偏微分方程和积分系统的混沌研究
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摘要
混沌(chaos)是非线性科学研究的中心课题之一,是非线性动力系统普遍存在的一种运动形式。同时,混沌研究对非线性动力学的发展起着全局性、本质性的影响。非线性动力学的某些研究一开始就与混沌探索联系在一起。但是,直到20世纪50年代末,混沌理论创立之前,混沌概念还是极其模糊的。即使现在,不同领域对混沌的理解也很不相同。但就一般而言,混沌是指在确定性的系统中,不需要附加任何随机因素亦可出现的类似随机的动力学行为。混沌系统的最大特点就在于系统的演化对初始条件十分敏感。因此,从长期意义上讲,系统的行为是不可预测的。
关于动力系统混沌的研究吸引了许多科学家和数学家的兴趣。1975年,Li与Yorke[1]研究了连续区间映射,得到了一个著名的结果:“周期3蕴含混沌”。该判别定理在研究一维离散动力系统的混沌问题时有非常重要的作用。在该文中他们首次给出混沌的一个数学定义。之后,出现了几个不同的混沌定义[2-5]。有些较强而有些较弱,依赖于对不同问题研究的需要。1978年,F.R.Marotto受Li与Yorke工作的启发,把Li-Yorke定理推广到n维情况,并给出了扩张不动点和返回扩张不动点的概念。Marotto定理[6,定理3.1]证明了返回扩张不动点导致Li-Yorke意义下混沌。最近,史玉明和陈关荣抓住了扩张不动点和返回扩张不动点的本质,把Marotto对R~n中连续可微映射定义的扩张不动点和返同扩张不动点的概念推广到了一般度量空间中[7]。并且她们建立了几个完备度量空间上离散动力系统的混沌判定定理[7,8]。许多动力系统模型导出的偏微分疗程或积分方程常在Banach空间中进行研究。
在力学和电学系统中对非线性振荡的研究一直以来都是科学家和工程师们研究的重要课题[9]。最近几年,这类研究更多的集中在了混沌现象的研究。但对于由偏微分方程控制的力学系统的混沌振荡的数学研究却不多。一般来说,偏微分方程的研究需要更深的数学知识和技巧。偏微分方程有很多不同的类型。在本文第二章,我们只考虑一端带有线性边值条件,一端带有非线性边值条件的振动弦的混沌振荡问题。
积分方程是近代数学的一个重要分支。它在力学、数学物理和工程等领域有重要应用。同时我们知道在对积分方程进行迭代求其近似解时,会出现一些奇怪的现象,即对初始值作微小改变,其近似解会有很大不同,甚至有时其近似解图象会极其混乱,即现在我们称为的混沌现象。本文的第三章建立积分系统的混沌判定定理,对上述现象可以给出一个合理的解释。
本文主要讨论两方面的问题:一是波动方程的混沌问题,二是Banach空间上的一类混沌离散系统的微扰问题。本文由三章组成,主要内容如下:
第一章概述了混沌研究的进展,给出了一些预备知识,其中包括了几个数学上常用的混沌定义,以及动力系统中的一些基本概念,并回顾了目前已经取得的关于离散动力系统混沌判定的一些结果。
第二章主要讨论波动方程的混沌振荡问题。波动方程是在工程和力学中一类非常重要的偏微分方程。本章考虑一维弦振动ω_(tt)-ω_(xx)=0,x∈(0,1)。它是一个无限维调和振荡,其中左端x=0处,满足一个线性边值条件;有端x=1处,满足非线性边值条件。首先,我们把这个问题转化为一阶双曲系统,利用特征线法把问题转化为一个特殊的离散系统,u_(n+1)=H_0(u_n)。从而,把问题转化为一维空间上的混沌问题进行研究。如果映射H_0在区间上混沌,我们就说该波动方程的边值问题混沌。另外,为了说明所获得结果,本章给出了一个例子,并给出了计算机仿真。
第三章主要研究Banach空间上的一类混沌离散系统的微扰问题。首先我们讨论一类混沌离散系统的微扰问题。因为混沌系统不是结构稳定的,即对一个混沌系统加一个小的扰动,扰动系统可能混沌,也可能不混沌。这里我们主要讨论了当原系统具有正则非退化返回扩张不动点时,加一个小扰动可使得系统依然还是一个混沌系统。然后,将所得结果应用于三类积分系统的混沌判定的研究。
Chaos is one of central topics research on the nonlinear science and is a universal dynamical behavior of nonlinear dynamical systems.Meanwhile, it has a global and essential effect on the development of nonlinear dynamics, and some original research works of nonlinear dynamics were connected with chaos.However,before the end of 1950's and the establishment of chaos theory,the concept of chaos was very ambiguous.Even now,there are different understandings of chaos in different fields.In general,chaos means a random-like behavior(intrinsic randomness)in deterministic systems without any stochastic factors.The most important characteristic of a chaotic system is that its evolution has highly sensitive dependence on initial conditions.So, from a long-term perspective,future behaviors of a chaotic system are unpredictable.
Research on chaos in dynamical systems has attracted a lot of attention from many scientists and mathematicians.In 1975,Li and Yorke[1]invesrigated a continuous map on a interval and obtained the well-known result: "period 3 implies chaos".This criterion plays an important rule in studying chaos problems of one-dimensional discrete dynamical systems.They are the first ones who introduced a mathematical definition of chaos.Later,there appeared several different definitions of chaos[2-5];some are stronger and some are weaker,depending on requirements in studying different problems. In 1978,F.R.Marotto was inspired by the work of Li-Yorke,generalized the Li-Yorke's theorem to n-dimensional real space,and introduced the concepts of expanding fixed point and snap-back repeller.The Marotto's theorem[6, Theorem 3.1]proved that a snap-back repeller implies chaos in the sense of Li-Yorke.Recently,Y.Shi and G.Chen captured the essential meanings of the expanding fixed point and snap-back repeller,and generalized these two concepts for the continuously differentiable maps in R~n to maps in general metric spaces.And they established several criteria of chaos in discrete dynamical systems on complete metric spaces[7,8].Many partial differential equations and intergral equations,which are deduced by models of dynamical systems, are often studied on Banach spaces.
Nonlinear vibrations in mechanical and electronic systems are always an important topic studied by scientists and engineers[9].In recent years,some study on this problem was focused on chaotic phenomena.But it seems that there are a little results obtained in the mathematical study of chaotic vibrations in mechanical systems governed by partial differential equations(PDEs) containing nonlinearities.In general,more mathematical knowledges and techniques are required to study dynamical behaviors of PDEs.PDEs have many different types.Here,we will only study chaotic vibrations of a vibrating string with nonlinear boundary conditions in Chapter 2.
Integral equations are an important branch of the advanced mathematics. It has very important applications to mechanics,mathematical physics, engineering,etc.Moreover,it is well-known that the integral equation can be iterated to find its numerical solution.But in some cases there may be some strange phenomenon that a small change on the initial value may result in a great difference in approximate solution,even the graph of the solution may be complicated,which is now called a chaotic phenomenon.In Chapter 3,we will establish several criteria of chaos for integral systems.These results may give an explanation for the above mentioned problem.
In this dissertation,we mainly study chaos problems on vibrations of the one-dimensional wave equation and small perturbation of a class of chaotic discrete systems on Banach spaces.This dissertation consists of three chapters. Their main contents are briefly introduced as follows.
In Chapter 1,we summarize the development of chaos and give some preliminaries,including several basic definitions of chaos that often used in mathematics,some other concepts in discrete dynamical systems,and several criteria of chaos.
In Chapter 2,we study the chaotic vibration of a one-dimensional wave equation.The wave equation is an important type of the partial differential equations in the study of engineering technology and mechanics.In this chapter, we consider the one-dimensional vibrating string satisfying w_(tt)-w_(xx)=0. x∈(0,1).It is an infinite-dimensional harmonic oscillator with the boundary conditions:at.the left end x=0,it satisfies a linear condition,while at the right,end x=1,it satisfies a nonlinear boundary condition.First,the problem is reformulated into an equivalent first-order hyperbolic system.By the method of characteristics,the problem is reduced to a discrete iteration problem:u_(n+1)=H_0(u_n),where H_0 is an interval map.This PDE problem is said to be chaotic if the map H_0 is chaotic.In addition,we provide an example with computer simulations for illustration.
In Chapter 3,we study a small perturbation of a class of chaotic discrete systems on Banach spaces.First,we discuss the small perturbation of a class of chaotic discrete systems.In general,a chaotic system is not structurally stable;that is,a chaotic system with a small perturbation may be chaotic or not.Here,we discuss a system is still chaotic by a small perturbation,where the system has a regular nondegenerate snap-back repeller.Second,we emply the obtaiu results to study chaotic problems of three classes of integral systems on Banach spaces and establish some criteria of chaos.
关于动力系统混沌的研究吸引了许多科学家和数学家的兴趣。1975年,Li与Yorke[1]研究了连续区间映射,得到了一个著名的结果:“周期3蕴含混沌”。该判别定理在研究一维离散动力系统的混沌问题时有非常重要的作用。在该文中他们首次给出混沌的一个数学定义。之后,出现了几个不同的混沌定义[2-5]。有些较强而有些较弱,依赖于对不同问题研究的需要。1978年,F.R.Marotto受Li与Yorke工作的启发,把Li-Yorke定理推广到n维情况,并给出了扩张不动点和返回扩张不动点的概念。Marotto定理[6,定理3.1]证明了返回扩张不动点导致Li-Yorke意义下混沌。最近,史玉明和陈关荣抓住了扩张不动点和返回扩张不动点的本质,把Marotto对R~n中连续可微映射定义的扩张不动点和返同扩张不动点的概念推广到了一般度量空间中[7]。并且她们建立了几个完备度量空间上离散动力系统的混沌判定定理[7,8]。许多动力系统模型导出的偏微分疗程或积分方程常在Banach空间中进行研究。
在力学和电学系统中对非线性振荡的研究一直以来都是科学家和工程师们研究的重要课题[9]。最近几年,这类研究更多的集中在了混沌现象的研究。但对于由偏微分方程控制的力学系统的混沌振荡的数学研究却不多。一般来说,偏微分方程的研究需要更深的数学知识和技巧。偏微分方程有很多不同的类型。在本文第二章,我们只考虑一端带有线性边值条件,一端带有非线性边值条件的振动弦的混沌振荡问题。
积分方程是近代数学的一个重要分支。它在力学、数学物理和工程等领域有重要应用。同时我们知道在对积分方程进行迭代求其近似解时,会出现一些奇怪的现象,即对初始值作微小改变,其近似解会有很大不同,甚至有时其近似解图象会极其混乱,即现在我们称为的混沌现象。本文的第三章建立积分系统的混沌判定定理,对上述现象可以给出一个合理的解释。
本文主要讨论两方面的问题:一是波动方程的混沌问题,二是Banach空间上的一类混沌离散系统的微扰问题。本文由三章组成,主要内容如下:
第一章概述了混沌研究的进展,给出了一些预备知识,其中包括了几个数学上常用的混沌定义,以及动力系统中的一些基本概念,并回顾了目前已经取得的关于离散动力系统混沌判定的一些结果。
第二章主要讨论波动方程的混沌振荡问题。波动方程是在工程和力学中一类非常重要的偏微分方程。本章考虑一维弦振动ω_(tt)-ω_(xx)=0,x∈(0,1)。它是一个无限维调和振荡,其中左端x=0处,满足一个线性边值条件;有端x=1处,满足非线性边值条件。首先,我们把这个问题转化为一阶双曲系统,利用特征线法把问题转化为一个特殊的离散系统,u_(n+1)=H_0(u_n)。从而,把问题转化为一维空间上的混沌问题进行研究。如果映射H_0在区间上混沌,我们就说该波动方程的边值问题混沌。另外,为了说明所获得结果,本章给出了一个例子,并给出了计算机仿真。
第三章主要研究Banach空间上的一类混沌离散系统的微扰问题。首先我们讨论一类混沌离散系统的微扰问题。因为混沌系统不是结构稳定的,即对一个混沌系统加一个小的扰动,扰动系统可能混沌,也可能不混沌。这里我们主要讨论了当原系统具有正则非退化返回扩张不动点时,加一个小扰动可使得系统依然还是一个混沌系统。然后,将所得结果应用于三类积分系统的混沌判定的研究。
Chaos is one of central topics research on the nonlinear science and is a universal dynamical behavior of nonlinear dynamical systems.Meanwhile, it has a global and essential effect on the development of nonlinear dynamics, and some original research works of nonlinear dynamics were connected with chaos.However,before the end of 1950's and the establishment of chaos theory,the concept of chaos was very ambiguous.Even now,there are different understandings of chaos in different fields.In general,chaos means a random-like behavior(intrinsic randomness)in deterministic systems without any stochastic factors.The most important characteristic of a chaotic system is that its evolution has highly sensitive dependence on initial conditions.So, from a long-term perspective,future behaviors of a chaotic system are unpredictable.
Research on chaos in dynamical systems has attracted a lot of attention from many scientists and mathematicians.In 1975,Li and Yorke[1]invesrigated a continuous map on a interval and obtained the well-known result: "period 3 implies chaos".This criterion plays an important rule in studying chaos problems of one-dimensional discrete dynamical systems.They are the first ones who introduced a mathematical definition of chaos.Later,there appeared several different definitions of chaos[2-5];some are stronger and some are weaker,depending on requirements in studying different problems. In 1978,F.R.Marotto was inspired by the work of Li-Yorke,generalized the Li-Yorke's theorem to n-dimensional real space,and introduced the concepts of expanding fixed point and snap-back repeller.The Marotto's theorem[6, Theorem 3.1]proved that a snap-back repeller implies chaos in the sense of Li-Yorke.Recently,Y.Shi and G.Chen captured the essential meanings of the expanding fixed point and snap-back repeller,and generalized these two concepts for the continuously differentiable maps in R~n to maps in general metric spaces.And they established several criteria of chaos in discrete dynamical systems on complete metric spaces[7,8].Many partial differential equations and intergral equations,which are deduced by models of dynamical systems, are often studied on Banach spaces.
Nonlinear vibrations in mechanical and electronic systems are always an important topic studied by scientists and engineers[9].In recent years,some study on this problem was focused on chaotic phenomena.But it seems that there are a little results obtained in the mathematical study of chaotic vibrations in mechanical systems governed by partial differential equations(PDEs) containing nonlinearities.In general,more mathematical knowledges and techniques are required to study dynamical behaviors of PDEs.PDEs have many different types.Here,we will only study chaotic vibrations of a vibrating string with nonlinear boundary conditions in Chapter 2.
Integral equations are an important branch of the advanced mathematics. It has very important applications to mechanics,mathematical physics, engineering,etc.Moreover,it is well-known that the integral equation can be iterated to find its numerical solution.But in some cases there may be some strange phenomenon that a small change on the initial value may result in a great difference in approximate solution,even the graph of the solution may be complicated,which is now called a chaotic phenomenon.In Chapter 3,we will establish several criteria of chaos for integral systems.These results may give an explanation for the above mentioned problem.
In this dissertation,we mainly study chaos problems on vibrations of the one-dimensional wave equation and small perturbation of a class of chaotic discrete systems on Banach spaces.This dissertation consists of three chapters. Their main contents are briefly introduced as follows.
In Chapter 1,we summarize the development of chaos and give some preliminaries,including several basic definitions of chaos that often used in mathematics,some other concepts in discrete dynamical systems,and several criteria of chaos.
In Chapter 2,we study the chaotic vibration of a one-dimensional wave equation.The wave equation is an important type of the partial differential equations in the study of engineering technology and mechanics.In this chapter, we consider the one-dimensional vibrating string satisfying w_(tt)-w_(xx)=0. x∈(0,1).It is an infinite-dimensional harmonic oscillator with the boundary conditions:at.the left end x=0,it satisfies a linear condition,while at the right,end x=1,it satisfies a nonlinear boundary condition.First,the problem is reformulated into an equivalent first-order hyperbolic system.By the method of characteristics,the problem is reduced to a discrete iteration problem:u_(n+1)=H_0(u_n),where H_0 is an interval map.This PDE problem is said to be chaotic if the map H_0 is chaotic.In addition,we provide an example with computer simulations for illustration.
In Chapter 3,we study a small perturbation of a class of chaotic discrete systems on Banach spaces.First,we discuss the small perturbation of a class of chaotic discrete systems.In general,a chaotic system is not structurally stable;that is,a chaotic system with a small perturbation may be chaotic or not.Here,we discuss a system is still chaotic by a small perturbation,where the system has a regular nondegenerate snap-back repeller.Second,we emply the obtaiu results to study chaotic problems of three classes of integral systems on Banach spaces and establish some criteria of chaos.
引文
[1]T.Li.J.A,Yorke."Period three implies chaos",Amer.Math.Monthly,1975,82:985-992.
[2]J.Banks,J.Brooks,G.Cairns,G.Davis,P.Stacey."On Devaney's definition of chaos",Amer.Math,Monthly,1992.99:332-334.
[3]R.Devaney."An Introduction to Chaotic Dynamical Systems",2nd ed.Redwood City:Addison-Wesley Publishing Company,1989.
[4]M.Martelli,M.Dang,T.Seph."Defining chaos".Math.Magazine,1998,71:112-122.
[5]C.Robinson."Dynamical Systems:Stability,Symbolic Dynamics and Chaos",Florida:CRC Press.1995
[6]F.R.Marotto."Snap-back repellers imply chaos in R~n",J.Math.Anal.Appl.,1978,71:112-122.
[7]Y.Shi,G.Chen."Chaos of discrete dynamical systems in complete metric spaces",Chaos,Solitons and Fractals,2004,22:555-571.
[8]Y.Shi,P.Yu."Study on chaos induced by turbulent maps in noncompact sets",Chaos,Solitons and Fractals.2006,28:1165-1180.
[9]J.J.Stoker."Nonlinear vibrations",Wiley-Interscience,New York,1950.MR 11:666a.
[10]D.Ruelle,F.Takens."On the nature of turbulence",Commun.Math.Phys.,1971,20:167-192.
[11]R.M.May."Simple mathematical models with very compliticated dynamics".Nature,1976,261:159-167.
[12]B.Aulbach.B.Kieninger,"On three definitions of chaos".Nonlinear Dynamics and Systems Theory,2001,1:23-37.
[13]Y.Shi,P.Yu." Chaos induced by regular snap-back repeller",J.Math.Anal.Appl.,2008,337:1480-1494.
[14]P.Touhey."Yet another definition of chaos",Amer.Math.Monthly,1997,104:411-414.
[15]S.Wiggins."Introduction to Applied Nonlinear Dynamical Systems and Chaos",Springer-Verlag,New York,1990.
[16]周作领.符号动力系统.上海科技教育出版社,上海,1997.
[17]W.Huang,X.Ye."Devaney's chaos or 2-scattering implies Li-Yorke's chaos",Topology and its Applications,2002,117:259-272.
[18]Y.Shi,P.Yu."On chaos of logistic maps",Dynamics of continuous,Discrete and Impulsive systems,ser.B,2007,14:175-196.
[19]Y.Shi G.Chen."Discrete chaos in Banach spaces",Science in China,Ser.A:Mathematics,Chinese version:2004b,34:595-609;English version:2005,48:222-238.
[20]G.R.Chen,X.Dong."From chaos to order-perspectives and method-dogies in controlling chaotic nonlinear dynamical systems",Int.J.Bifurcation and Chaos,1993,3:1363-1409.
[21]F.C.Moon."Chaotic Vibrations",Wiley-Interscience.New York,1987.
[22]G.Chen,S.B.Hsu,J.Zhou."Chaotic vibration of the one-dimensional wave equation due to a self-excitation boundary condition,Part Ⅰ,controlled hysteresis",Transactions Amer.Math,Soc.,1998,350:4265-4311.
[23]G.Chen,S.Hsu,J.Zhou."Chaotic vibration of the one-dimensional wave equation due to a self-excitation boundary condition,Part Ⅱ,energy injection,period doubling and homoclinic orbits",Int,J.Bifurcation and Chaos.1998,8:423-445.
[24]G.Chen,S.Hsu,J.Zhou."Chaotic vibration of the one-dimensional wave equation due to a self-excitation boundary conditionIbid,Part Ⅲ,natural hysteresis memory effects",Int.J.Bifurcation and Chaos,1998,8:447-470.
[25]G.Chen.S.Hsu,J.Zhou."Snapback repellers as a cause of chaotic vibration of the wave equation with a van der Pol boundary condition and energy injection at the middle of the span",J.Math,Phys.,1998,39:6459-6489.
[26]Y.Huang."Growth rates of total variations of snapshots of the 1D linear wave equation with composite nonlinear boundary reflection relations",Int.J.Bifurcation,and Chaos,2003,13:1183-1195.
[27]J.Fourier."The Analytical Theory of Heat",Dover,New York,1955,reprinted from the 1878 ED.
[28]N.H.Abel."Oeuvres Completes.2 vols",Christiania,Oslo,1823,1:11-27.
[29]D.IIilbert."Grundzuge ciner Allgemeinen Theorie der Linearen Integralagleichungen",Teubner,Leipzig,1912.reprinted by Chelsea,New York,1953.
[30]K.Shiraiwa,M.Kurata,"A generalization of a theorem of Marotto",Proc.Janpan Acad.,1979,55:286-289.
[31]S.Smale."Diffeomorphisms with many periodic points.Differntial and Combinatorial Topology",Princetin Univerdily Press,Press,Princeton,NJ,1968,63-80.
[32]Y.Shi,P.Yu.G.Chen."Chaotification of diserete dynamical systems in Banach spaces",Int.J.Bifurcation and Chaos,2006,16:2615-2636.
[2]J.Banks,J.Brooks,G.Cairns,G.Davis,P.Stacey."On Devaney's definition of chaos",Amer.Math,Monthly,1992.99:332-334.
[3]R.Devaney."An Introduction to Chaotic Dynamical Systems",2nd ed.Redwood City:Addison-Wesley Publishing Company,1989.
[4]M.Martelli,M.Dang,T.Seph."Defining chaos".Math.Magazine,1998,71:112-122.
[5]C.Robinson."Dynamical Systems:Stability,Symbolic Dynamics and Chaos",Florida:CRC Press.1995
[6]F.R.Marotto."Snap-back repellers imply chaos in R~n",J.Math.Anal.Appl.,1978,71:112-122.
[7]Y.Shi,G.Chen."Chaos of discrete dynamical systems in complete metric spaces",Chaos,Solitons and Fractals,2004,22:555-571.
[8]Y.Shi,P.Yu."Study on chaos induced by turbulent maps in noncompact sets",Chaos,Solitons and Fractals.2006,28:1165-1180.
[9]J.J.Stoker."Nonlinear vibrations",Wiley-Interscience,New York,1950.MR 11:666a.
[10]D.Ruelle,F.Takens."On the nature of turbulence",Commun.Math.Phys.,1971,20:167-192.
[11]R.M.May."Simple mathematical models with very compliticated dynamics".Nature,1976,261:159-167.
[12]B.Aulbach.B.Kieninger,"On three definitions of chaos".Nonlinear Dynamics and Systems Theory,2001,1:23-37.
[13]Y.Shi,P.Yu." Chaos induced by regular snap-back repeller",J.Math.Anal.Appl.,2008,337:1480-1494.
[14]P.Touhey."Yet another definition of chaos",Amer.Math.Monthly,1997,104:411-414.
[15]S.Wiggins."Introduction to Applied Nonlinear Dynamical Systems and Chaos",Springer-Verlag,New York,1990.
[16]周作领.符号动力系统.上海科技教育出版社,上海,1997.
[17]W.Huang,X.Ye."Devaney's chaos or 2-scattering implies Li-Yorke's chaos",Topology and its Applications,2002,117:259-272.
[18]Y.Shi,P.Yu."On chaos of logistic maps",Dynamics of continuous,Discrete and Impulsive systems,ser.B,2007,14:175-196.
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