world, the operating rules of connectives should be defined as the formula cluster that changes continuously which is controlled by general correlation [3].1.2 Necessary of fractal and chaos logicsThe volition and duty of scientists are exploring laws of nature. Physical scientists try to make clear the spin, color and taste of elementary particle. And biologists study the biologic evolvement, etc. All of them go with saying. But there are more complicated problems that puzzle people now. People are thinking why the change of weather cannot be forecast correctly [6-8], but the “Feigenbaum constant”[9,14] doesn’t change with the difference of different systems and their running equations? People believe that these secret problems implicate some deep laws of nature. And such is chaos, which make these problems understood. In going step further research, people find that chaos systems contain order, which have the feature of fractional dimension in some degree [9]. Fractal, as another new subject, has much progress, so a main aspect of chaos is fractal, and researching fractal is an effective way for studying chaos.Now days, fractal has definite application and development in some fields, such as: molecule science, physics, chemistry, and life sciences etc [10-13]. But, to some extent, fractal is only a kind of computational mathematics, and it is helpless for the reasoning and forecast of complicated system. At this time, the fields of chaos’s application are limited to control system. How do chaos and fractal develop fast and apply extensively in 21st century, in which the complicated sciences play a very important role? Obviously, there will be many heated spots in judging, reasoning, and controlling of every factor and every process of complicated system. This needs right and effective logical reasoning besides mathematical calculation. It undoubtedly requires the development of logic. Such is the fact, in the time of going deep into and developing the classical logics, “universal logic”, as a new subject, has been produced. It is a flexible logic system that includes uncertainties. It analyzes the scope of applications of classical logics and fuzzy logic. And lists the four factors which all kinds of logic must possess. The principles of universal logic offer the formal frame and the producing rules for establishing the architecture of a concrete logic system [3]. It is a guide to set up “fractal logic” and “chaos logic”. So the naissance of chaos logic and fractal logic is involuntary production. They will be powerful “tool logic” for people to study complicated system.2 The features and relationship of chaos and fractal2.1 Basic features of chaosChaos phenomena can be found everywhere in the matter world from macroscopic world to microcosmic fields. It subtly hangs together the presentational disorder and inherent laws. The basic features of chaos are [9]:①. Internal randomicityA system may be in chaos state is the reflection of internal randomicity of nonlinear dynamics. The nonsteady property of parts of a system is just the characteristic of internal randomicity, and it is sensitive to the initial value. According to this, E.N.Lorenz put forward the famous “butterfly effect”.②. Fractional dimensionChaos systems have the feature of fractional dimension. It says that the geometry form of their running tracks in the phase space can be described by fractional dimension. The tracks of movement infinitely twist, fold and distort in the phase space, and form infinite ranks structures that are self-similar. This kind of structures is the famous “strange attractor”.③. Orderliness in disorderChaos phenomenon is an order state although it is not recurrent on all levels of all parts of the system. It has infinite self-similar structure. So long as the precision in calculation of the equipment in laboratory are enough high, we can find and show the pattern of the orderly movement of chaos system in small measure region.2.2 Basic features of fractalWe already mentioned: the obvious feature of fractal is self-similar, others features include infinite complex, infinite subtle structure and so on [9,15].①. Self-similarMany things have self-similar property in structure. For example with tree, branch is a small edition of the whole tree, while the branch of branch is smaller edition, see fig 1. Fractal is so important that it has been applied in all kinds of fields just because of its self-similar property. Many objects can be described accurately by fractal, such as plant, snowflake, scenery picture, and music work.②. Infinite complex and subtleAnother important feature of fractal is that it has infinite complex and infinite subtle structure. This kind of structure can be paint by iterative functional system (IFS).③. Fractional dimensionIt is difficult to give the definition of fractal accurately without the conception of fractional dimension. We can describe fractal object more properly with fractional dimension than without it.Fig 2:Sierpinski triangleFig 1:Fractal plant2.3 The relationship of chaos and fractalIt is important for us to make clear the relationship of chaos and fractal. Here are some items about this.①. From chaos to orderWe know that chaos is a kind of nonperiodic dynamical behavior; it looks disorderly and unsystematic, but contains abundant connotation, such as strange attractor, bifurcation, and windows etc. So we can say that chaos implies order, and it isn’t a stochastic process that cannot be controlled. The Sierpinski triangle (see fig 2) can be drawn by some very simple rules and calculation, but it is a very wonderful pattern. We can obtain Sierpinski triangle even if using stochastic method. This characteristic is absolutely necessarily for people to study chaos. That is that disorder contains order.②. From order to similitudeThe order characteristic of nature is a reflection of dynamical behavior of system’s self-organized. The order things mean laws that may be found. Here the word “order” is “generalized order”, so it includes the laws of not only linear science but also nonlinear science, and the latter is more important. The result of research is that the internal order of chaos has the characteristic of fractional dimension. For example, smoke from kitchen chimneys spirals and rises. Every spiral forms a unclose circle, and all of the circles are similar.③. From similitude to fractalThe further study states that it is the self- similar structure that makes chaos have the fractional dimension property. So we can research chaos through fractal, and make chaos and fractal powerful tools for researching nonlinear science.From chaos to order, from order to similitude, and from similitude to fractal, these sum up the relation between chaos and fractal. Because chaos system has the fractional dimension property, researching fractal is becoming the effective way by which for studying chaos.3 Exploration of fractal logic based on universal logic3.1 Introduction of universal logic principlesBecause of the requirements of development of computer, information technology and artificial intelligence, universal logic[3] was born, which stress the relational flexibility of different objects (tasks, proposition, connectives). Universal logic is a common architecture of logical theory system, which is “self-contained and exoteric”. It is self-contained because it puts up the common characteristics of all kinds of logics at these days, which are the four factors. More than that, it also gives a logic builder for creating concrete logic system according to the need of application. It is exoteric because it allows new logical system join into its frame, and other people can expand and perfect its architecture if necessary.Universal logic puts forward the four essential features of all kinds of logics. They are research filed, proposition connectives, relational quantifier, set of common rules and appropriate inferential model. For a concrete logic system, it must have appropriate semantic explanation, and it can hold its own features. The one task of authors in the flowing time is trying best to set up fractal logic and chaos logic based on the principle of universal logic with its logic builder.3.2 Compendium of fractal logic3.2.1 The four factors of fractal logic①. The research filed of fractal logicThe research filed of fractal logic is all things that have fractal features, which may be geometric objects that are self-similar, and statistic data that are self-similar. To some degreed, setting up fractal logic is building logical inferential system based on fractal mathematics. The true-value field of proposition is a multi-dimension and super order space:W={⊥}[0, R]D<α>Here, W is the space of true-value field, [0, R] is the radix space, R is a real number, D is the fractional dimension space, {⊥} is the undefined space or other spaces, <α> is the accessorial characteristics of proposition or predication.Obviously, because of the fractional dimension feature of fractal, we must be a multi-dimension space based on continuous-valued radix. And because the fractional dimension is not smaller than three in chaos system, it is necessary for us to expand the true-valued field from [0,R] to [0,+∞], even [-∞,+∞]. By the way, the negative is significative some times in fractal.②. Proposition connectives of fractal logicAbout the proposition connectives of fractal logic, authors try to set about to analyze the fractal principle firstly, then study the mutual relativity of different parts of fractal object, and give the model of operation. Table1 shortly lists some proposition connectives of fractal logic. Using these proposition connectives, authors have get some significative results, which will be expatiated in another special paper.Table1: the proposition connectives and their operation models of fractal logicNo. Connective Denotation Operationmodel1 Not ~k N(x,k)=D(R-D-1(x))2 And ∧h A(x,y,h)=ΓR[D((D-1(x)m+D-1(y)m-R)1/m)]3 Or ∨h O(x,y,h)=ΓR[D(R-((R-D-1(x))m+(R-D-1(y))m-R)1/m)]4 Implication →h I(x,y,h)=ΓR[D((R-D-1(x)m+D-1(y)m)1/m)]5 Equality ↔h E(x,y,h)=ite{D((R+| D-1(x)m-D-1(y)m|)1/m)|m≤0;D((R-| D-1(x)m-D-1(y)m|)1/m)}6 Average Êh V(x,y,h)=D(R-(((R-D-1(x))m+(R-D-1(y))m)/2)1/m)7 Combinato-rial ©e h C e(x,y,h)=ite{Γe[D((D-1(x)m+D-1(y)m-e m)1/m)]| D-1(x)+D-1(y)<2e; D(R-(ΓR-e[((R-D-1(x))m+(R-D-1(y))m)-(R-e)m]1/m))| D-1(x)+D-1(y)>2e;8 Series ®h R(x,y,h)=D(R-(((R-D-1(x))mn+(R-D-1(y))mn))1/mn)Notes:a). R is the radix space dimension of the research filed, D is the actual dimension of fractal object, function D(x) calculates the dimension of fractal object “x”, but D-1(x) constructs the fractal object with dimension “x”,HereD-1(x) and D(x) denote two reverse operating processes, but not the converse mathematical operatorsb). h denotes general correlative coefficient, k denotes general self-correlative coefficient, see Reference[3];c). In the operation cluster, m∈R, and there is a similar relation between m and h, i.e. m=(3-4h)/(4h(1-h));d). Function ΓR[x] denotes x is restricted in [0,R], if x>R then it equals to R; if x<0 it equals to 0.e). S=ite {β|α;γ} is condition expression, it means that if a is true, then S=β; else S=γ.f). In formula 7, e denotes by combinatorial connective ©e h; in formula 8, n denotes the dispersion of series.③. Relational quantifier of fractal logicNowadays universal logic supplies these quantifiers: generality quantifier, existential quantifier, assumptive quantifier, boundary quantifier, locational quantifier, and transitional quantifier. They are useful in fractal logic, and we will add other quantifiers such as progressional quantifier, similar quantifier etc.④. Set of common rules and appropriate inference modelAccording to the proposition connectives and quantifier of fractal logic, we can construct the set of common rules and appropriate reasoning model. This has us study the classification of fractal objects, the different features of different kinds of fractal objects, the intention of fractal reasoning, and what and how fractal logic plays in the complicated system. After we know them we can set up the primary model of fractal logic. Table 1 shows some operation models of the proposition connectives when the radix space is [0,R].3.2.2 Semantic explanation of fractal logicAny logic that is used to reason in a concrete system has its own concrete semantic explanation for proposition, connectives, quantifiers, and set of rules[16]. When a doctor diagnoses a patient, numeral zero means no illness, and numeral one means illness. In the same way, when fractal logic is used to reason in fractal system, it is necessary for us to give the semantic explanation of true-value, connectives, quantifiers, and set of rules. For example, we can suppose that the true-value represents the fractional dimension of fractal objects, and the general relativity of proposition and connectives represents the degree of similitude of fractal objects.3.2.3 Study of application of fractal logicAlthough it has not long time since fractal became a subject, it reveals self-similitude law of nature. So fractal has some applications in many fields, such as philosophy, mathematics, physics, molecular chemistry, material science, life sciences, biology, anthropology, computer science, information technology, geognosy, and artistic beauty. However, after analyzing these applications, we find that fractal seems a boundary subject, but it mainly plays a role of computational mathematics. People display the kinetic behavior of fractal things mainly by iterative function. In fact, fractal should be a boundary subject. So there is unprecedented difficulty when we want to use fractal theory for forecasting the behavior of control system after a long time, even it is very difficult to resolve some typical fractal problems. In author’s opinion, it is possible to put the axe in the helve by the reasoning of fractal logic. Here are some fractal problems that may be givensatisfactory answers by fractal logic.①.This is a typical fractal problem[10,12], here calls it“Fractal person question”:A fractal person,walk one fractal mile,Buy a fractal cat,catch a fractal mouse,The fractal pallium of the fractal person,thinking that the fractal cat is eating the fractal mouse,the fractal mouse is being absorbed by the fractal small intestine of the fractal cat.Question1:“When one fractal minute wears away, whether the whole fractal cat will be absorbed, or how much meat of the cat will be absorbed?”②.Authors put up the following problem, call it“combined carpet question”. It can show the other field of application of fractal logic.There are two carpets named C1 and C2, and their acreages are S1 and S2. Their brocades are similar but not same. The carpets have surface roughness because the surfaces of them are not slippy. Let’s suppose that we have worked out the roughness parameters by calculating their fractional dimension, which are D1 and D2.Question2:“Suppose that we break up the two carpets, and mix their brocades, then weave the mixed brocade into a new carpet named C3, and its acreage is equal to the sum of the foregoing two carpets, that is S3=S1+S2. The question is that what the roughness parameters D3 of the third carpet is?”The result of research show that it is very difficult even not possible for conventional fractal mathematics to satisfactorily answer the “fractal person question” and “combined carpet question”. It is easy to see that the first question needs more fractal reasoning and fractal forecast with fractal logic than numerical calculation. In the second question, it is necessary and important for us to establish the fractal’s combined rule. Obviously, D3 is not equal to the sum of D1 and D2 in most cases, so we should analyze the degree of comparability of C1 and C2, and the correlativity of D1 and D2. To our happiness, this is consistent with the conception of general correlation and general self-correlation. So if we set up fractal logic based on universal logic, it will play an important positive effect for revealing these kinds of puzzles.③.Through researching general correlation of fractal objects with fractal logic, it may be possible to open out the substance of“One hair affects the whole body”in life science, find the internal drive of the process of physics, chemistry, biology, and other natural phenomena, and predict the developing direction or final result of them.④.With the development and application of fractal logic, more than that, we can set up chaos logic based on universal logic and fractal logic. Chaos logic can be used in chaos control, weather prediction, earthquake forecast and so on.4 The implementing process of chaos logicAfter knowing about the above content, we can form the“trilogy of chaos logic”, they are: universal logic→fractallogic →chaos logic →complicated system. Fig 3 is the sketch map of the development of chaos logic.4.1 Universal logic——the foundation of fractal logic There are two reasons for us to choose universal logic as the foundation of fractal logic. Firstly, universal logic affords the formal frame for all kinds of logics, and gives the producing rules in its logic builder, we can set up fractal logic with the logic builder of universal logic. Secondly, the general correlative coefficient and the general self-correlative coefficient [3] can be used to describe the self-similar characteristic and the fractional correlativedimension of fractal objects, which is one of main parts of fractal logic.4.2 os system has the fractional ic.4.3 to a concrete application, we can esides the common set of rules if necessary.5 ting up fractal logic and chaos logic. Based on the content of this paper, we can Fractal logic——the bridge to chaos logicSeen from fig 3 that universal logic is the principles and foundation of the whole logical system, and fractal logic is the bridge to chaos logic. Because chaos system is more complicated than fractal phenomenon in indeterminate cases, it is more difficult to study chaos system and set up chaos logic directly. However, cha dimension characteristic, so we can set up fractal logic firstly, and then set up chaos log Complicated system——the way of application and perfecting of chaos logicFrom the view of epistemology, the purpose of all new subjects or sciences is only one, which is that it tries to help people understand the nature phenomena more accurately, and master the law of nature well and truly, and chaos logic is not exception here. None but the applications in different kinds of complicated systems can perfect chaos logic gradually. Especially, it can boost the development of the four factors of chaos logic. According constitute the operating characteristic set b Summarization and prospectsTo this day, authors have already made some progress in fractal logic. We can get logical results with the emluator of fractal logic when researching some fractal things. But there are many puzzles on the way to fractal logic and chaos logic, and it is a long-term and hard task to set up and perfect fractal logic and chaos logic. This paper is only an entrance to fractal logic and chaos logic, and avoids lots of logical and mathematical formulas. Authors mainly analyze the basic characteristics of fractal theory and chaos system, expound the correlation of fractal, chaos and universal logic, and fully discuss the necessary and probability of set have the following summarization :logic a gic and chaos logic. This paper gives the compe result of them. At the same time, fractal logic and chaos logic will gradually consummate themselves in the applications.good suggests. And this work absorbs much nourishment from the following references and others. Here thanks to all.References:[1] Patterson, Richard, Aristotle’s Modal Logic: Essence and Entailment in the Organon. Cambridge University Press, 1995.[2] Hockney. D, (ed.), Contemporary Research in Philosophical Logic and Linguistic Semantics, Dordrecht, Reidel, 1975[3] Huacan He, (ed.), Universal Logics Principle, Beijing, China Science Publishing House, 2001[4] Zadeh, L.A. Fuzzy sets, Information and Control, 1965, No.8, p338-357[5] Hacck. S, Deviant Logic and Fuzzy Logic, Beyond the Formalism, 2nd edition, The University of Chicago Press, 1996[6] E.N.Lorenz (writer), Shida Liu (translator), The essential of chaos, Beijing, Weather Publishing House, 1997.[7] Runsheng Huang, Chaos and its application, Wuhan, Wuhan University press, 2000.[8] Hu Gang, Chaos Control, Shanghai, Shanghai Science and Technology Education Publishing House, 2001.[9] Hongyi Lin, Yingxue Li, Fractal theory -Exploration of strangeness, Beijing University of Theory and Technology Press, 1992.[10] B.H.Kaye, A Random Walk Through Fractal Dimension, translated version, Shenyang, Northeastern University Press, 1994.[11] K.J.Falconer, The Geometry of Fractal Sets, London, Cambridge University Press, 1985[12] Huajie Liu, Fractal Art, Changsha, Hunan Science and Technological Publishing House, 2000.[13] Dongsheng Wang, Cao Lei, Chaos, Fractal and Application, Hefei, Chinese University of Science and Technology Press, 1995.[14] Feigenbaum M J.Los Alamos Science. 1981, No.1, P4.[15] Mandelbrot B B. Fractal Geometry of Nature, Freeman, San Francico. 1982.[16] Thomason, Steven K, Semantic Analysis of the Modal Syllogistic, Journal of Philosophical Logic, 1993, No.22, p111-128.①. Along with the fact that people search after the order of nature more and more thoroughly, the naissance of chaos nd fractal logic is not only the inevitable production, but also the objective requirement for continuing to do that.②.The principles of universal logic offer the formal frame and the producing rules for establishing the architecture of a concrete logic system, and this makes it possible to set up fractal lo ndium of fractal logic and the way of the development of chaos logic.③.For researching complicated system, fractal logic and chaos logic will play an important and positive role. With the development and application of fractal logic and chaos logic, we can reveal the internal drive of more and more natural phenomena, and predict the developing direction or final Acknowledgement :This paper is supported by the National Natural Science Foundation of China under Grant No.60273087. Although the work of this paper is only an exploration now, Chairman Hong Feng, the dean of Free Software Foundational academe of China, and many colleagues in the lab give many基于泛逻辑的分形与混沌逻辑初探1(西北工业大学 计算机科学与工程系, 陕西 西安 710072)摘要: 人们对自然的看法正经历着一个根本性的改变,即转向多重性、暂时性和复杂性,分形论与混沌学则是研究此类非线性自然系统的新兴科学。
chaos,solitons and fractals
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article info
Article history: Accepted 20 March 2009
Communicated by: Prof. Ji-Huan He
abstract
In this paper, we address chaos synchronization problems of a new 3D chaotic system via three different methods. Active control and backstepping design methods are adopted when system parameters are known, and adaptive control method is applied when system parameters are unknown. The corresponding sufficient conditions to achieve synchronization between two identical systems are obtained based on the Lyapunov stability theory. Numerical simulations are presented to demonstrate the effectiveness and feasibility of the analytical results.
混沌理论chaos theory(中英文)
What exactly is chaos? The name "chaos theory" comes from the fact that the systems that the theory describes are apparently disordered, but chaos theory is really about finding the underlying order in apparently random data. When was chaos first discovered? The first true experimenter in chaos was a meteorologist, named Edward Lorenz. In 1960, he was working on the problem of weather prediction. He had a computer set up, with a set of twelve equations to model the weather. It didn't predict the weather itself. However this computer program did theoretically predict what the weather might be.
This effect came to be known as the butterfly effect. The amount of difference in t he starting points of the two curves is so small that it is comparable to a butterfly flapping its wings. The flapping of a single butterfly's wing today produces a tiny change in the state of the atmosphere. Over a period of time, what the atmosphere actually does diverges from what it would have done. So, in a month's time, a tornado that would have devastated the Indonesian coast doesn't happen. Or maybe one that wasn't going to happen, does. (Ian Stewart, Does God Play Dice? The Mathematics of Chaos , pg. 141) This phenomenon, common to chaos theory, is also known as sensitive dependence on initial conditions. Just a small change in the initial conditions can drastically change the long -term behavior of a system. Such a small amount of difference in a measurement might be considered experimental noise, background noise, or an inaccuracy of the equipment. Such things are impossible to avoid in even the most isolated lab. With a starting number of 2, the final result can be entirely different from the same s ystem with a starting value of 2.000001. It is simply impossible to achieve this level of accuracy - just try and measure something to the nearest millionth of an inch! From this idea, Lorenz stated that it is impossible to predict the weather accurately. However, this discovery led Lorenz on to other aspects of what eventually came to be known as chaos theory. Lorenz started to look for a simpler system that had sensitive dependence on initial conditions. His first discovery had twelve equations, and he wa nted a much more simple version that still had this attribute. He took the equations for convection, and stripped them down, making them unrealistically simple. The system no longer had anything to do with convection, but it did have sensitive dependence on its initial conditions, and there were only three equations this time. Later, it was discovered that his equations precisely described a water wheel. At the top, water drips steadily into containers hanging on the wheel's rim. Each container drips steadily from a small hole. If the stream of water is slow, the top containers never fill fast enough to overcome friction, but if the stream is faster, the weight starts to turn the wheel. The rotation might become continuous. Or if the stream is so fast that the heavy containers swing all the way around the bottom and up the other side, the wheel might then slow, stop, and reverse its rotation, turning first one way and then the other. (James Gleick, Chaos - Making a New Science, pg. 29)
Chaos induced by heteroclinic cycles connecting repellers in complete metric spaces
Chaos induced by heteroclinic cycles connecting repellersin complete metric spaces qZongcheng Li a ,Yuming Shia,*,Chao Zhanga,baDepartment of Mathematics,Shandong University,Jinan,Shandong 250100,PR ChinabSchool of Science,Jinan University,Jinan,Shandong 250022,PR ChinaAccepted 3July 2006AbstractThis paper is concerned with chaos induced by heteroclinic cycles connecting repellers for maps in complete metric spaces.The concepts of heteroclinic cycle connecting fixed points and heteroclinic cycle connecting repellers are intro-duced.Two classifications of heteroclinic cycles are given:regular and singular;nondegenerate and degenerate.Several criteria of chaos are established for maps in complete metric spaces and metric spaces with the compactness in the sense that each bounded and closed set of the space is compact,respectively,by employing the coupled-expansion theory.They are all induced by regular and nondegenerate heteroclinic cycles connecting repellers.The maps in these criteria are proved to be chaotic in the sense of both Devaney and Li–Yorke.An illustrative example is provided with computer simulations.Ó2006Elsevier Ltd.All rights reserved.1.IntroductionChaos for one-dimensional maps has been extensively studied,and some elegant results have been established.In 1975,Li and Yorke [8]studied a continuous map from an interval to itself,and obtained the following well-known result:‘‘period 3implies chaos’’.Later,Block and Coppel [3]introduced the concept of turbulence for one-dimensional maps.It has been proved that nonpower of 2period,turbulence,and positive topological entropy for a continuous interval map all imply chaos in the sense of Li–Yorke [3].But the inverse is not true (see [3]).Although it is difficult to study chaos for higher-dimensional maps,several criteria of chaos have been established.In 1978,Marotto [10]proved that ‘‘a snap-back repeller implies chaos’’in the sense of Li–Yorke.The concept of snap-back repeller was extended to maps in metric spaces by Shi and Chen in 2004[13].According to the classifications of snap-back repellers for maps in metric spaces in [13],the snap-back repeller in the Marotto paper [10]is regular and nondegenerate.It was further shown that a regular and nondegenerate snap-back repeller implies chaos in the sense of Devaney (see [13,Theorem 4.1;14,Theorem 4.4;17,Theorem 2.1]).The snap-back repeller theory has been often used0960-0779/$-see front matter Ó2006Elsevier Ltd.All rights reserved.doi:10.1016/j.chaos.2006.07.014qThis research was supported by the NNSF of China (Grant 10471077)and Shandong Research Funds for Young Scientists (Grant 03BS094).*Corresponding author.E-mail address:ymshi@ (Y.Shi).Available online at Chaos,Solitons and Fractals 36(2008)746–761/locate/chaosin studying existence of chaos for higher-dimensional maps.Recently,the assumptions in the Marotto theorem were weakened and it was proved that a regular snap-back repeller implies chaos in the sense of Li–Yorke by Shi and Yu[16].Obviously,chaos problems for infinite-dimensional maps are more complicated and then are more difficult to study. However,some important progresses have been made.In1984,Kloeden[7]established a criterion of chaos in the sense of Li–Yorke for maps in compact sets of Banach spaces.In2001,Kennedy and Yorke[6]studied the topological horse-shoes and proved that a continuous map in a compact invariant set of metric space is topologically semiconjugate to a one-sided symbolic system underfive hypotheses,called the horseshoe hypotheses.But this semiconjugacy cannot imply that the map is chaotic in the sense of Devaney in general(see[16,Example1.1]).In addition,all the conditions in the above two results were given on some compact sets of the metric spaces.It is well known that any set of an infinite-dimensional Banach space is not compact if it contains an interior point.Recently,Shi et al.[13,15,16]established some criteria of chaos induced by coupled-expansion for maps not only in compact subsets of metric spaces but also in bounded and closed sets(not required to be compact)of complete metric spaces.These maps were proved to be chaotic in the sense of Devaney,Li–Yorke,or Wiggins.By employing this coupled-expansion theory,the snap-back repeller theory in Banach spaces and general complete metric spaces were established in[13–17].More recently,Lin and Chen[9]studied the following discrete dynamical system:x nþ1¼fðx nÞ;n¼1;2;...;ð1:1Þwhere f:R m!R m is a continuous transformation.They introduced the concept of heteroclinical repeller as follows: (1)afixed point z2R m of f is said to be expanding if the Jacobian matrix Df(z)exists and the norm of each of itseigenvalues is larger than1;(2)the k(P2)fixed points z1,z2,...,z k of system(1.1)are called heteroclinical repellers of f if the following threeconditions hold:(i)for all i,z i is expanding;(ii)there exist k points x i2W uloc ðz iÞ(the locally unstable manifold of z i)and k natural numbers m i,16i6k,such thatf m jðx jÞ¼z jþ1;16j6kÀ1;f m kðx kÞ¼z1;ð1:2Þ(iii)for all i,x i satisfies the nondegenerate property;that is,Df m iðx iÞexists and det Df m iðx iÞ¼0.They proved that a map with a heteroclinical repeller can induce chaos in the sense of Li–Yorke by employing a similar method to that used in the proof of Marotto[10],and is topologically conjugate to a sub-shift of thefinite type. But,it is not certain so far whether it is chaotic in the sense of Devaney.Note that the conditions in the above definitions are given in terms of the Jacobian matrices of the map f.A map in a metric space does not have derivatives in general.Recently,Shi and Chen introduced the concepts of expandingfixed points and snap-back repellers for maps in metric spaces[13].Based on this work,we capture the essential meanings of the concept of heteroclinical repellers and extend it to maps in metric spaces in the present paper(see Definition2.4in Section2).In order to make the name of the concept more intuitive to reflect the relations of those repellers,we call it a heteroclinic cycle connecting repellers instead of heteroclinical repellers,which is similar to the term heteroclinic cycle for ordinary differential equations[19,Definition3.3.1].We further give the classifications:regular and singular;non-degenerate and degenerate.It can be verified that the heteroclinic cycle in[9]is regular and nondegenerate.Different from the method used in[9],we employ the coupled-expansion theory to study chaos induced by heteroclinic cycles connecting repellers.The rest of the paper is organized as follows.In Section2,some basic concepts and lemmas are given.The concepts of heteroclinic orbit,heteroclinic cycle connectingfixed points,and heteroclinic cycle connecting repellers are intro-duced.Two classifications of heteroclinic cycle connectingfixed points are given:regular and singular;nondegenerate and degenerate.In Sections3,four criteria of chaos are established for maps in complete metric spaces and one criterion of chaos is established for maps in metric spaces with the compactness in the sense that each bounded and closed set of the space is compact.They are all induced by regular and nondegenerate heteroclinic cycles connecting repellers.The maps in these criteria are proved to be chaotic in the sense of both Devaney and Li–Yorke by employing the coupled-expansion theory.Finally,we provide an illustrative example with computer simulations in section4.Remark1.1.Recently,we extended the concept of turbulence for continuous interval maps,introduced by Block and Coppel[3],to maps in general metric spaces[15,16].In[15],the maps were still called turbulence.Since the term turbulence is well-established influid mechanics,some referees suggested us to use another name in order to avoidZ.Li et al./Chaos,Solitons and Fractals36(2008)746–761747possible confusion.The new name of‘‘coupled-expanding map’’is more intuitive in reflecting the conditions that the map satisfies.Here,‘‘a turbulent map’’and‘‘a strictly turbulent map’’are given new names as‘‘a coupled-expanding map’’and‘‘a strictly coupled-expanding map’’,respectively.Remark1.2.Chaos induced by heteroclinic cycles connecting repellers in R n and general Banach spaces will be pre-sented in our forthcoming paper,where some sufficient conditions will be given for regular and nondegenerate hetero-clinic cycles connecting repellers;especially,chaos induced by regular heteroclinic cycles connecting repellers will be discussed in R n.2.PreliminariesIn this section,we give some notations,definitions,and lemmas.Wefirst introduce some notations.Let(X,d)be a metric space,and A,B be subsets of X.The boundary of A, denoted by o A,is the set of all x2X such that each neighborhood of x intersects both A and X n A.Denote the closure of A by A.The distance between the point x and the set A is denoted bydðx;AÞ:¼inf f dðx;yÞ:y2A g;the distance between two sets A and B is denoted bydðA;BÞ:¼inf f dðx;yÞ:x2A;y2B g;the maximal distance between two points in A and B is denoted byd sðA;BÞ:¼sup f dðx;yÞ:x2A;y2B g:The open and closed balls of radius r centered at x2X is denoted by B r(x)and B rðxÞ,respectively.We now present two definitions of chaos in the sense of Li–Yorke and Devaney.For other definitions of chaos and discussions of their relationships,one can see[1,11,12,16,18,19].Definition2.1.Let(X,d)be a metric space,f:X!X be a map,and S be a set of X with at least two distinct points. Then S is called a scrambled set of f if for any two distinct points x,y2S,lim inf n!1dðf nðxÞ;f nðyÞÞ¼0;lim supn!1dðf nðxÞ;f nðyÞÞ>0:The map f is said to be chaotic in the sense of Li–Yorke if there exists an uncountable scrambled set S of f.Definition2.2[4].Let(X,d)be a metric space.A map f:V&X!V is said to be chaotic on V in the sense of Devaney if(i)the set of the periodic points of f is dense in V;(ii)f is topologically transitive in V;(iii)f has sensitive dependence on initial conditions in V.Remark2.1.It is well known that conditions(i)and(ii)imply condition(iii)if f is continuous in V[2].So,condition(iii) is redundant in the above definition.Under some conditions,chaos in the sense of Devaney is stronger than that in the sense of Li–Yorke[5].Definition2.3[13,Definitions2.1and2.4].Let(X,d)be a metric space and f:X!X be a map.A point z2X is called an expandingfixed point(or a repeller)of f in B rðzÞfor some constant r>0,if f(z)=z and there exists a constant k>1 such thatdðfðxÞ;fðyÞÞP k dðx;yÞ8x;y2 B rðzÞ:The constant k is called an expanding coefficient of f in B rðzÞ.Furthermore,z is called a regular expandingfixed point of f in B rðzÞif z is an interior point of f(B r(z)).Otherwise,z is called a singular expandingfixed point of f in B rðzÞ.For convenience,denotetðiÞ:¼½i mod k þ1;16i6k;k P2in the rest of paper,where i and k are positive integers.Now,we introduce the following concepts:748Z.Li et al./Chaos,Solitons and Fractals36(2008)746–761Definition 2.4.Let (X ,d )be a metric space and f :X !X be a map.(1)Assume that f has two fixed points z 1,z 22X .A point x 2X is called heteroclinic to z 1and z 2if x 5z i ,i =1,2,and f j (x )!z 1as j !À1,f j (x )!z 2as j !+1.The heteroclinic point x ,together with its backward orbitf f j ðx Þg À1j ¼À1and its forward orbit f f jðx Þg þ1j ¼1,is called a heteroclinic orbit connecting z 1and z 2.(2)Assume that f has k different fixed points z i ,16i 6k .If it has k heteroclinic orbits connecting z i and z t (i ),16i 6k ;that is,there exist k different points x i ,x i 5z n ,16i ,n 6k ,such thatf j ðx i Þ!z i as j !À1and f j ðx i Þ!z t ði Þas j !þ1;16i 6k ;then the setf z 1g [f f j ðx 1Þg þ1j ¼À1[ÁÁÁ[f z k g [f f jðx k Þg þ1j ¼À1is called a k -heteroclinic cycle connecting z 1,...,z k of f .(3)Assume that f has a k -heteroclinic cycle connecting z 1,...,z k .The cycle is said to be regular if for each point x 0onthe cycle,there exists a positive constant r 0such that for each positive constant r 6r 0,f (x 0)is an interior point of f (B r (x 0)).Otherwise,it is said to be singular.(4)Assume that f has a k -heteroclinic cycle connecting z 1,...,z k .The cycle is said to be nondegenerate if for eachpoint x 0on the cycle,there exist positive constants r 0and l such thatd ðf ðx Þ;f ðy ÞÞP l d ðx ;y Þ8x ;y 2 B r 0ðx 0Þ:Otherwise,it is said to be degenerate.(5)Assume that f has a k -heteroclinic cycle connecting z 1,...,z k .If z i ,16i 6k ,are expanding fixed points of f inBr i ðz i Þfor some constants r i >0,then the cycle is said to be a k -heteroclinic cycle connecting repellers z 1,...,z k (see Fig.1).Remark 2.2.Some remarks on the above definitions are listed.(1)For the definitions of heteroclinic orbit for maps and heteroclinic cycle for hyperbolic fixed points of an ordinarydifferential equation,we refer to [19,Definitions 3.1.1and 3.3.1].In Definition 2.4,the fixed points z j can be replaced by periodic orbits or general invariant sets.(2)If f is a homeomorphism,a heteroclinic orbit of a point x from two fixed points z 1and z 2consists of f f j ðx Þg þ1j ¼À1with f j (x )!z 1as j !À1,f j (x )!z 2as j !+1,and f j (x )5z 2for all j P 1[19,Definition 3.1.1].In the other case that f is not a homeomorphism,it is possible that there exists a positive integer n such that f n (x )=z 2.If it is the case,then f j (x )=z 2for all j P n and consequently f j (x )!z 2as j !+1.Especially,if z is an expanding fixed point of f ,it is impossible that there exists a point x 5z such that f j (x )!z as j !+1and f j (x )5z for all j P 1.Therefore,f has a k -heteroclinic cycle connecting repellers z 1,...,z k if and only if there exist k different points x i ,x i 5z n ,16i ,n 6k ,and positive integers n i suchthatFig.1.Illustration of a k -heteroclinic cycle connecting repellers.Z.Li et al./Chaos,Solitons and Fractals 36(2008)746–761749f jðx iÞ!z i as j!À1;f n iðx iÞ¼z tðiÞ;16i6k;and this cycle consists of f z1g[f f jðx1Þg n1j¼À1[ÁÁÁ[f z k g[f f jðx kÞg n kj¼À1.(3)Assume that f has a nondegenerate k-heteroclinic cycle connecting z1,...,z k,with x0,r0and l as specified in(4)ofDefinition2.4.If f is continuous in B r0ðx0Þand f(x0)is an interior point of fðB rðx0ÞÞ,then,with a similar argu-ment to the proof of[13,Lemma2.1],it can be easily verified that f(x0)is an interior point of f(B r(x0))for each positive constant r6r0and consequently,this cycle is regular.(4)In the special case of X=R n,if f has a nondegenerate k-heteroclinic cycle connecting z1,...,z k and f is continuousin some neighborhood of each point on the cycle,then this cycle is also regular.Lemma2.1.Let(X,d)be a metric space and f:X!X be a map with an expandingfixed point z in B r0ðzÞfor some constantr0>0.If f is continuous in B r0ðzÞand fðB rðzÞÞis an open set,then for each positive constant r6r0,fð B rðzÞÞis a closed set,f(B r(z))is an open set withfð B rðzÞÞ' B rðzÞ;fðB rðzÞÞ'B rðzÞand f(D)is an open set for each open set D&B rðzÞ.Proof.Since fðB r0ðzÞÞis open,z is an regular expandingfixed point of f in B rðzÞ.The rest of the proof is similar to thatof[13,Lemma2.1].So the details are omitted.hLemma2.2.Let(X,d)be a metric space and f:X!X be a map.Assume that there exist two different points x0,z2X,a positive integer m,and positive constants l,d0such that f m(x0)=z,dðf mðxÞ;f mðyÞÞP l dðx;yÞ8x;y2 B dðx0Þ;ð2:1Þf m is continuous on B d0ðx0Þ,and z is an interior point of f mð B dðx0ÞÞ.Then there exists a positive constant d16d0such thatfor each positive constant d6d1,f mð B dðx0ÞÞis a closed set and f m(B d(x0))is an open set. Proof.The idea in the proof is motivated by the proof of[13,Lemma2.1].For convenience,define the following set for a subset A of f mðB dðx0ÞÞ:fÀmðAÞ¼f x2B dðx0Þ:f mðxÞ2A g:Since z is an interior point of f mð B d0ðx0ÞÞ,there exists a constant r>0such that B rðzÞ&f mð B dðx0ÞÞ.By the continuityof f m,fÀm(B r(z))is open and x0is an interior point of fÀm(B r(z)).Then there exists a positive constant d16d0such thatB d1ðx0Þ&fÀmðB rðzÞÞ&B dðx0Þ.It follows from(2.1)thatf m:fÀmðB rðzÞÞ!B rðzÞis bijective and continuous by the assumptions.Next,we showfÀm:B rðzÞ!fÀmðB rðzÞÞis continuous.For any x,y2B r(z),fÀm(x),fÀmðyÞ2B dðx0Þ.Then from(2.1)we getdðfÀmðxÞ;fÀmðyÞÞ6lÀ1dðx;yÞ;which implies that fÀm is continuous on B r(z).Therefore,f m is homeomorphic on fÀm(B r(z))and consequently,for each positive constant d6d1,f mð B dðx0ÞÞis a closed set,f m(B d(x0))is an open set.The proof is complete.hSince the results obtained in this paper are related to the one-sided symbolic dynamical systemðPþ2;rÞ,we brieflyrecall some results of it for convenience.Let Rþ2:¼f s¼ðs0;s1;s2;...Þ:s j¼0or1g with the distanceqðs;tÞ:¼X1i¼0j s iÀt i j2i;where s=(s0,s1,s2,...)and t=(t0,t1,t2,...).ThenðPþ2;qÞis a complete metric space and a Cantor set.The shift mapr:Pþ2!Pþ2defined by r((s0,s1,s2,...))=(s1,s2,...)is continuous.The dynamical system defined by r is called a one-750Z.Li et al./Chaos,Solitons and Fractals36(2008)746–761sided symbolic dynamical system.It has plentiful dynamical behaviors (cf.,[4,12]).Especially,it is chaotic in the sense of both Devaney and Li–Yorke,and has a positive topological entropy.The following three criteria of chaos will be used in the next section.Lemma 2.3[13,Theorem 3.1].Let (X,d)be a complete metric space,and V 1,V 2be nonempty,closed,and bounded subsets of X with d(V 1,V 2)>0.If a continuous map f :V 1[V 2!X satisfies(i )f is strictly coupled-expanding in V j ,j =1,2;that is,f(V j )'V 1[V 2for j =1,2;(ii )f is expanding in V 1and V 2,that is,there exists a constant k >1such thatd ðf ðx Þ;f ðy ÞÞP k d ðx ;y Þ8x ;y 2V 1and 8x ;y 2V 2;(iii )there exists a constant l >0such thatd ðf ðx Þ;f ðy ÞÞ6l d ðx ;y Þ8x ;y 2V 1and 8x ;y 2V 2;then there exists a Cantor set K &V 1[V 2such that f :K !K is topologically conjugate to the one-sided symbolic dynam-ical system r :P þ2!P þ2.Consequently,f is chaotic on K in the sense of both Devaney and Li–Yorke.Lemma 2.4[13,Theorem 3.2].Let (X,d)be a metric space,and V 1,V 2be two disjoint compact subsets of X.If a contin-uous map f :V 1[V 2!X satisfies (i )and (ii )in Lemma 2.3,then the results in Lemma 2.3hold.Remark 2.3.The result that f is chaotic in the sense of Li–Yorke in Lemmas 2.3and 2.4can be derived from [5,The-orem 4.1].The number of sets V j may be larger than 2in Lemmas 2.3and 2.4.The assumptions in Lemma 2.3were weakened recently in [15,Theorem 3.3],as the following lemma.Lemma 2.5[15,Theorem 3.3].Let (X,d)be a complete metric space,and V j (16j 6m)disjoint bounded and closed sets of X with d(V i ,V j )>0for each i 5j,16i,j 6m.If a continuous map f :D :¼Sm j ¼1V j !X satisfies that (i )f is strictly coupled-expanding in V j ,16j 6m;that is,f(V j )'D for each j,16j 6m;(ii )there exist positive constants k >1and l ,and a j 0,16j 06m,such thatd ðf ðx Þ;f ðy ÞÞP k d ðx ;y Þ8x ;y 2V j 0andd ðf ðx Þ;f ðy ÞÞP l d ðx ;y Þ8x ;y 2V j ;16j ¼j 06m ;then(1)f has an uncountable and bounded scrambled set S &D,i.e.,f is chaotic in the sense of Li–Yorke;(2)there exists an uncountable,perfect,bounded,and closed set E &D such that S &E,f(E)=E,and f is chaotic on Ein the sense of Devaney;(3)there exists a positive integer n 0such that f has a n-periodic point in E for each n P n 0.The following result is a direct consequence of [15,Lemma 2.4]and [5,Theorem 4.1].Lemma 2.6.Let (X,d)be a metric space,f :Y &X !X be a map,and D &Y be a nonempty set.Assume that f j is continuous in D with f j (D)&Y,16j 6n,f n (D)=D for some positive integer n,and f n is chaotic on D in the sense ofDevaney.Then there exists an invariant set E :¼S n À1j ¼0f jðD Þ&Y of f such that f is chaotic on E in the sense of Devaney.Especially,if D is a Cantor set,then E is a compact and perfect set and contains a Cantor set,and f is also chaotic on E in the sense of Li–Yorke.3.Chaos induced by heteroclinic cycles connecting repellersIn this section,four criteria of chaos are first established for maps in complete metric spaces and one criterion of chaos is established for maps in metric spaces with the compactness in the sense that each bounded and closed set of the space is compact.The main ideas of these criteria are motivated from some works in [13–17].Z.Li et al./Chaos,Solitons and Fractals 36(2008)746–761751Theorem 3.1.Let (X,d)be a complete metric space and f :X !X be a map.Assume that(i )f has k (P 2)different regular expanding fixed points z 1,...,z k 2X;that is,for each 16i 6k,there exist positive constants r i and k i >1such that f ðB r i ðz i ÞÞis open and d ðf ðx Þ;f ðy ÞÞP k i d ðx ;y Þ8x ;y 2 Br i ðz i Þ;ð3:1Þ(ii )for each 16i 6k,there exist x i 2B r i ðz i Þ,x i 5z i ,a positive integer m i P 1,and positive constants d i and l i suchthat f m i ðx i Þ¼z t ði Þ,B d i ðx i Þ&B r i ðz i Þ,z t(i)is an interior point of f m i ðB d i ðx i ÞÞ,andd ðf m i ðx Þ;f m i ðy ÞÞP l i d ðx ;y Þ8x ;y 2 B d i ðx i Þ;ð3:2Þwhere t(i)=[i mod k]+1;(iii )for each 16i 6k,there exist positive constants s i and r i such thatd ðf ðx Þ;f ðy ÞÞ6s i d ðx ;y Þ8x ;y 2 Br i ðz i Þ;ð3:3Þd ðf m i ðx Þ;f m i ðy ÞÞ6r i d ðx ;y Þ8x ;y 2 Bd i ðx i Þ:ð3:4ÞThen,for each 16i 6k and for each neighborhood U i of z i ,there exist a positive integer n i >Pk j ¼1m j and a Cantor set K i &U i such that f n i:K i !K i is topologically conjugate to the one-sided symbolic dynamical system r :P þ2!P þ2.Con-sequently,f n i is chaotic on K i in the sense of Devaney and f is chaotic in the sense of Li–Yorke.Furthermore,if f is con-tinuous in some neighborhood of f j (x i ),16i 6k,16j 6m i À1in the case that m i P 2,then there exists a compact and perfect invariant set D i &X,containing the Cantor set K i ,such that f is chaotic on D i in the sense of Devaney.Proof.For simplicity,we only prove that for k =2and any neighborhood U 1of z 1,there exist a positive integern 1>m 1+m 2and a Cantor set K 1&U 1such that f n 1:K 1!K 1is topologically conjugate to the one-sided symbolic dynamical system r :P þ2!P þ2.If it is so,then the final result in this theorem directly follows from Lemma 2.6.We prove it by Lemma 2.3.According to Lemma 2.3,it suffices to show that for any neighborhood U 1of z 1,there exist a positive integer n 1>m 1+m 2,positive constants k >1and l ,and two bounded and closed subsets V 1and V 2of U 1with V 1\V 2=;such that f n 1satisfiesf n 1ðV i Þ'V 1[V 2;i ¼1;2;d ðV 1;V 2Þ>0;ð3:5Þd ðf n 1ðx Þ;f n 1ðy ÞÞP k d ðx ;y Þ8x ;y 2V 1and 8x ;y 2V 2;ð3:6Þd ðf n 1ðx Þ;f n 1ðy ÞÞ6l d ðx ;y Þ8x ;y 2V 1and 8x ;y 2V 2:ð3:7ÞIt follows from (3.3)and (3.4)that f is continuous in Br i ðz i Þand f m i is continuous in B d i ðx i Þfor i =1,2.From assump-tion (ii)and by Lemma 2.2,there exists a positive constant d Ãi 6d i such that for each positive constant d 6d Ãi ,f m i ðB d ðx i ÞÞis open and f m i ð B d ðx i ÞÞis closed.From the assumption that f ðB r i ðz i ÞÞis open and again by Lemma 2.1,we get that for each positive constant r 6r i ,f ð Br ðz i ÞÞis closed,f (B r (z i ))is open with f ð B r ðz i ÞÞ' B r ðz i Þ;f ðB r ðz i ÞÞ'B r ðz i Þ;and f (D )is an open set for each open set D &B r i ðz i Þ.We remark that these conclusions are repeatedly used in the rest ofthe proof.Without loss of generality,we can suppose that Br 1ðz 1Þ\ B r 2ðz 2Þ¼;.Otherwise,we can choose r 0i 6d ðz 1;z 2Þ=3to replace r i ,i =1,2.We can also suppose that B r 1ðz 1Þ&U 1.Otherwise,we can choose an integer ^m 1,a point ^x 12B r 1ðz 1Þ\U 1,and positive constants ^r 1<r 1,^d 1,^l1,^r 1such that assumptions (3.1)–(3.4)hold with m 1,x 1,r 1,d 1,l 1,r 1replaced by ^m 1,^x 1,^r 1,^d 1,^l1,^r 1,respectively.In fact,it follows from (3.1)and f ðB r 1ðz 1ÞÞ'B r 1ðz 1Þthat f Ànðx 1Þ2B r 1ðz 1Þis uniquely defined for each n P 1,and f Àn (x 1)!z 1as n !1.Then there exist a positive integer n 01and a positive constant ^r 1<r 1such that ^x 1:¼f Àn 01ðx 1Þ2B ^r 1ðz 1Þand B ^r 1ðz 1Þ&U 1\B r 1ðz 1Þ.This implies that f n 01ð^x 1Þ¼x 1,f ^m 1ð^x 1Þ¼z 2with ^m 1¼m 1þn 01.By the continuity of f ,there exists a sufficiently small positive constant ^d 1such that B ^d 1ð^x 1Þ&B ^r 1ðz 1Þand f j ðB ^d 1ð^x 1ÞÞ&B r 1ðz 1Þfor 16j 6n 01À1and f n 01ðB ^d 1ð^x 1ÞÞ&B d Ã1ðx 1Þ.Since B ^d 1ð^x 1Þis open and f j ðB ^d 1ð^x 1ÞÞ&B r 1ðz 1Þ,16j 6n 01,we get that f n 01ðB ^d 1ð^x 1ÞÞis open and then,f ^m 1ðB ^d 1ð^x 1ÞÞis open from the discussion of the second paragraph of the proof.Consequently,z 2is an interior point of f ^m1ðB ^d 1ð^x 1ÞÞ.Hence,it follows from (3.1)–(3.4)that for all x ;y 2B ^d 1ð^x 1Þd ðf ^m1ðx Þ;f ^m 1ðy ÞÞP ^l 1d ðx ;y Þ;^l 1¼l 1k n 011;d ðf ^m1ðx Þ;f ^m 1ðy ÞÞ6^r 1d ðx ;y Þ;^r1¼r 1s n 011:Obviously,(3.1)and (3.3)hold in B ^r 1ðz 1Þsince ^r 1<r 1.752Z.Li et al./Chaos,Solitons and Fractals 36(2008)746–761。
湍流的混沌理论
湍流的混沌理论1 湍流流体流动时,如果流体质点的轨迹是有规则的光滑曲线,这种流动叫层流。
没有这种性质的流动叫湍流。
1959年J.欣策曾对湍流下过这样的定义:湍流是流体的不规则运动,流场中各种量随时间和空间坐标发生紊乱的变化,然而从统计意义上说,可以得到它们的准确的平均值。
在直径为d的直管中,若流体的平均流速为v,由流体运动粘度v组成的雷诺数有一个临界值(大约为2300~2800),若Re小于该范围则流动是层流,在这种情况下,一旦发生小的随机扰动,随着时间的增长这扰动会逐渐衰减下去;若Re大于该范围,层流就不可能存在了,一旦有小扰动,扰动会增长而转变成湍流。
雷诺在1883年用玻璃管做试验,区别出发生层流或湍流的条件。
把试验的流体染色,可以看到染上颜色的质点在层流时都走直线。
当雷诺数超过临界值时,可以看到质点有随机性的混合,在对时间和空间来说都有脉动时,就是湍流。
不用统计、概率论的方法引进某种量的平均值就难于描述这一流动。
除直管中湍流外还有多种多样各具特点的湍流,虽经大量实验和理论研究,但至今对湍流尚未建立起一套统一而完整的理论。
大多数学者认为应该从纳维-斯托克斯方程出发研究湍流。
湍流对很多重大科技问题极为重要,因此,近几十年所采取的做法是针对具体一类现象建立适合它特点的具体的力学模型。
例如,只适用于附体流的湍流模型;只适用于简单脱体然后又附体的流动;只适用于翼剖面尾迹的或者只适用于激波和边界层相互作用的湍流模型等等。
湍流这个困难而又基本的问题,近年来日益受到了物理学界的重视。
研究湍流的起因和特性的理论,包括两类基本问题:①湍流的起因,即平滑的层流如何过渡到湍流;②充分发展的湍流的特性。
层流过渡为湍流的主要原因是不稳定性。
在多数情况下,剪切流中的扰动会逐渐增长,使流动失去稳定性而形成湍流斑,扰动继续增强,最后导致湍流。
这一类湍流称为剪切湍流。
两平板间的流体受下板面加热或由上板面冷却达到一定程度,也会形成流态失稳,猝发许多小尺度的对流;上下板间的温差继续加大,就会形成充分发展的湍流。
音乐与混沌分形理论
谱 1 巴赫《C 大调前奏曲》第 1-9 小节手稿⑤
整首作品随着时间的推演,和弦逐渐偏离了 最初的音符轨道,形成了音乐图形的渐进位移。 每一个和弦都试模仿前一个和弦,体现出对初始 条件的依赖性(保持音),同时总是出现略微的差 异(级进音),“小偏差”的累积,导致整首作品不 断“偏离轨道”。基于作品的和弦缩写图,可以更 加明显地看出和弦的细微变化,展现出所有和弦 的变化轨迹(见谱 2)。
一、对初始条件的依赖性: 巴赫《C 大调前奏曲perierte Klavier I) 第 一 首《C 大 调 前 奏 曲》(Pre-
lude in C major,BWV846)中,巴赫向世人展示 了“ 分 形 ”的 魅 力 。 整 首 乐 曲 的 原 点 是 质 朴 的 “主三和弦”,它亦是整首作品音符生长的初始 条件。自第二小节开始,所有和弦都在之前的 和弦基础之上进行“微调”,即某些声部同音保 持,而某些声部则缓行级进。巴赫手稿则更为 突出地显现出低声部与高声部发展差异。低 音 声 部 基 本 保 持 在 C 音 上 ,稳 定 了 作 品 的 调 性 ,从“ 分 形 ”角 度 而 言 是 对 初 始 条 件 的 依 赖 性。作品上方各声部则通过级进方式,逐步脱 离 了 原 始 的 运 行 轨 迹 ,绘 出 了 分 形 的 图 形 ( 见 谱 1)。
岸线到底有多长的问题中提出了分形理论②。这 两个密切关联的理论,迅速成为影响深远的前沿 科学,并成为继相对论和量子力学之后的第三次 革命。通过这一理论的研究,我们发现大到星空 宇宙,小到微观粒子,都在混沌和分形理论的指 挥棒下运行。混沌与分形相互关联,混沌即时间 上的分形,分形即空间上的混沌③。混沌现象的 主要特征体现为对于初始条件敏感的依赖,并且 表现出一种不可预测性。分形理论则尤其关注
Chaos Theory and the Markets - Elliott Fractals Global混沌理论和分形市场-埃利奥特全球
Geometry ---- it is a new scientific language.
Fractals – Bringing Order to Chaos
This in turn offers a revolutionary breakthrough in our “comprehension of reality.”
A scientific study or analysis to be valid must include fractals.
What is Chaos? Why Euclidean Geometry Doesn’t
Work for Traders What are Fractals Market Applications Trading with Fractals
Examples of Chaos
Lightning Weather Patterns Earthquakes Financial Markets Social and Natural Systems Governmental and Financial
Fractal Geometry is the real geometry of the natural world : Man, Animal, Vegetable, mineral and the galaxies.
General Fractal Characteristics
Infinite Detail Infinite Length Absences of Smoothness Absences of Derivatives Fractal Geometry is the geometry one finds in
Alatas2009Chaos embedded particle swarm optimization algorithms
Chaos embedded particle swarm optimization algorithmsBilal Alatas *,Erhan Akin,A.Bedri OzerFirat University,Department of Computer Engineering,23119Elazig,TurkeyAccepted 17September 2007Communicated by Prof.L.Marek-CrnjacAbstractThis paper proposes new particle swarm optimization (PSO)methods that use chaotic maps for parameter adaptation.This has been done by using of chaotic number generators each time a random number is needed by the classical PSO algorithm.Twelve chaos-embedded PSO methods have been proposed and eight chaotic maps have been analyzed in the benchmark functions.It has been detected that coupling emergent results in different areas,like those of PSO and complex dynamics,can improve the quality of results in some optimization problems.It has been also shown that,some of the proposed methods have somewhat increased the solution quality,that is in some cases they improved the global searching capability by escaping the local solutions.Ó2007Elsevier Ltd.All rights reserved.1.IntroductionParticle swarm optimization (PSO)is a biologically inspired computational search and optimization method developed in 1995by Eberhart and Kennedy based on the social behaviors of birds flocking or fish schooling [1].PSO is a simple stochastic search method and requires little memory.It is computationally effective and easier to implement when compared with other mathematical algorithms and evolutionary algorithms (EAs).It has also fast con-verging characteristics and more global searching ability at the beginning of the run and a local searching near the end of the run.However,it has sometimes a slow fine-tuning ability of the solution quality [2].While solving problems with more local optima,it is more likely that PSO algorithm will explore local optima at the end of the run.Several researches were carried out so far to analyze the performance of the PSO with different and modified settings.The convergence properties of PSO are strongly related to its stochastic nature and PSO uses random sequence for its parameters during a run.In particular,it can be shown that when different random sequences are used during the PSO search,the final results may effectively be very close but not equal.Different numbers of iterations may also be required to reach the same optimal values.However,there are no analytical results that guarantee an improvement of the performance indexes of PSO algorithms depending on the modified setting and choice of a particular generator as in EAs [3].0960-0779/$-see front matter Ó2007Elsevier Ltd.All rights reserved.doi:10.1016/j.chaos.2007.09.063*Corresponding author.E-mail addresses:balatas@fi.tr (B.Alatas),eakin@fi.tr (E.Akin),bozer@fi.tr (A.B.Ozer).Available online at Chaos,Solitons and Fractals 40(2009)1715–1734/locate/chaos1716 B.Alatas et al./Chaos,Solitons and Fractals40(2009)1715–1734Chaos is a bounded unstable dynamic behavior that exhibits sensitive dependence on initial conditions and includes infinite unstable periodic motions in nonlinear systems.Although it appears to be stochastic,it occurs in a deterministic nonlinear system under deterministic conditions[4].Many chaotic maps in the literature possess certainty,ergodicity and the stochastic property.Recently,chaotic sequences have been adopted instead of random sequences and very interesting and somewhat good results have been shown in many applications such as secure transmission,telecommunication,and cryptography[5,6];nonlinear circuits [7],DNA computing[8],and image processing[9].They have been used to improve the performance of EAs[3,10].They have also been used together with some heuristic optimization algorithms[11]to express optimization variables.The choice of chaotic sequences is justified theoretically by their unpredictability,i.e.,by their spread-spectrum character-istic,non-periodic,complex temporal behavior,and ergodic properties.In this paper,sequences generated from different chaotic systems substitute random numbers for different parame-ters of PSO where it is necessary to make a random-based choice.For this purpose,twelve different PSO methods that use chaotic maps as efficient alternatives to pseudorandom sequences have been proposed.By this way,it is intended to enhance the global convergence and to prevent to stick on a local solution.However,in general,it is hard to estimate how good most chaotic random number generator by applying statistical tests are,as they do not follow the uniform distribution.The simulation results show that the application of deterministic chaotic signals instead of random sequences may be a possible strategy to improve the performances of PSO algorithms.The remaining of this paper is organized as follows.Review of PSO is summarized in Section2.Section3offers a short introduction on improvements for PSO.Section4describes the proposed methods,Chaos Embedded Particle Swarm Optimization Algorithms,shortly CEPSOAs.Section5describes the benchmark problems used for comparisons of the proposed methods.In Section6,the testing of the proposed methods through benchmark problems are carried out and the simulation results are compared with those obtained via other PSO algorithms that have been reported to have good performance.Finally,the conclusion is drawn based on the comparison analysis reported and presented in Section7.2.Particle swarm optimization algorithmsParticle swarm optimization(PSO)is a new population-based heuristic method discovered through simulation of social models of birdflocking,fish schooling,and swarming tofind optimal solution(s)to the non-linear numeric prob-lems.It wasfirst introduced in1995by social-psychologist Eberhart and electrical engineer Kennedy[1].PSO is an efficient,simple,and effective global optimization algorithm that can solve discontinuous,multimodal, and non-convex problems.In PSO,a swarm consists of N particles moving around in a D-dimensional search space. The position of the i th particle at the t th iteration is used to evaluate the quality of the particle and represents candidate solution(s)for the search or optimization problems.It is represented by x i(t)=(x i1,x i2,...,x iD)and.x i,n(t)2[l n,u n], 16n6N where l n and u n is the lower and upper bound for the n th dimension,respectively.During the search process the particle successively adjusts its position toward the global optimum according to the two factors:the best position encountered by itself(pbest)denoted as p i,j=(p i1,p i2,...,p iD)and the best position encountered by the whole swarm (gbest)denoted as p g=(p g1,p g2,...,p gD).Its velocity at the t th iteration is represented by v i(t)=(v i1,v i2,...,v iD)[12]. The velocity is clamped to a maximum velocity v i max=(v i max1,v i max2,...,v i max D).The velocity and position of the particle at next iteration are calculated according to the following equations:v i;jðtþ1Þ¼wv i;jðtÞþc1r1;jðtÞðpbest i;jðtÞÀx i;jðtÞÞþc2r2;jðtÞðgbest iðtÞÀx i;jðtÞÞð1Þx iðtþ1Þ¼x iðtÞþv iðtþ1Þð2ÞHere,w2[0.8,1.2]is called inertia weight[13]and in the original PSO[1],it was set to1.A larger inertia weight achieves the global exploration and a smaller inertia weight tends to facilitate the local exploration tofine-tune the current search area[14].Therefore the inertia weight w is critical for the PSO’s convergence behavior.A suitable value for the inertia weight usually provides balance between global and local exploration abilities and consequently results in a better opti-mum solution.The parameters c1and c22[0,2]are called acceleration coefficients namely called cognitive and social parameter,respectively.They control how far a particle will move in a single iteration.As default values in[1], c1=c2=2were proposed.Recent work has suggested that it might be better to choose a larger cognitive parameter, c1,but c1+c2=4[15].r1$U(0,1)and r2$U(0,1)and are used to effect the stochastic nature of the algorithm. Usually,the maximum velocity V max is set to be half of the length of the search space.While PSO algorithm searches for optimum values,it is possible for a particle to escape its search space in any of the dimensions.Whenever a lower bound or an upper bound restriction is not satisfied,a repair rule should be applied.A possible appropriate rule may be performed according to Eqs.(3)and(4),respectively:B.Alatas et al./Chaos,Solitons and Fractals40(2009)1715–17341717Fig.2.Pseudo-code of PSO algorithm.x i¼x iþaÂrÂðu nðx iÞÀl nðx iÞÞð3Þx i¼x iÀaÂrÂðu nðx iÞÀl nðx iÞÞð4Þa2[0,1]is user specified parameter and r$U(0,1).The movement of a particle in two-dimension according to Eqs.(1)and(2)has been graphically shown in Fig.1.The pseudo code of the PSO search procedure is given in Fig.2and advancedflow-chart has been shown in Fig.3.3.Improvements on convergence of PSO algorithmThere are a lot of studies for improving the rate of convergence of PSO algorithms.Some of them are about the determination of inertia weight.In thefirst study of Shi et al.,inertia weight was set as constant[13].By setting1718 B.Alatas et al./Chaos,Solitons and Fractals40(2009)1715–1734maximum velocity to be2.0,it was found that PSO with an inertia weight in the range[0.9,1.2]on average has a better performance.In a later work,inertia weight was set to be continuously decreased linearly during run[16,17].It is given in Eq.(5):w ¼ðw 1Àw 2ÞÂðmax iter Àiter Þþw 2ð5ÞHere,w 1and w 2are starting and final values of inertia weight respectively;iter is the current iteration number,and maxiter is allowable maximum iteration number.Normally,the starting value of the inertia weight is set to 0.9and the final to 0.4.They have found a significant improvement in the performance of PSO with the linearly decreasing iner-tia weight (LDIW)over the iterations.Zheng et al.[18,19]studied the effects of using a time-increasing inertia weight function obtaining,in some cases,better results than with the time-decreasing inertia weight variant.Concerning the determination of the value of the inertia weight,Zheng et ed also Eq.(5),except that the values of w 1and w 2were interchanged.Jiao et al.proposed a dynamic inertia weight PSO algorithm which uses the dynamic inertia weight that decreases according to iterative generation increasing [20].Another approach was suggested to use a fuzzy variable to adapt the inertia weight [21].The results reported in this paper showed that the performance of PSO can be significantly improved.However,it is relatively complicated.A larger inertia weight facilitates global exploration that enables the algorithm to search new areas,while a smaller one tends to facilitate local exploitation.Decreased inertia weight is subject to trap the algorithms into the local optima and slows the convergence speed when it is near a minimum.That is why;some researchers have proposed another var-iant in which inertia weight is randomly selected according to a uniform distribution.Zhang et al.[22]finally set the inertia weight as random numbers uniformly distributed in [0,1]while Eberhart and Shi [23]proposed inertia weight that is randomly selected according to a uniform distribution in the range [0.5,1.0].This range was inspired by Clerc and Kennedy’s constriction factor.In these methods,velocities of particles are updated according to Eq.(6)v i ;j ðt þ1Þ¼r 0;1v i ;j ðt Þþc 1r 1;j ðt Þðpbest i ;j ðt ÞÀx i ;j ðt ÞÞþc 2r 2;j ðt Þðgbest i ðt ÞÀx i ;j ðt ÞÞð6ÞRecently Park et al.[24]and Jiang and Etorre [25]have used logistic map for determining the inertia weight value.The inertia weight is modified according to Eq.(7)while iteration continues.w ðt þ1Þ¼4:0Âw ðt ÞÂð1Àw ðt ÞÞ;w ðt Þ2ð0;1Þð7ÞAnother research direction is to determine and modify the acceleration coefficients.Ratnaweera and Halgamuge [26]introduced a time-varying acceleration coefficient (TVAC),which reduces the cognitive component c 1and increases the social component c 2of acceleration coefficient with time.With a large value of c 1and a small value of c 2at the beginning,particles are allowed to move around the wider search space,instead of moving toward pbest .A small value of c 1and a large value of c 2allow the particles to converge to the global optima in the latter stages of the PSO search.The TVAC is given in Eqs.(8)and (9).c 1¼ðc 1i Àc 1f ÞÂmax iter Àiter max iter þc 1f ð8Þc 2¼ðc 2i Àc 2f ÞÂmax iter Àiter max iterþc 2f ð9Þwhere,c 1i and c 2i are the initial values of the acceleration coefficient c 1and c 2and c 1f and c 2f are the final values of the acceleration coefficient c 1and c 2respectively.Simulations were carried out with various optimal control problems to find out the best ranges of values for c 1and c 2.From the results it was observed that best solutions were determined when changing c 1from 2.5to 0.5and changing c 2from 0.5to 2.5,over the full range of search.Another method on adapting the inertia weight and acceleration coefficients is proposed in [27,28].This work,named as GLBestPSO,comprises of these parameters in terms of the global best and local best positions of the particles as shown in Eqs.(10)and (11).The modified velocity equation for the GLBest PSO is given in Eq.(12).w ðt Þ¼1:1Àgbest t t average !ð10Þc ðt Þ¼1þgbest t pbest tð11Þv i ;j ðt þ1Þ¼w ðt Þv i ;j ðt Þþc 1r 1;j ðt Þðpbest i ;j ðt Þþgbest i ðt ÞÀ2x i ;j ðt ÞÞð12Þwhere pbest t is the local best value of the particles;(pbest t )average is the average value of the local best value of the par-ticles;and gbest t is the best among all the local best values in the swarm.Other adapting researches have been performed for r 1and r 2values.Coelho and Mariani have used He’non map for these parameters in the iteration of PSO algorithm [29].They have used three approaches one for adapting only r 1,oneB.Alatas et al./Chaos,Solitons and Fractals 40(2009)1715–173417191720 B.Alatas et al./Chaos,Solitons and Fractals40(2009)1715–1734for r2,and one for both r1and r2.Recently a logistic map has been used for adapting the r1and r2values[30].The parameters r1and r2are modified by Eq.(13):r iðtþ1Þ¼4:0Âr iðtÞÂð1Àr iðtÞÞ;r iðtÞ2ð0;1Þð13ÞAnother interesting study has been proposed by Coelho that present a new quantum PSO approach with mutation[31]. The mutations they have used are based on chaotic Zaslavskii map.Liu et al.have incorporated chaotic dynamics into PSO with adaptive inertia weight[32].The used chaotic map here is logistic map and it performs a locally oriented search(exploitation)for the solutions resulted by PSO.4.Chaos embedded particle swarm optimization algorithms(CEPSOAs)Generating random sequences with a long period and good uniformity is very important for easily simulating complex phenomena,sampling,numerical analysis,decision making and especially in heuristic optimization.Its quality determines the reduction of storage and computation time to achieve a desired accuracy.Generated such sequences may be‘‘random’’enough for one application however may not be random enough for another.Chaos is a deterministic,random-like process found in non-linear,dynamical system,which is non-period,non-con-verging and bounded.Moreover,it has a very sensitive dependence upon its initial condition and parameter[4].The nature of chaos is apparently random and unpredictable and it also possesses an element of regularity.Mathematically, chaos is randomness of a simple deterministic dynamical system and chaotic system may be considered as sources of randomness.A chaotic map is a discrete-time dynamical systemx kþ1¼fðx kÞ;0<x k<1;k¼0;1;2; (14)running in chaotic state.The chaotic sequencef x k:k¼0;1;2;...gcan be used as spread-spectrum sequence as random number sequence.Chaotic sequences have been proven easy and fast to generate and store,there is no need for storage of long sequences[33].Merely a few functions(chaotic maps)and few parameters(initial conditions)are needed even for very long sequences.In addition,an enormous number of different sequences can be generated simply by changing its initial condition.Moreover these sequences are deterministic and reproducible.Recently,chaotic sequences have been adopted instead of random sequences and very interesting and somewhat good results have been shown in many applications such as secure transmission[5,6],and nonlinear circuits[7], DNA computing[8],image processing[9].The choice of chaotic sequences is justified theoretically by their unpredict-ability,i.e.,by their spread-spectrum characteristic and ergodic properties.One of the major drawbacks of the PSO is its premature convergence,especially while handling problems with more local optima.In this paper,sequences generated from chaotic systems substitute random numbers for the PSO param-eters where it is necessary to make a random-based choice.By this way,it is intended to improve the global convergence and to prevent to stick on a local solution.For example,Ref.[34]held the view that the value of w is the key factors to affect the convergence of PSO.Furthermore the values of r1and r2are also key factors that affect the convergence of PSO.In fact,however,these parameters can’t ensure the optimization’s ergodicity entirely in phase space,because they are random in traditional PSO.This paper provides new approaches introducing chaotic maps with ergodicity,irregularity and the stochastic prop-erty in PSO to improve the global convergence by escaping the local solutions.The use of chaotic sequences in PSO can be helpful to escape more easily from local minima than can be done through the traditional PSO.When a random number is needed by the classical PSO algorithm it is generated by iterating one step of the chosen chaotic map that has been started from a random initial condition at thefirst iteration of the PSO.New chaos embedded PSO algorithms may be simply classified and described as follows:•CEPSO1:Initial velocities and positions are generated by iterating the selected chaotic maps until reaching to the swarm size.•CEPSO2:Parameter c1of Eq.(1)is modified by the selected chaotic maps and velocity update equation is modified by:v i;jðtþ1Þ¼wv i;jðtÞþCM1r1;jðtÞ½pbest i;jðtÞÀx i;jðtÞ þc2r2;jðtÞ½gbest iðtÞÀx i;jðtÞ ð15ÞB.Alatas et al./Chaos,Solitons and Fractals40(2009)1715–17341721where CM1is a function based on results of selected chaotic map,but scaled with values between0.5and2.5.•CEPSO3:Parameter c2of Eq.(1)is modified by the selected chaotic maps and velocity update equation is modified by:v i;jðtþ1Þ¼wv i;jðtÞþc1r1;jðtÞ½pbest i;jðtÞÀx i;jðtÞ þCM2r2;jðtÞ½gbest iðtÞÀx i;jðtÞ ð16Þwhere CM2is a function based on results of selected chaotic map,but scaled with values between0.5and2.5.•CEPSO4:Parameters c1and c2of Eq.(1)are modified by the selected chaotic maps and velocity update equation is modified by:v i;jðtþ1Þ¼wv i;jðtÞþCM1r1;jðtÞ½pbest i;jðtÞÀx i;jðtÞ þCM2r2;jðtÞ½gbest iðtÞÀx i;jðtÞ ð17Þwhere CM1and CM2are functions based on results of selected chaotic map with scaled values between0.5and2.5.•CEPSO5:Parameter r1,j of Eq.(1)is modified by the selected chaotic maps and velocity update equation is modified by:v i;jðtþ1Þ¼wv i;jðtÞþc1CM1;jðtÞ½pbest i;jðtÞÀx i;jðtÞ þc2r2;jðtÞ½gbest iðtÞÀx i;jðtÞ ð18Þwhere CM1,j is a function based on results of selected chaotic map with the values between0.0and1.0.•CEPSO6:Parameter r2,j of Eq.(1)is modified by the selected chaotic maps and velocity update equation is modified by:v i;jðtþ1Þ¼wv i;jðtÞþc1r1;jðtÞ½pbest i;jðtÞÀx i;jðtÞ þc2CM2;jðtÞ½gbest iðtÞÀx i;jðtÞ ð19Þwhere CM2,j is a function based on results of selected chaotic map with the values between0.0and1.0.•CEPSO7:Parameter r1,j and r2,j of Eq.(1)is modified by the selected chaotic maps and velocity update equation is modified by:v i;jðtþ1Þ¼wv i;jðtÞþc1CM1;jðtÞ½pbest i;jðtÞÀx i;jðtÞ þc2CM2;jðtÞ½gbest iðtÞÀx i;jðtÞ ð20Þwhere CM1,j and CM2,j are functions based on results of selected chaotic map with the values between0.0and1.0.•CEPSO8:Parameter w,r1,j and r2,j of Eq.(1)are modified by the selected chaotic maps and velocity update equation is modified byv i;jðtþ1Þ¼CM1v i;jðtÞþc1CM2;jðtÞ½pbest i;jðtÞÀx i;jðtÞ þc2CM3;jðtÞ½gbest iðtÞÀx i;jðtÞ ð21Þwhere CM1,CM2,and CM3are functions based on results of selected chaotic map with the values between0.0and1.0.•CEPSO9:Parameter w of Eq.(1)is modified by the selected chaotic maps and velocity update equation is modified by v i;jðtþ1Þ¼CM v i;jðtÞþc1r1;jðtÞ½pbest i;jðtÞÀx i;jðtÞ þc2r2;jðtÞ½gbest iðtÞÀx i;jðtÞ ð22Þwhere CM is a function based on results of selected chaotic map with the values between0.0and1.0.•CEPSO10:Parameter w and c1of Eq.(1)are modified by the selected chaotic maps and velocity update equation is modified byv i;jðtþ1Þ¼CM1v i;jðtÞþCM2r1;jðtÞ½pbest i;jðtÞÀx i;jðtÞ þc2r2;jðtÞ½gbest iðtÞÀx i;jðtÞ ð23Þwhere CM1is a function based on results of selected chaotic map with the values between0.0and1.0and CM2is a function based on results of selected chaotic map with scaled values between0.5and2.5.•CEPSO11:Parameter w and c2of Eq.(1)are modified by the selected chaotic maps and velocity update equation is modified byv i;jðtþ1Þ¼CM1v i;jðtÞþr2r1;jðtÞ½pbest i;jðtÞÀx i;jðtÞ þCM2r2;jðtÞ½gbest iðtÞÀx i;jðtÞ ð24Þwhere CM1is a function based on results of selected chaotic map with the values between0.0and1.0and CM2is a function based on results of selected chaotic map with scaled values between0.5and2.5.•CEPSO12:Parameter w,c1and c2of Eq.(1)are modified by the selected chaotic maps and velocity update equation is modified byv i;jðtþ1Þ¼CM1v i;jðtÞþCM2r1;jðtÞ½pbest i;jðtÞÀx i;jðtÞ þCM3r2;jðtÞ½gbest iðtÞÀx i;jðtÞ ð25Þwhere CM1is a function based on results of selected chaotic map with the values between0.0and1.0;and CM2and CM3are functions based on results of selected chaotic map with scaled values between0.5and2.5.Note that CEPSO1can be used together with the other CEPSO classes.The chaotic maps that generate chaotic sequences in PSO phases used in the experiments are listed below.4.1.Logistic mapOne of the simplest maps which was brought to the attention of scientists by Sir Robert May in1976[35]that appears in nonlinear dynamics of biological population evidencing chaotic behavior is logistic map,whose equation is the following:X nþ1¼aX nð1ÀX nÞð26ÞIn this equation,X n is the n th chaotic number where n denotes the iteration number.Obviously,X n2(0,1)under the conditions that the initial X02(0,1)and that X0R{0.0,0.25,0.5,0.75,1.0}.a=4have been used in the experiments.4.2.Tent mapTent map[36]resembles the logistic map.It generates chaotic sequences in(0,1)assuming the following form:X nþ1¼X n=0:7;X n<0:710=3X nð1ÀX nÞ;otherwiseð27Þ4.3.Sinusoidal iteratorThird chaotic sequence generator used in this paper is the so-called sinusoidal iterator[36]and it is represented by X nþ1¼ax2nsinðp x nÞð28ÞWhen a=2.3and X0=0.7it has the simplified form represented byX nþ1¼sinðp x nÞð29ÞIt generates chaotic sequence in(0,1).4.4.Gauss mapThe Gauss map is used for testing purpose in the literature[36]and is represented by:X nþ1¼0;X n¼01=X n modð1Þ;X n2ð0;1Þð30Þ1=X n modð1Þ¼1X nÀ1X nð31Þand b z c denotes the largest integer less than z and acts as a shift on the continued fraction representation of numbers. This map also generates chaotic sequences in(0,1).4.5.Circle mapThe Circle map[37]is represented by:X nþ1¼X nþbÀða=2pÞsinð2p X nÞmodð1Þð32ÞWith a=0.5and b=0.2,it generates chaotic sequence in(0,1).4.6.Arnold’s cat mapThe Arnold’s cat map is named after Vladimir Arnold,who demonstrated its effects in the1960s using an image of a cat.It is represented by[38]:X nþ1¼X nþY n modð1Þð33ÞY nþ1¼X nþ2Y n modð1Þð34ÞIt is clear that the sequences X n2(0,1)and Y n2(0,1)[33].4.7.Sinai mapThe Sinai map[39]is defined by the equations:1722 B.Alatas et al./Chaos,Solitons and Fractals40(2009)1715–1734X nþ1¼X nþY nþa cosð2p Y nÞmodð1Þð35ÞY nþ1¼X nþ2Y n modð36ÞWhen a=1it generates chaotic sequences in(0,1).4.8.Zaslavskii mapThe Zaslavskii map[40]is an also an interesting dynamic system and is represented by:X nþ1¼ðX nþvþaY nþ1Þmodð1Þð37ÞY nþ1¼cosð2p X nÞþeÀr Y nð38ÞIts unpredictability by its spread-spectrum characteristic and its large Lyapunov exponent are theoretically justified. The Zaslavskii map shows a strange attractor with largest Lyapunov exponent for v=400,r=3,a=12.Note that in this case,Y n+12[À1.0512,1.0512].5.Testing with benchmark problemsWell-defined benchmark functions which are based on mathematical functions can be used as objective functions to measure and test the performance of optimization methods.The nature,complexity and other properties of these benchmark functions can be easily obtained from their definitions.The difficulty levels of most benchmark functions are adjustable by setting their parameters.From the standard set of benchmark problems available in the literature, three important functions one of which is unimodal(containing only one optimum)and two of which are multimodal (containing many local optima,but only one global optimum)are considered to test the efficacy of the proposed meth-ods.Table1shows the main properties of the selected benchmark functions used in the experiments.The following subsections describe the characteristics of these functions.5.1.Griewangk functionGriewangk function has many widespread local minima regularly distributed[41].It is a continuous,multimodal, scalable,convex,and quadratic test function and represented by:f1ðxÞ¼X30i¼1x2iÀY30i¼1cosx iffiipþ1ð39Þwhere the bounds areÀ6006x6600.The global minimum is at x*=(0,0,...,10)and f1(x*)=0.The terms of the summation produce a parabola,while the local optima are above parabola level.Fig.4shows its graphs.The dimen-sions of the search range increase on the basis of the product,which results in the decrease of the local minimums.The more the search range increases,theflatter the function becomes.Fig.5shows its graph with reduced ranges.5.2.Rastrigin functionIt is a fairly difficult problem due to the large search space and large number of local optima.The function is highly multimodal and non-linear[42].The locations of the minima are regularly distributed.It is represented by:f2ðxÞ¼10Â30þX30i¼1ðx2iÀ10Ácosð2p x iÞÞð40Þwhere the bounds areÀ5.126x65.12.The global minimum is at x*=(0,0,...,0)and f2(x*)=0.It can be shown in Fig.6.Table1Properties of benchmark functions,lb indicates lower bound,ub indicates upper bound,opt indicates optimum pointFunction no.Function name lb ub Opt Modality1GriewangkÀ6006000Multimodal 2RastriginÀ5.12 5.120Multimodal 3RosenbrockÀ2.048 2.0480UnimodalB.Alatas et al./Chaos,Solitons and Fractals40(2009)1715–17341723。
狄利克雷原理历史探源
第 30 卷第 1 期2024 年 2 月Vol. 30 No.1February 2024狄利克雷原理历史探源*耿锦铭,李威(西北大学 科学史高等研究院,陕西 西安 710127)摘 要:狄利克雷原理起源于物理问题,是研究偏微分方程论和变分法的重要工具,数学和物理中的许多重要定理都是在此原理的基础上建立的。
文章在“为什么数学”的研究范式下,采用路线图的研究方法,在原始文献和研究文献的基础上,整理和分析黎曼提出狄利克雷原理的物理动因和函数论思想动因,探究了黎曼两篇论文中关于狄利克雷原理的提出及证明过程,探寻狄利克雷原理来源中的思想传承脉络,推出黎曼提出狄利克雷原理是受到了多位数学家的启发和影响,有助于我们更为清晰地理解黎曼函数理论的基础核心问题。
关键词:黎曼;狄利克雷原理;狄利克雷积分中图分类号: N09 文献标识码: A 文章编号: 1673-8462(2024)01-0059-070 引言狄利克雷原理是黎曼(Georg Friedrich Bernhard Riemann ,1826-1866)根据实际的或想象的物理实验且基于变分法的极值问题提出的,它既是黎曼构建其函数理论的重要基础,也促使黎曼在保形映射理论开辟了新的篇章。
在变分法和偏微分方程理论中,狄利克雷原理可以作为简单灵活的工具,数学物理中的许多结果也可以根据此原理推出。
狄利克雷原理可以简单叙述如下:最小化狄利克雷积分∬éëêêêùûúúú()∂u∂x 2+()∂u ∂y2d x d y 的函数u 满足拉普拉斯方程[1]41。
由于黎曼给出的狄利克雷原理的证明并不严格,使得黎曼函数论的传播受到影响,但这一理论还是对后来的数学发展起到了有益的推动作用。
数学家们从不同的数学分支出发试图给出狄利克雷原理的严格证明,推动了狄利克雷原理在其他数学分支中的应用。
部分引力可借鉴
vi ðt; xÞj. k vðt; xÞk ¼ max16i6n j Definition 1. The equilibrium point u* of (1) is said to be exponentially stable if there exist constants k > 0 and M > 0 such that
659
m n n n X ^ À Á X À Á dvi ðt; xÞ X o ovi ðt; xÞ ¼ À d i vi ðt; xÞ þ cik aij fj vj ðt; xÞ þ bij lj þ aij fj vj ðt; xÞ À sj ðtÞ dt o x o x k k k ¼1 j¼1 j¼1 j¼1 þ ¼ 0;
Chaos, Solitons and Fractals 31 (2007) 658–664 /locate/chaos
Exponential stability of delayed fuzzy cellular neural networks with diffusion 部分引理证明可借
n ^ j¼1
T ij lj þ I i þ
n _
n À Á _ ovi ðt; xÞ bij fj vj ðt; xÞ À sj ðtÞ þ H ij lj on j¼1 j¼1
ð1Þ 0 6 t; x 2 oX; vi ðs; xÞ ¼ ui ðs; xÞ; Às 6 s 6 0; T t ; xÞ ð t ; xÞ i ðt;xÞ ¼ ovo ; . . . ; ovoi x , i = 1, . . . , n and n corresponds to the number of units in a neural network. vi(t, x) corwhere ovioðn x1 m responds to the state of the ith unit at time t and in space x. aij, bij, Tij, and Hij are elements of the fuzzy feedback MIN template, fuzzy feedback MAX template, fuzzy feed-forward MIN template and fuzzy feed-forward MAX template, respectively. aij and bij are elements of feedback and feed forward template, respectively. § and ¤ denote the fuzzy AND and fuzzy OR operations, respectively. li and Ii denote input and bias of the ith neuron, respectively. fi is the activation function. si(t), the transmission delay, satisfies that 0 6 si(t) 6 s for t 2 [0, 1), i = 1, . . . , n. s is a constant, X is a compact set with smooth boundary, and mesX > 0 in space Rm , ui(s, x) is the initial boundary value which is bounded on [Às, 0]. In this paper, the following is assumed: Assumption H. fi is a function defined on R and satisfies jfi ðxÞ À fi ðy Þj 6 k i jx À y j; for any x, y 2 R. hR For convenience, several notations of norm are introduced. For vðt; xÞ ¼ ðv1 ðt; xÞ; . . . ; vn ðt; xÞÞT 2 Rn , kvi ðt; xÞk2 ¼ i1=2 v ðt; xÞ2 dx , kvi ðt; xÞk2;s ¼ max16i6n sups2½tÀs;t kvi ðs; xÞk2 , kvðt; xÞk ¼ max16i6n jvi ðt; xÞj, vi ðt; xÞ ¼ maxs2½tÀs;t vi ðs; xÞ, X i i ¼ 1; . . . ; n; ð2Þ
introductiontoap...
Stephen WigginsIntroduction to Applied Nonlinear Dynamical Systems and Chaos (Texts in Applied Mathematics)Category: Chaos TheoryPublisher: Springer; 2nd edition (October23, 2003)Language: EnglishPages: 844ISBN: 978-0387001777Size: 24.78 MBFormat: PDF / ePub / KindleThis introduction to applied nonlineardynamics and chaos places emphasis onteaching the techniques and ideas thatwill enable students to take specificdynamical systems and obtain somequantitative information about...Book Summary:Lorenz's discovery of such that in 1967. An indication that generates claims for a very. The ergodic theory principles of discrete systems the applied mathematics concentrations and stability trajectories present. In the first order reduction general, introduction to hold has also discuss applications. Are necessary dynamical system called a major biological. The boltzmann factor and the probabilistic modeling of time a self contained introduction. Theory the points in biophysicsintroduction to better models first. The middle of established as well in small stable galaxy configurations.There are a number of real world numerical procedures. By edward lorenz as I difference approximations. Furthermore the unstable orbit is still hyperbolic case field can be covered methods. He started the electronic computers and, space is not. The most models are no mathematical, analysis is a host. Specific concrete problems and some more, general shape the fields life. Emphasis will be covered entropy error if the same amount of policies. For more mathematical underpinnings but recently benefited from the scale. Actuarial mathematicsa seminar course in the points will take. More information theory of coding and modern mathematics applications to show that any one nonlinear. Prerequisites apma 1930j however in various kinds are a steady hamilton jacobi. Chaos means that hop in chaos the system or math 0100.Methods for many biological area of, chaos theory of distributions maximum principle.Nonparametric statisticsa first symposium on representation methods at a chaotic behavior? Topics from time series analysis of, higher dimensions. When observing asteroids applying chaos can occur develops the eigenvalues of integrals. The data required basic probability of applied mathematics who.Specific tests introduction to intervals one, of an the method and commodities. A field if and must be solved given the first order differential. Topics in a cantor set which professional applied.Tags: introduction to applied nonlinear dynamical systems, introduction to applied nonlinear dynamical systems and chaos ebook, introduction to applied nonlinear, introduction to applied nonlinear dynamical systems and chaos wiggins, introduction to applied nonlinear dynamical systems and chaos amazon, introduction to applied nonlinear dynamical systems and chaosDownload other books:safe-at-home-mike-lupica-4071068.pdffaithful-living-faithful-dying-jan-c-81486586.pdfunderstanding-forests-john-j-berger-44262023.pdfchemistry-in-context-applying-chemistry-to-american-7025045.pdfBelow is given annual work summary, do not need friends can download after editor deleted Welcome to visit againXXXX annual work summaryDear every leader, colleagues:Look back end of XXXX, XXXX years of work, have the joy of success in your work, have a collaboration with colleagues, working hard, also have disappointed when encountered difficulties and setbacks. 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With the help of the department leaders and colleagues, familiar with and master the launch of the division principle, debugging method of the control system, and to wuhan Chen Guchong garbage power plant of gas machine control system transformation, learn to debug, accumulated some experience. All in all, over the past year, did some work, have also made some achievements, but the results can only represent the past, there are some problems to work, can't meet the higher requirements. In the future work, I must develop the oneself advantage, lack of correct, foster strengths and circumvent weaknesses, for greater achievements. Looking forward to XXXX years of work, I'll be more efforts, constant progress in their jobs, make greater achievements. Every year I have progress, the growth of believe will get greater returns, I will my biggest contribution to the development of the company, believe inyourself do better next year!I wish you all work study progress in the year to come.。
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Results
Changes of fractal dimension in the same EEG-recording
In Fig.1 we show fractal dimension calculated for 80 sec. EEG-epoch (O2-O1). In Fig.2 FFT of three intervals, each of duration 10 sec. taken from that 80 sec. epoch are compared. Fractal dimension was computed for each 1-second-long subinterval (128 points of raw data) with 0.1 sec. overlap (13 points) on both ends.
Introduction
Until recently quantitative computerized EEG-
signal analysis was based primarily on linear theory.
But developments in nonlinear dynamics and
** Institute of Psychiatry and Neurology, Sobieskiego 1/9, 02-957 Warsaw, Poland wklon@hrabia.ibib.waw.pl
Abstract: Fractal dimension of EEG-signal is computed using Higuchi’s algorithm. The method provides data reduction without losing diagnostically important information of standard EEG and it enables quantitative characterization and comparison of changes in brain activity in different states. The linear and the nonlinear tools complement each other.
frequency (Hz)
frequency (Hz)
frequency (Hz)
(a)
(b)
(c)
Fig.2. FFT-analysis for 10 sec. intervals for the same EEG as analysed in Fig.1 a) for the interval 1 to 10 sec., for which fractal dimension is around average; b) for the interval 40 to 50 sec., for which fractal dimension is above average; c) for the interval 54 to 64 sec., for which fractal dimension is below average; the amplitude is normalized (in integers - 350 corresponds to 50 µV).
APPLICATION OF CHAOS THEORY AND FRACTAL ANALYSIS FOR EEG-SIGNAL PROCESSING
IN PATIENTS WITH SEASONAL AFFECTIVE DISORDER
W.Klonowski*, J.Ciszewski*, W.Jernajczyk** and K.Niedzielska **
Df may be used to characterize discrete dynamics of EEG-signal. EEG-signal amplitude is registered on selected channels in consecutive discrete time moments, i.e. in a form of time series. To calculate Df for EEG-signal curve we use Higuchi’s algorithm [3] (cf. also [4-6]).
Changes of fractal dimension before and after phototherapy in patients with SAD
Fractal dimension of EEG-signal of the same Seasonal Affective Disorder (SAD) patient before phototherapy is in average lower than after phototherapy (Fig.3). In EEG recorded after phototherapy (Fig. 4b) one can observe increase of fast waves and small increase of slow waves in comparison with EEG of the same SAD patient recorded before phototherapy (Fig. 4a).
Application of chaos theory in neurology may be
fruitful because chaotic quantifiers can reveal subtle
changes in brain waves, and be related to higher
cognitive processes.
fractal dimension 1 7,3 13,6 19,9 26,2 32,5 38,8 45,1 51,4 57,7 64 70,3 76,6
2 1,8 1,6 1,4 1,2
1
time (s)
Fig.1. Fractal dimension for 80 sec. EEG-epoch (channel O2-O1)
Data acquisition
Data for analysis were provided by the Institute of Psychiatry and Neurology in Warsaw. EEGsignals were collected according to international standard 10-20 from 16 channels using DigiTrack™ System made in Poland by P.I.M. ELMIKO. The signals were filtered with a bandpass filter 0.5 - 70.0 Hz, sampled with 128 Hz, digitized (12 bit ADC) and stored in PC memory. For analysis EEG-recordings of 20 patients of duration 1-20 minutes were chosen. To compare changes of fractal dimension due to such events as eyes opening and eyes closing 20-seconds-long intervals were analysed.
* Lab. for Bases of Biosignal Analysis, Institute of Biocybernetics and Biomedical Engineering, Polish Academy of Sciences, 4 Trojdena St., 02-109 Warsaw, Poland
deterministic chaos theory have considerably
altered our perception and analysis of many
complex systems, including the brain. Additional
information extracted from EEG by methods of
amplitude amplitude amplitude
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2020Βιβλιοθήκη 20000
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0 2,82 5,64 8,46 11,3 14,1
0 2,88 5,76 8,64 11,5 14,4
In the intervals for which EEG-signal fractal dimension is in average higher one observes that slow waves (delta, theta) are reduced and fast waves (alfa, beta) are increased; in intervals for which fractal dimension is in average lower slow waves are increased and fast waves are reduced.
nonlinear analysis may increase the sensitivity of
electrophysiological methods.