亚洲免费av电影一区二区三区,日韩爱爱视频,51精品视频一区二区三区,91视频爱爱,日韩欧美在线播放视频,中文字幕少妇AV,亚洲电影中文字幕,久久久久亚洲av成人网址,久久综合视频网站,国产在线不卡免费播放

        ?

        Four kinds of gradient representations of autonomous Birkhoffian systems

        2017-05-25 00:37:21CUIJinchaoLIAOCuicuiMEIFengxiang
        關(guān)鍵詞:國家自然科學(xué)基金責(zé)任編輯梯度

        CUI Jin-chao,LIAO Cui-cui,MEI Feng-xiang

        (1.School of Science,Jiangnan University,Wuxi Jiangsu214122,China; 2.School of Aerospace Engineering,Beijing Institute of Technology,Beijing100081,China)

        Four kinds of gradient representations of autonomous Birkhoffian systems

        CUI Jin-chao1,LIAO Cui-cui1,MEI Feng-xiang2

        (1.School of Science,Jiangnan University,Wuxi Jiangsu214122,China; 2.School of Aerospace Engineering,Beijing Institute of Technology,Beijing100081,China)

        In order to study the integration and the stability of autonomous Birkhoffian systems,we propose four kinds of gradient systems to represent the autonomous Birkhoffian systems.By analysing the relationship between the gradient systems and the Birkhoffian systems,we obtain the conditions that the Birkhoffian systems can be transformed into a kind of four gradient systems.Then,we use the properties of gradient system to investigate the problems of integration and stability of the Birkhoffian systems.Finally,we give some examples to illustrate the application of the theory.

        Birkhoffian systems;gradient system;integration;stability

        0 Introduction

        In Ref.[1],the author pointed out that it is especially suitable to study the gradient systems by using of Lyapunov functions.More general linear-gradient system has already been put forward in Ref.[2].In addition to usual gradient system,there are three other kinds of gradient systems,i.e.,the skew-gradient system,the symmetric negative def i nite gradientsystem,and the semi-def i nite gradient system.Thus,there are four kinds of gradient systems, which have important values for us to study integral and stability of solution of the system. For the applications of gradient system on mechanical systems,there have been some results already,such as Refs.[3-8].Based on those researches,the paper mainly tries to transform an autonomous Birkhoffian system[9-10]into one of the four kinds of gradient systems with some conditions,and then use the properties of four kinds of gradient system to study the problems of integration and stability of the Birkhoffian systems.

        1 Four kinds of gradient systems and Birkhof f’s equations

        The differential equations of motion of the general gradient systems are

        whereV(x)is called potential function andx=(x1,x2,···,xm).The general gradient systems consist of the following importance properties∶

        1)The functionVis a Lyapunov function of the system(1)and˙V=0 if and only ifx=(x1,x2,···,xm)is an equilibrium point.

        2)For the linear system of the gradient system(1),there are only real characteristic roots at any equilibrium points.

        Similarly,the skew-gradient systems can be formula as

        wherebij(x)=-bji(x)and the unified subscript denotes summation convention as well as in the following text.In addition,V=V(x)is called the energy function.The skew-gradient systems have the following importance properties∶

        1)The functionV=V(x)is an integration of the system(2).

        2)IfVis a Lyapunov function,the solution of the system(2)is stable.

        The differential equations of the gradient systems with symmetric negative def i nite matrices are[2]

        whereSij=Sij(x)is symmetric negative def i nite matrix.This system has the property as follows∶IfVis a Lyapunov function,the solution of the system(3)is stable.

        The differential equations of the gradient systems with negative semi-def i nite matrices are[2]

        whereaij=aij(x)is symmetric negative semi-def i nite matrix.The property of system(4)as follows∶ifVis a Lyapunov function,the solution of the system(4)is stable.

        In order to convenience our statement,the above four kinds of gradient systems are separately called as the first to the fourth class gradient system.

        2 The autonomous Birkhof f’s equations

        The gradient representations of autonomous Birkhoffian systems have the form

        whereB=B(a),a=(a1,a2,···,a2n)and we have

        For the system(5),if there exist matrices(bμν(a)),(Sμν(a)),(aμν(a))and a functionV=V(a)satisfying the following conditions

        it can be transformed into one of the four kinds of gradient systems which be def i ned at the end of section 2.Clearly,Eq.(8)can be satisfied easily,but not the case of Eqs.(7),(9),and (10).

        3 Illustrative examples

        Consider a Birkhoffian system with[9]

        Try to transform it into a gradient system and study the stability of solution. From the Birkhoffian equation(5),we know that

        LetV=B,we have

        which is the second kind of gradient system.Vis an integral of the system but cannot be a Lyapunov function.Consider the second and the fourth identities of the above equation and they can be rewritten as

        where

        which is not only an integral but also a Lyapunov function,therefore the solutionsa2=a4=0 are stable.

        Birkhoffian system is

        Try to transform it into a gradient system and study stability of solution.

        From the Birkhoffian equation(5),we know that

        It can be rewritten as

        where the matrix is anti-symmetric andVsatisfy

        which is the second kind of gradient system.Vis an integral of the system and a Lyapunov function.Therefore,the solutionsa1=a2=0 are stable.

        Birkhoffian system is

        Try to transform it into a gradient system and study stability of solution.

        The Birkhoffian equation(5)gives

        or

        where the matrix is symmetric negative def i nite and V is

        which is the third kind of gradient system.V is not a Lyapunov function.The characteristic equation has positive real roots.Therefore,the solutions a1=a2=0 are not stable.

        4 Conclusions

        We know that the autonomous Birkhoffian systems consist of the following properties,i.e., Birkhoffian function B is integral of the system,and if B can also be a Lyapunov function,the solutions of the system are stable,which is obtained in view of gradient system in this paper.It is easy to see that the autonomous Birkhoffian system is the second kind system naturally,but it is difficult to become three other kinds of gradient systems.Even through an autonomous Birkhoffian system can be transformed into three other kinds of gradient systems,V is also very difficult to become the Lyapunov function.In this case,the first-order approximation theory can be used if we want to study the stability of solution.

        [1]HIRSCH M W,SMALE S,DEVANEY R L.differential Equations,Dynamical Systems,and an Introduction to Chaos[M].3rd ed.Waltham:Academic Press,2012.

        [2]MCLACHLAN R I,QUISPEL G R W,ROBIDOUX N.Geometric integration using discrete gradients[J].Philosophical Transactions of the Royal Society of London:Series A,1999,357:1021-1045.

        [3]LOU Z M,MEI F X.A second order gradient representation of mechanics system[J].Acta Physica Sinica,2012, 61:337-340.

        [4]MEI F X.On the gradient system[J].Mechanics in Engineering,2012,34:89-90.

        [5]MEI F X,WU H B.A gradient representation for generalized Birkhoffian system[J].Dynamics and Control, 2012,10:289-292.

        [6]MEI F X,WU H B.Generalized Hamilton system and gradient system[J].Science China Physics,Mechanics and Astronomy,2013,43:538-540.

        [7]MEI F X,CUI J C,WU H B.A gradient representation and a fractional gradient representation of Birkhof f system[J].Transactions of Beijing Institute of Technology,2012,32:1298-1300.

        [8]MEI F X.Analytical MechanicsⅡ[M].Beijing:Beijing Institute of Technology Press,2013.

        [9]GUO Y X,LIU C,LIU S X.Generalized Birkhoffian realization of nonholonomic systems[J].Communications in Mathematics,2010,18:21-35.

        [10]SANTILLI R M.Foundations of Theoretical MechanicsⅡ[M].New York:Springer,1983.

        (責(zé)任編輯:李藝)

        自治Birkhoff系統(tǒng)的四類梯度表示

        崔金超1,廖翠萃1,梅鳳翔2
        (1.江南大學(xué)理學(xué)院,江蘇無錫214122;2.北京理工大學(xué)宇航學(xué)院,北京100081)

        提出四類梯度系統(tǒng),并研究自治Birkhof f系統(tǒng)的梯度表示.給出系統(tǒng)成為梯度表示和分?jǐn)?shù)維梯度的條件,利用梯度系統(tǒng)的性質(zhì)來研究Birkhof f系統(tǒng)的積分和解的穩(wěn)定性,舉例說明結(jié)果的應(yīng)用.

        Birkhof f系統(tǒng);梯度系統(tǒng);積分;穩(wěn)定性

        2016-04-19

        國家自然科學(xué)基金(11272050,11401259);江南大學(xué)自主科研資助項(xiàng)目(JUSRP11530)

        崔金超,男,講師,研究方向?yàn)榧s束力學(xué)系統(tǒng)的穩(wěn)定性.E-mail:cjcwx@163.com.

        O316

        A

        10.3969/j.issn.1000-5641.2017.03.010

        1000-5641(2017)03-0094-05

        猜你喜歡
        國家自然科學(xué)基金責(zé)任編輯梯度
        一個改進(jìn)的WYL型三項(xiàng)共軛梯度法
        常見基金項(xiàng)目的英文名稱(一)
        一種自適應(yīng)Dai-Liao共軛梯度法
        一類扭積形式的梯度近Ricci孤立子
        我校喜獲五項(xiàng)2018年度國家自然科學(xué)基金項(xiàng)目立項(xiàng)
        English Abstracts
        2017 年新項(xiàng)目
        國家自然科學(xué)基金項(xiàng)目簡介
        English Abstracts
        English Abstracts
        一本色道亚州综合久久精品| 久草热8精品视频在线观看| 精品国产免费Av无码久久久| 亚洲精品一区二区三区国产| 青青草免费在线爽视频| 国模无码一区二区三区不卡| 欧美丰满大爆乳波霸奶水多| 久久久久久国产福利网站| 亚洲精品一区二区三区在线观| 乱老年女人伦免费视频| 精品久久久久久777米琪桃花| 国产精品久久婷婷婷婷| 亚洲六月丁香色婷婷综合久久| 日日摸夜夜添夜夜添高潮喷水| 无码人妻丰满熟妇片毛片| 日本a级大片免费观看| 国产精品自产拍av在线| 亚洲综合网国产精品一区| 亚洲精品综合欧美一区二区三区| 99re6久精品国产首页| 青青草在线免费观看视频| 欧美成人午夜免费影院手机在线看| 18无码粉嫩小泬无套在线观看| 中文字幕有码在线视频| 黑人玩弄极品人妻系列视频| 日韩一区国产二区欧美三区 | 午夜影视啪啪免费体验区入口| 亚洲中文乱码在线视频| 中文字幕人妻在线中字| 国产在线精品一区二区不卡| 果冻蜜桃传媒在线观看| 中文字幕隔壁人妻欲求不满| 影音先锋女人av鲁色资源网久久| 久久亚洲高清观看| 国产精品女同一区二区免| 人妻 偷拍 无码 中文字幕| 日本a级特黄特黄刺激大片 | av网站免费观看入口| 国产在线精品一区二区三区直播| 亚洲最大天堂无码精品区| 中文字幕亚洲精品第一页|