徐偉 岳曉樂(lè) 韓群,2

(1.西北工業(yè)大學(xué)應(yīng)用數(shù)學(xué)系,西安 710072) (2.華中農(nóng)業(yè)大學(xué)理學(xué)院,武漢 430070)

胞映射方法及其在非線性隨機(jī)動(dòng)力學(xué)中的應(yīng)用*

徐偉1?岳曉樂(lè)1韓群1,2

(1.西北工業(yè)大學(xué)應(yīng)用數(shù)學(xué)系,西安 710072) (2.華中農(nóng)業(yè)大學(xué)理學(xué)院,武漢 430070)

介紹了與非線性隨機(jī)動(dòng)力學(xué)研究密切相關(guān)的幾類(lèi)胞映射方法的研究和進(jìn)展,主要有廣義胞映射圖方法、基于短時(shí)高斯逼近的胞映射方法、并行胞映射方法等.簡(jiǎn)述了胞映射方法在隨機(jī)動(dòng)力學(xué)中的應(yīng)用情況,重點(diǎn)關(guān)注隨機(jī)響應(yīng)、分岔、離出和碰撞振動(dòng)系統(tǒng).給出了胞映射方法面臨的挑戰(zhàn),以及未來(lái)研究可能的發(fā)展方向.

胞映射方法, 并行算法, 離出, 短時(shí)高斯逼近

引言

在自然界和實(shí)際工程問(wèn)題當(dāng)中,往往受到像大氣湍流、地面強(qiáng)風(fēng)、海浪運(yùn)動(dòng)和路面不平整等因素的影響,這些作用都帶有隨機(jī)性,由此誘發(fā)了非線性隨機(jī)動(dòng)力學(xué)的研究[1-2],已成為數(shù)學(xué)、力學(xué)、物理、經(jīng)濟(jì)金融等領(lǐng)域較活躍的科學(xué)前沿,其主要研究?jī)?nèi)容包括隨機(jī)響應(yīng)、隨機(jī)分岔、隨機(jī)控制和隨機(jī)離出問(wèn)題等[3-4],研究方法有從確定性系統(tǒng)推廣而來(lái)的隨機(jī)攝動(dòng)法、隨機(jī)多尺度法、隨機(jī)平均法等近似解析方法,也有像胞映射方法、路徑積分法等有效的數(shù)值方法[5].由于解析方法難以處理強(qiáng)非線性或強(qiáng)隨機(jī)激勵(lì)問(wèn)題,借助一定的數(shù)值方法,可以揭示很多非常復(fù)雜動(dòng)力學(xué)現(xiàn)象,因此數(shù)值方法對(duì)非線性隨機(jī)動(dòng)力學(xué)的研究具有不可替代的作用.目前,隨著計(jì)算機(jī)水平的快速發(fā)展,計(jì)算性能越來(lái)越高,速度越來(lái)越快,成本逐步降低,這為數(shù)值方法的應(yīng)用研究提供了更廣闊的舞臺(tái).

胞映射方法[6]是分析非線性動(dòng)力系統(tǒng)全局特性的有效數(shù)值工具,因其快速、準(zhǔn)確和適用范圍廣等特點(diǎn)而備受關(guān)注,尤其是改進(jìn)后的廣義胞映射方法,通過(guò)引入Markov鏈?zhǔn)沟糜?jì)算優(yōu)勢(shì)更加明顯,不僅能夠計(jì)算系統(tǒng)的拓?fù)浣Y(jié)構(gòu)等定性性質(zhì)(如吸引子和吸引域的空間分布),而且還能從統(tǒng)計(jì)意義上反映動(dòng)力系統(tǒng)的定量性質(zhì),因此廣義胞映射方法在隨機(jī)動(dòng)力系統(tǒng)中逐步展現(xiàn)出了良好的應(yīng)用前景[7].當(dāng)前,由于一些實(shí)際問(wèn)題的需求,系統(tǒng)規(guī)模由低維走向高維,隨機(jī)激勵(lì)從高斯變成非高斯,數(shù)值方法在科學(xué)研究中扮演著越來(lái)越重要的角色,因此,開(kāi)展廣義胞映射方法在非線性隨機(jī)動(dòng)力學(xué)中的應(yīng)用研究具有重要的科學(xué)意義.

1 非線性隨機(jī)動(dòng)力學(xué)中的胞映射方法研究進(jìn)展

胞映射方法最初由Hsu[6]于上世紀(jì)80年代提出,主要用于強(qiáng)非線性系統(tǒng)的全局分析,基本思想是將動(dòng)力系統(tǒng)的連續(xù)狀態(tài)空間離散成胞狀態(tài)空間,其中包含大量的規(guī)則幾何體,稱(chēng)之為胞,并用整數(shù)序號(hào)標(biāo)示.每個(gè)胞都是一些狀態(tài)點(diǎn)的集合,以胞中的點(diǎn)為起始點(diǎn)向前積分,構(gòu)造點(diǎn)到點(diǎn)的映射,而積分終點(diǎn)所在的胞稱(chēng)為像胞.點(diǎn)與點(diǎn)之間的映射關(guān)系就轉(zhuǎn)化成了胞與胞之間的映射關(guān)系,用胞動(dòng)力系統(tǒng)來(lái)分析原始動(dòng)力系統(tǒng)的性質(zhì).相比傳統(tǒng)的點(diǎn)映射,胞映射方法的顯著特點(diǎn)是它數(shù)值積分時(shí)不用記錄長(zhǎng)時(shí)間的軌線,只需要計(jì)算一個(gè)映射時(shí)間步長(zhǎng)后的軌線終點(diǎn),從而達(dá)到提高計(jì)算效率的目的.胞映射方法自提出之后,眾多學(xué)者參與胞映射的研究工作,發(fā)展了一系列改進(jìn)版本,較有代表性的有簡(jiǎn)單胞映射方法[6,8]、廣義胞映射方法[7,8]、圖胞映射方法[9-11]、胞參照點(diǎn)映射方法[12,13]、插值胞映射方法[14]、面向集合法[15]等等.

眾多改進(jìn)方法中,廣義胞映射方法由于能夠同時(shí)分析動(dòng)力系統(tǒng)的定性性質(zhì)和定量性質(zhì),地位變得尤為重要.其基本思想是在尋找像胞時(shí),每個(gè)胞內(nèi)需選取多個(gè)采樣點(diǎn),分別從每個(gè)采樣點(diǎn)出發(fā)計(jì)算一條軌線,每個(gè)胞可以有多個(gè)像胞,并以一定的概率轉(zhuǎn)移到它的像胞,這樣可以得到任意兩個(gè)胞之間的一步轉(zhuǎn)移概率.胞與胞的轉(zhuǎn)移關(guān)系等價(jià)于一個(gè)有限的Markov鏈,因此可運(yùn)用Markov鏈的相關(guān)理論對(duì)胞映射動(dòng)力系統(tǒng)進(jìn)行分析,并用于隨機(jī)動(dòng)力系統(tǒng)的響應(yīng)分析.Sun利用廣義胞映射方法研究隨機(jī)系統(tǒng)的首次穿越和響應(yīng)分析[16,17],Wu和Zhu[18]研究了一類(lèi)脈沖激勵(lì)下捕食和被捕食者模型的隨機(jī)響應(yīng)分析,Yue等[19,20]分別給出了有界噪聲和泊松白噪聲激勵(lì)下典型動(dòng)力系統(tǒng)的響應(yīng)概率密度函數(shù),Hong等[21,22]運(yùn)用研究了幾類(lèi)模糊動(dòng)力系統(tǒng)瞬態(tài)和穩(wěn)態(tài)隸屬分布函數(shù)的演化過(guò)程.

非線性隨機(jī)動(dòng)力系統(tǒng)響應(yīng)的概率密度函數(shù)可以通過(guò)求解相應(yīng)的Fokker-Planck-Kolmogorov(FPK)方程來(lái)獲得,通過(guò)構(gòu)造從某些給定初始條件出發(fā)的短時(shí)解是求解問(wèn)題的一種有效途徑.為了提高非線性系統(tǒng)短時(shí)解的精度,Sun和Hsu[23]成功構(gòu)造了高斯白噪聲激勵(lì)下非線性系統(tǒng)FPK方程的一種短時(shí)高斯解,將系統(tǒng)的短時(shí)概率密度函數(shù)近似為高斯分布,其均值和方差可通過(guò)數(shù)值求解高斯閉包后的矩方程得到,然后運(yùn)用這種短時(shí)高斯解計(jì)算廣義胞映射方法中的胞轉(zhuǎn)移概率,并研究了幾類(lèi)典型系統(tǒng)的隨機(jī)振動(dòng)分析.基于短時(shí)高斯逼近胞映射方法,Sun[24]還研究了非線性系統(tǒng)的最優(yōu)控制問(wèn)題,以及含干摩擦系統(tǒng)的非線性系統(tǒng)的隨機(jī)振動(dòng)問(wèn)題[25].針對(duì)含周期和高斯白噪聲激勵(lì)的非線性系統(tǒng),Han等[26]通過(guò)短時(shí)高斯逼近方法計(jì)算含周期激勵(lì)系統(tǒng)的轉(zhuǎn)移概率矩陣,提高了廣義胞映射方法在計(jì)算響應(yīng)概率密度函數(shù)時(shí)的效率,研究發(fā)現(xiàn)確定性混沌鞍會(huì)影響隨機(jī)響應(yīng)概率密度函數(shù).Li等[27,28]通過(guò)引入演化概率向量的概念,利用廣義胞映射方法研究了Mathieu-Duffing振子在加性和乘性噪聲共同作用下的分岔行為.

圖論分析方法的引入,進(jìn)一步擴(kuò)展了廣義胞映射方法在定性分析中應(yīng)用范圍.Xu等從全局的角度分析了隨機(jī)吸引子和隨機(jī)鞍的演化特性,提出了新的隨機(jī)分岔定義,研究了高斯白噪聲激勵(lì)下幾類(lèi)典型非線性動(dòng)力系統(tǒng)的隨機(jī)分岔行為[29-31].Yue和Xu[32,33]研究了有界噪聲激勵(lì)下單勢(shì)井Duffing振子和SD振子的隨機(jī)分岔,發(fā)現(xiàn)隨機(jī)吸引子和隨機(jī)鞍形狀及大小改變的方向與系統(tǒng)不穩(wěn)定流形的形狀始終保持一致.Hong和Sun[34,35]通過(guò)提出模糊胞映射方法,研究了模糊噪聲激勵(lì)下非線性動(dòng)力系統(tǒng)的分岔現(xiàn)象.Liu等利用廣義胞映射圖方法研究了高斯白噪聲激勵(lì)下分段線性系統(tǒng)的全局動(dòng)力學(xué)[36],以及噪聲誘導(dǎo)周期激勵(lì)系統(tǒng)的逃逸問(wèn)題[37].最近,基于分?jǐn)?shù)階導(dǎo)數(shù)的短記憶原理,廣義胞映射圖方法又成功地應(yīng)用于分?jǐn)?shù)階動(dòng)力系統(tǒng)全局特性的研究中[38-40].

胞映射方法自提出以來(lái),面臨的最主要問(wèn)題就是計(jì)算精度與計(jì)算速度之間的矛盾,隨著計(jì)算機(jī)水平的提高,并行技術(shù)為胞映射方法提供了新的途徑.Kreuzer和Lagemann[41]通過(guò)分析給出了胞映射方法中引入并行技術(shù)的可行性,并應(yīng)用于二維動(dòng)力系統(tǒng)中.Eason和Dick[42]基于多核CPU提出了并行多自由度胞映射方法,用于分析維數(shù)較高的動(dòng)力系統(tǒng).Belardinelli和Lenci[43]研究了一種基于分布式計(jì)算的胞映射方法,針對(duì)高維非線性系統(tǒng)的大范圍吸引域提出了有效的并行算法.Sun和其合作者[44]基于GPU多線程實(shí)現(xiàn)技術(shù),結(jié)合簡(jiǎn)單胞映射、廣義胞映射、迭代細(xì)分算法和有效后處理方法提出了并行胞映射方法,該方法被運(yùn)用到高維非線性動(dòng)力系統(tǒng)的全局分析中,表現(xiàn)出了非常高的計(jì)算效率.隨后,他們運(yùn)用并行胞映射方法計(jì)算了非線性向量函數(shù)的簡(jiǎn)單零點(diǎn)問(wèn)題[45].Yue[46]基于并行和細(xì)分技術(shù),有效地提高了胞映射方法的計(jì)算精度和速度,該方法能夠處理動(dòng)力系統(tǒng)的復(fù)雜吸引域邊界.在基于演化概率向量的胞映射方法基礎(chǔ)上,Li等[47]進(jìn)一步利用GPU并行技術(shù)提高計(jì)算效率,研究了噪聲誘導(dǎo)一類(lèi)二自由度分段線性系統(tǒng)的轉(zhuǎn)遷行為.

本文僅介紹了與隨機(jī)動(dòng)力學(xué)密切相關(guān)的幾類(lèi)胞映射方法的最新進(jìn)展,其他有關(guān)的改進(jìn)方法的進(jìn)展情況,可參考文獻(xiàn)及專(zhuān)著[48-51].

2 胞映射方法在隨機(jī)動(dòng)力系統(tǒng)中的若干應(yīng)用研究

2.1 隨機(jī)響應(yīng)與分岔

考慮n維隨機(jī)動(dòng)力系統(tǒng)[23]

(1)

其中t為時(shí)間變量,ζ(t)為隨機(jī)激勵(lì),f為向量值函數(shù),D為給定的狀態(tài)空間區(qū)域.x(t)為穩(wěn)態(tài)Markov過(guò)程,令p(x,t)為x(t)在時(shí)刻t的概率密度函數(shù),則有

(2)

(3)

令t0=(m-1)Δt和t=mΔt,那么在式(3)條件下式(2)又可以寫(xiě)為：

(4)

運(yùn)用廣義胞映射方法時(shí),將連續(xù)狀態(tài)空間離散成胞化空間,得到系統(tǒng)狀態(tài)胞之間的轉(zhuǎn)移關(guān)系,上述式(4)變?yōu)椋?/p>

(5)

矩陣形式為：

p(n+1)=P·p(n)

(6)

其中P={pji}表示一步轉(zhuǎn)移概率矩陣,p(n)指的是n步迭代之后系統(tǒng)在狀態(tài)空間中的概率分布,也即系統(tǒng)在時(shí)刻tn=t0+nΔT時(shí)的概率分布.若給定初始概率分布p(0),可以計(jì)算得到系統(tǒng)在tn時(shí)刻的瞬態(tài)響應(yīng)的概率分布：

p(n)=P·p(n-1)=P2·p(n-2)=…=Pn·p(0)

(7)

數(shù)值計(jì)算時(shí)不需要求出Pn,只需計(jì)算n次矩陣P和向量p(i),i=0,1,2,…,n-1的乘積即可.

對(duì)于含周期激勵(lì)的系統(tǒng)[26],在計(jì)算一步轉(zhuǎn)移概率時(shí),轉(zhuǎn)移概率求解時(shí)的時(shí)間長(zhǎng)度直接取為系統(tǒng)的周期,雖然這樣在形式上顯得很簡(jiǎn)單,但是運(yùn)算過(guò)程卻相當(dāng)費(fèi)時(shí),短時(shí)高斯逼近方法構(gòu)造FPK方程的短時(shí)解是提高胞映射方法計(jì)算效率的有效途徑.考慮含周期和高斯白噪聲激勵(lì)的n維非線性隨機(jī)動(dòng)力系統(tǒng),其對(duì)應(yīng)的It隨機(jī)微分方程為

dX(t)=f(X,t)dt+σ(X)dB(t)

(8)

求解短時(shí)高斯逼近解時(shí),首先考慮響應(yīng)過(guò)程X(t)的一階矩和二階矩

m(t)=E[X(t)]

C(t)=E[(X-m)(X-m)T]

(9)

(10)

其中m(kτ)和C(kτ)是方程(9)滿足初始條件m[(k-1)τ]=x0和C[(k-1)τ]=0時(shí)的短時(shí)解.需要指出的是計(jì)算條件概率需要進(jìn)行數(shù)值積分,用到了Gauss-Legendre積分方法[52-53].

圖1 系統(tǒng)(11)的全局特性圖 μ1=μ2=0.0; μ1=0.09,μ2=0.133; μ1=0.09,μ2=0.134Fig.1 Global properties of system (11) μ1=μ2=0.0; μ1=0.09,μ2=0.133; μ1=0.09,μ2=0.134

考慮隨機(jī)激勵(lì)下的光滑非連續(xù)(SD)振子[54-55]：

(11)

其中β為阻尼系數(shù),f和ω為諧和激勵(lì)的幅值和頻率,α為光滑參數(shù),ξ1(t)和ξ2(t)為兩個(gè)有界噪聲過(guò)程,取參數(shù)β=0.2,α=0.42,f=0.22,ω=0.85時(shí),利用廣義胞映射圖方法得到確定性系統(tǒng),以及噪聲激勵(lì)下系統(tǒng)的全局特性,如圖1,發(fā)現(xiàn)隨機(jī)分岔的發(fā)生是隨機(jī)吸引子與隨機(jī)鞍發(fā)生碰撞的結(jié)果,且隨機(jī)吸引子和隨機(jī)鞍的形狀和大小的改變方向和系統(tǒng)的不穩(wěn)定流形形狀始終保持一致[33].

考慮SD振子受周期和高斯白噪聲共同作用,即μ1=0.01,μ2=0.0,ξ1(t)為高斯白噪聲,取β=0.04,α=0.6,f=0.83,ω=1.0606時(shí),確定系統(tǒng)的全局特性如圖2所示.圖3進(jìn)一步給出了不同時(shí)刻瞬態(tài)聯(lián)合概率密度函數(shù)的等高線圖,可以發(fā)現(xiàn)瞬態(tài)聯(lián)合概率密度函數(shù)的形狀逐漸演化成了確定系統(tǒng)混沌鞍的形狀[26].

圖2 確定性系統(tǒng)(11)的吸引子A(星號(hào))和混沌鞍S(點(diǎn))Fig.2 Attractor A (asterisk) and chaotic saddle S (dots) of system (11)

圖3 系統(tǒng)(11)在不同時(shí)刻瞬態(tài)聯(lián)合概率密度函數(shù)的等高線圖(a) t=1T；(b) t=2T；(c) t=3T；(d) t=4TFig.3 The contour plots of transient joint probability density function of system (10) at different times(a) t=1T；(b) t=2T；(c) t=3T；(d) t=4T

2.2 首次穿越時(shí)間

首次穿越時(shí)間是研究非線性隨機(jī)動(dòng)力系統(tǒng)離出問(wèn)題的重要指標(biāo)之一,關(guān)注的是系統(tǒng)響應(yīng)會(huì)在何時(shí)從一個(gè)穩(wěn)態(tài)出發(fā)首次穿過(guò)其安全域的邊界到達(dá)另一個(gè)穩(wěn)態(tài)的安全域.需要計(jì)算系統(tǒng)首次穿越時(shí)間的統(tǒng)計(jì)分布,包括首次穿越時(shí)間的概率密度函數(shù)及平均首次穿越時(shí)間.近年來(lái),許多學(xué)者研究了隨機(jī)游走模型的首次穿越時(shí)間[56,57]和雙穩(wěn)動(dòng)力系統(tǒng)的平均首次穿越時(shí)間[58],但是這些研究對(duì)象往往都局限在一階系統(tǒng).二階非線性系統(tǒng)首次穿越時(shí)間的精確解一般很難獲取,因此需要借助一些近似方法或數(shù)值方法來(lái)進(jìn)行研究.胞映射方法是研究首次穿越時(shí)間問(wèn)題一種有效的數(shù)值方法[16],比直接數(shù)值模擬方法有明顯優(yōu)勢(shì).

x,x1∈Rm

(12)

式中L[·]表示后向Kolmogorov方程的微分算子.

dy1…dyn-1

(13)

若假設(shè)系統(tǒng)的響應(yīng)為齊次Markov過(guò)程,則有：

i=1,2,…,n

(14)

然后基于廣義胞映射方法求解含吸收邊界條件的一步轉(zhuǎn)移概率矩陣,并在初始分布中考慮對(duì)應(yīng)的初始條件,即可得到首次穿越時(shí)間的概率密度,以及平均首次穿越時(shí)間.通過(guò)比較,發(fā)現(xiàn)胞映射方法和直接數(shù)值模擬結(jié)果吻合較好,如圖4所示[59].

圖4 對(duì)稱(chēng)雙穩(wěn)系統(tǒng)首次穿越時(shí)間的概率密度函數(shù)f(t)實(shí)線表示廣義胞映射方法的結(jié)果,圓圈和星號(hào)表示直接模擬的結(jié)果Fig.4 First-passage time probability density functions f(t) in the asymmetric bistable system solid lines are results from the generalized cell mapping method, circles and stars are results from direct Monte Carlo simulation

2.3 離出位置分布

離出位置分布是刻畫(huà)非線性隨機(jī)動(dòng)力系統(tǒng)離出問(wèn)題的另一個(gè)重要指標(biāo).系統(tǒng)在噪聲激勵(lì)下的逃逸路線與確定性吸引域邊界的交點(diǎn)被稱(chēng)為離出點(diǎn).離出點(diǎn)的位置與首次穿越時(shí)間一樣具有隨機(jī)性,因此需要計(jì)算離出位置統(tǒng)計(jì)分布的概率密度函數(shù).Bobrovsky和Schuss[60]運(yùn)用漸進(jìn)展開(kāi)法研究時(shí)觀察到了離出位置分布的一種非常規(guī)偏移,也被稱(chēng)作為鞍點(diǎn)回避現(xiàn)象[61,62].2013年,Khovanov等[63]發(fā)展了一種非線性非局部穩(wěn)定性方法,借此描述了可激發(fā)FHN系統(tǒng)中噪聲誘導(dǎo)逃逸問(wèn)題,并計(jì)算了離出位置分布.這兩種解析方法在運(yùn)用過(guò)程中也難免存在一些局限性[64],經(jīng)常需要借助數(shù)值方法來(lái)計(jì)算離出位置分布,胞映射方法即是其中一種有效的數(shù)值方法[37-65].

考慮Kramers離出問(wèn)題[66,67],布朗粒子在噪聲作用下從一個(gè)勢(shì)阱越過(guò)勢(shì)壘到達(dá)另一個(gè)勢(shì)阱.其運(yùn)動(dòng)可表示為下面的標(biāo)準(zhǔn)化隨機(jī)微分方程：

(15)

其中X和Y分別表示粒子運(yùn)動(dòng)的位移和速度,β是標(biāo)準(zhǔn)化電離常數(shù),取值為2.0,ε是一個(gè)表示溫度的標(biāo)準(zhǔn)化小參數(shù),W(t)是標(biāo)準(zhǔn)的高斯白噪聲.圖5給出了確定系統(tǒng)的全局行為,以及隨噪聲強(qiáng)度ε取不同時(shí)系統(tǒng)離出位置分布.可以發(fā)現(xiàn)分布曲線隨著ε的減小而慢慢往右移動(dòng),且峰值變高,這種變化的原因在于,隨著噪聲的越來(lái)越弱,系統(tǒng)最大可能的離出位置會(huì)慢慢逼近鞍點(diǎn)S.

圖5 系統(tǒng)(15)全局特性及離出位置分布(a)全局性質(zhì)(ε=0), A為吸引子,B為A的吸引域; (b)離出位置分布,實(shí)線表示廣義胞映射方法的結(jié)果,圓圈表示直接數(shù)值模擬的結(jié)果Fig.5 Global properties and exit location distribution of system (15)(a) global properties, A is attractor and B is the basin of attraction of attractor A; (b) exit location distribution, solid lines are results from the generalized cell mapping method, circles are results from direct Monte Carlo simulation

2.4 碰撞振動(dòng)系統(tǒng)

對(duì)于含高碰撞損失或碰撞約束不在平衡位置的碰撞振動(dòng)系統(tǒng),一般的近似解析方法不再適用,只能依賴數(shù)值方法來(lái)求解,路徑積分法就是其中一種.Dimentberg 等[68]基于短時(shí)高斯近似的路徑積分法研究了具有高碰撞損失的碰撞振動(dòng)系統(tǒng),驗(yàn)證了方法的有效性,但該結(jié)果只針對(duì)運(yùn)動(dòng)約束在平衡位置的情形.Li[69]試圖利用通過(guò)改進(jìn)廣義胞映射方法,使其適用于一般噪聲激勵(lì)下碰撞振動(dòng)系統(tǒng)的響應(yīng)概率密度函數(shù)求解.廣義胞映射方法在非光滑系統(tǒng)定性分析中已經(jīng)取得了一些研究進(jìn)展[36,70-73].

考慮隨機(jī)激勵(lì)下的單自由度碰撞振動(dòng)系統(tǒng)[69]：

(16)

(17)

圖6 系統(tǒng)(16)的穩(wěn)態(tài)聯(lián)合概率密度函數(shù)(a)q=-0.5;(b)q=-0.8;(c)q=-1.0;(d)q=-1.2Fig.6 The stationary joint probability density function of system (16)(a)q=-0.5;(b)q=-0.8;(c)q=-1.0;(d)q=-1.2

另外,胞映射方法在隨機(jī)最優(yōu)控制等領(lǐng)域還有一些重要的研究成果與進(jìn)展,詳情可參考綜述文獻(xiàn)[50-51].

3 結(jié)語(yǔ)

胞映射方法作為一種高效的數(shù)值方法,不僅在研究確定性系統(tǒng)全局分析中發(fā)揮巨大作用,還在隨機(jī)動(dòng)力系統(tǒng)中展現(xiàn)出了良好的應(yīng)用前景,且已有一定的研究基礎(chǔ).本文重點(diǎn)討論了胞映射方法在隨機(jī)動(dòng)力學(xué)領(lǐng)域的最新研究進(jìn)展,以及所取得的一些成果,包括隨機(jī)響應(yīng)與分岔、離出問(wèn)題以及在隨機(jī)非光滑系統(tǒng)中的應(yīng)用.用胞映射方法來(lái)研究隨機(jī)動(dòng)力系統(tǒng)有明顯的優(yōu)勢(shì),但又面臨眾多挑戰(zhàn),比如：

(1)高維問(wèn)題一直以來(lái)是眾多研究者所密切關(guān)注的難題,目前的改進(jìn)方法已取得重大研究進(jìn)展,特別是并行胞映射方法的提出大大提升了計(jì)算維數(shù).對(duì)高維非線性隨機(jī)動(dòng)力系統(tǒng)而言,如何改進(jìn)短時(shí)高斯逼近策略,并結(jié)合并行技術(shù)應(yīng)該是有效的途徑.

(2)胞映射方法已經(jīng)在眾多非線性動(dòng)力學(xué)領(lǐng)域有重要應(yīng)用,如逃逸問(wèn)題、碰撞振動(dòng)系統(tǒng)、分?jǐn)?shù)階系統(tǒng)等,未來(lái)的研究可以結(jié)果高效胞映射方法進(jìn)一步拓展在隨機(jī)動(dòng)力系統(tǒng)中的應(yīng)用范圍,發(fā)現(xiàn)新的動(dòng)力學(xué)現(xiàn)象.

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*The project supported by the National Natural Science Foundation of China(11472212, 11672230).

? Corresponding author E-mail:weixu@nwpu.edu.cn

17 March 2017,revised 18 April 2017.

CELL MAPPING METHOD AND ITS APPLICATIONS IN NONLINEAR STOCHASTIC DYNAMICAL SYSTEMS*

Xu Wei1?Yue Xiaole1Han Qun1,2

(1.DepartmentofAppliedMathematics,NorthwesternPolytechnicalUniversity,Xi′an710072,China) (2.CollegeofScience,HuazhongAgriculturalUniversity,Wuhan430070,China)

This paper introduces the development of several cell mapping methods closely associated with nonlinear stochastic dynamical systems, including generalized cell mapping method with digraph, cell mapping method based on short-time Gaussian approximation, parallel cell mapping method etc. The applications of cell mapping method in stochastic dynamical systems are also shown, focusing on stochastic response、bifurcation、exit and vibro-impact system. Finally, some challenges and possible future researches about cell mapping method are presented.

cell mapping method, parallel algorithm, exit, short-time Gaussian approximation

*國(guó)家自然科學(xué)基金資助項(xiàng)目(11472212, 11672230)

10.6052/1672-6553-2017-023

2017-03-17收到第1稿,2017-4-18收到修改稿.

? 通訊作者 E-mail:weixu@nwpu.edu.cn