張 卜,寧麗娟

(陜西師范大學(xué) 數(shù)學(xué)與信息科學(xué)學(xué)院,陜西 西安 710062)

關(guān)聯(lián)的乘性和加性驅(qū)動(dòng)的三穩(wěn)系統(tǒng)穩(wěn)態(tài)分析

張 卜,寧麗娟

(陜西師范大學(xué) 數(shù)學(xué)與信息科學(xué)學(xué)院,陜西 西安 710062)

運(yùn)用劉維方程和諾維科夫定理, 研究乘性和加性高斯白噪聲共同激勵(lì)下一維三穩(wěn)系統(tǒng)的穩(wěn)態(tài)概率密度函數(shù).結(jié)果表明，關(guān)聯(lián)強(qiáng)度λ和乘性噪聲強(qiáng)度P均能誘導(dǎo)相變的產(chǎn)生，而加性噪聲強(qiáng)度Q不能誘導(dǎo)相變的產(chǎn)生.通過(guò)數(shù)值模擬穩(wěn)態(tài)概率密度函數(shù)驗(yàn)證了所得結(jié)論的準(zhǔn)確性.

三穩(wěn)系統(tǒng); 噪聲; 相變; 穩(wěn)態(tài)概率密度函數(shù)

0 引 言

噪聲廣泛存在于自然界的各個(gè)領(lǐng)域,包括生物、 物理、 化學(xué)、 醫(yī)學(xué)等.傳統(tǒng)觀念認(rèn)為噪聲會(huì)影響信息傳遞的精確性,總是消極的.噪聲在產(chǎn)生雜亂的運(yùn)動(dòng),破壞序,破壞功能,抹去相與相之間的差別,導(dǎo)致均勻,起到了破壞的相變作用，是造成系統(tǒng)無(wú)序的根源.所以,人們想盡各種辦法消去噪聲對(duì)系統(tǒng)的影響.但研究發(fā)現(xiàn),這種無(wú)規(guī)律的隨機(jī)干擾并不總是對(duì)系統(tǒng)起到消極破壞的作用，在一定的非線性條件下它在產(chǎn)生相干運(yùn)動(dòng)和建立“序”上起到了十分積極的作用.如色關(guān)聯(lián)的色噪聲驅(qū)動(dòng)的雙穩(wěn)杜芬模型的穩(wěn)態(tài)分析[1],噪聲可以抑制腫瘤細(xì)胞的增長(zhǎng)[2]等.通過(guò)研究帶有噪聲的非線性系統(tǒng),發(fā)現(xiàn)許多確定性方程不可能產(chǎn)生的相變?cè)谠肼曌饔孟聟s成為可能.噪聲誘導(dǎo)下非線性動(dòng)力學(xué)系統(tǒng)的相變就是其中的一個(gè)重要問(wèn)題[3].研究發(fā)現(xiàn),當(dāng)系統(tǒng)的某個(gè)參數(shù)達(dá)到某一臨界值時(shí),穩(wěn)態(tài)概率密度函數(shù)曲線的結(jié)構(gòu)發(fā)生變化,即峰值數(shù)目發(fā)生變化.文獻(xiàn)[4]研究了非高斯色噪聲對(duì)FHN神經(jīng)元系統(tǒng)的影響，文獻(xiàn)[5]發(fā)現(xiàn)噪聲在基因轉(zhuǎn)錄調(diào)控過(guò)程中可以誘導(dǎo)蛋白質(zhì)濃度發(fā)生變化，文獻(xiàn)[6]研究了乘性噪聲和加性噪聲之間色關(guān)聯(lián)的單模激光系統(tǒng)的定態(tài)性質(zhì)，文獻(xiàn)[7]對(duì)色關(guān)聯(lián)白噪聲驅(qū)動(dòng)的雙穩(wěn)系統(tǒng)的穩(wěn)態(tài)進(jìn)行分析，文獻(xiàn)[8]研究了帶有對(duì)稱利維噪聲的腫瘤免疫系統(tǒng)的隨機(jī)分岔，文獻(xiàn)[9]研究了白噪聲驅(qū)動(dòng)下雙穩(wěn)系統(tǒng)的穩(wěn)態(tài)性質(zhì)，文獻(xiàn)[10]研究了色關(guān)聯(lián)作用下單模激光損失模型的光強(qiáng)定態(tài)問(wèn)題，文獻(xiàn)[11]研究了色關(guān)聯(lián)的白噪聲驅(qū)動(dòng)的雙穩(wěn)系統(tǒng)的隨機(jī)共振，文獻(xiàn)[12]研究了高斯白噪聲對(duì)神經(jīng)網(wǎng)絡(luò)模型隨機(jī)共振的影響.

上述研究中,大部分都局限在帶有雙勢(shì)阱系統(tǒng)和單勢(shì)阱系統(tǒng)模型中,對(duì)三勢(shì)阱系統(tǒng)的分析并不多.文獻(xiàn)[13]研究了色關(guān)聯(lián)的三勢(shì)阱系統(tǒng)下時(shí)滯誘導(dǎo)相變，文獻(xiàn)[14]研究了色關(guān)聯(lián)乘性和加性色噪聲驅(qū)動(dòng)的多穩(wěn)態(tài)系統(tǒng)的穩(wěn)態(tài)特性，文獻(xiàn)[15]研究了非高斯噪聲激勵(lì)的三勢(shì)阱系統(tǒng)的隨機(jī)響應(yīng).本文研究關(guān)聯(lián)乘性和加性的高斯白噪聲驅(qū)動(dòng)的多穩(wěn)系統(tǒng)的穩(wěn)態(tài)特性.分別討論了關(guān)聯(lián)強(qiáng)度λ,加性噪聲強(qiáng)度Q和乘性噪聲強(qiáng)度P對(duì)穩(wěn)態(tài)概率密度函數(shù)的影響,研究噪聲誘導(dǎo)相變現(xiàn)象的產(chǎn)生.

1 三穩(wěn)態(tài)系統(tǒng)的概率密度函數(shù)

考慮關(guān)聯(lián)的乘性和加性高斯白噪聲共同驅(qū)動(dòng)的三穩(wěn)系統(tǒng)，由下列的一維朗之萬(wàn)方程來(lái)表示：

(1)

(2)

其中，V(x)是確定性的勢(shì)函數(shù)，ξ(t)和η(t)分別為乘性和加性高斯白噪聲，統(tǒng)計(jì)性質(zhì)為

(3)

利用劉維方程[16]和諾維科夫定理[17]，從方程(1)～(3)得到近似的?？?普朗克方程為

(4)

方程(4)對(duì)應(yīng)的穩(wěn)態(tài)概率密度函數(shù)為

(5)

式(5)中N為歸一化常數(shù),廣義勢(shì)函數(shù)Φ(x)的表達(dá)式為

Φ(x)=A1x4/4+A2x3/3+A3x2/2+A4x+A5.

(6)

其中

通過(guò)數(shù)值計(jì)算方程(5),討論關(guān)聯(lián)強(qiáng)度λ,加性噪聲強(qiáng)度Q和乘性噪聲強(qiáng)度P對(duì)穩(wěn)態(tài)概率密度函數(shù)的影響.這里取α=-1,β=4,γ=1,以下結(jié)論均在這組參數(shù)下討論得出.

圖2為關(guān)聯(lián)強(qiáng)度λ取不同值時(shí),穩(wěn)態(tài)概率密度函數(shù)ρst(x)關(guān)于x的函數(shù)圖像.當(dāng)參數(shù)P=0.5,Q=0.5時(shí),可以看出,隨著λ的增大,穩(wěn)態(tài)概率密度函數(shù)由三峰結(jié)構(gòu)變?yōu)閱畏褰Y(jié)構(gòu),即λ可以誘導(dǎo)相變現(xiàn)象產(chǎn)生.在出現(xiàn)三峰的情況下,左峰比右峰高,隨著λ的增強(qiáng),穩(wěn)態(tài)概率密度函數(shù)由三峰結(jié)構(gòu)變?yōu)殡p峰結(jié)構(gòu).伴隨著λ的進(jìn)一步增大,左邊的增到最大,右峰完全消失,變成了單峰結(jié)構(gòu).

圖3為穩(wěn)態(tài)概率密度函數(shù)ρst(x)在不同加性噪聲強(qiáng)度Q下的變化曲線.當(dāng)參數(shù)P=0.5,λ=0.1時(shí),可以看出,隨著Q增加,穩(wěn)態(tài)概率密度函數(shù)的左右峰均下降,中峰上升,但還保留著三峰結(jié)構(gòu).表明隨著Q的增加,穩(wěn)態(tài)概率密度函數(shù)峰的個(gè)數(shù)沒(méi)有發(fā)生變化,即Q不可以誘導(dǎo)相變現(xiàn)象的產(chǎn)生.

圖4給出了穩(wěn)態(tài)概率密度函數(shù)ρst(x)作為x的函數(shù)，乘性噪聲強(qiáng)度P取不同值.當(dāng)參數(shù)Q=0.5,λ=0.1時(shí)，可以看出，隨著P的增大，穩(wěn)態(tài)概率密度函有選舉權(quán)的左峰比右峰下降的速度更快，而中峰卻慢慢上升，峰的結(jié)構(gòu)由雙峰變成了三峰.表明隨著P的增加，穩(wěn)態(tài)概率密度函數(shù)峰的結(jié)構(gòu)發(fā)生了變化，即P可以誘導(dǎo)相變現(xiàn)象產(chǎn)生.

為了進(jìn)一步驗(yàn)證白噪聲作用下三穩(wěn)系統(tǒng)的穩(wěn)態(tài)概率密度函數(shù)，對(duì)其進(jìn)行模擬.這里對(duì)方程(1)采取向前Eular算法，利用Box-Muller算法[18]產(chǎn)生高斯白噪聲.結(jié)果如圖5～7所示.可以看到白噪聲參數(shù)對(duì)穩(wěn)態(tài)概率密度函數(shù)的影響和理論結(jié)果是一致的.

圖4 穩(wěn)態(tài)概率密度函數(shù)在不同乘性噪聲強(qiáng)度下的變化曲線

Fig.4 Plot of the stationary probability disti-bution function for different multiplic-ative noise intensities

圖6 穩(wěn)態(tài)概率密度函數(shù)在不同加性噪聲強(qiáng)度下的數(shù)值模擬圖

Fig.6 Simulation results for the stationary probability distribution function with different additive noise intensities

2 結(jié) 論

本文利用理論近似和數(shù)值模擬研究了乘性和加性高斯白噪聲共同激勵(lì)下的一維三穩(wěn)系統(tǒng)的動(dòng)力學(xué)性質(zhì).發(fā)現(xiàn)關(guān)聯(lián)強(qiáng)度λ,加性噪聲強(qiáng)度Q和乘性噪聲強(qiáng)度P對(duì)概率密度函數(shù)有較大的影響,并得出以下結(jié)論:

(1) 關(guān)聯(lián)強(qiáng)度λ的增加能引起穩(wěn)態(tài)概率函數(shù)峰結(jié)構(gòu)的變化,使得從三峰結(jié)構(gòu)轉(zhuǎn)變?yōu)閱畏?從而峰數(shù)發(fā)生了變化,即λ能夠引起相變;

(2) 加性噪聲強(qiáng)度Q的改變使得穩(wěn)態(tài)概率密度函數(shù)的左峰和右峰下降,中峰上升,峰的結(jié)構(gòu)卻沒(méi)有發(fā)生變化,還保持著三峰結(jié)構(gòu),即Q不能誘導(dǎo)相變的產(chǎn)生;

(3) 乘性噪聲強(qiáng)度P的增大能夠引起穩(wěn)態(tài)概率密度函數(shù)從雙峰結(jié)構(gòu)向三峰結(jié)構(gòu)變化,左峰和右峰下降,中峰上升,峰數(shù)發(fā)生變化,即P能夠引起相變的發(fā)生.

[1] JIN Yanfei,XU Wei,LI Wei,et al.Steady-state analysis of a bistable duffing model driven by additive and multiplicative colored noises with a colored correlated noise[J].Jouranl of Dynamics and Control,2005,3(2): 60-65.

[2] WANG Canjun,QUN Wei,MEI Dongcheng.Assoicated relaxation time and the correlation function for a tumor cell growth system subjected to color noises[J].Physical Letters A,2008,372(13):2176-2182.

[3] 胡崗.隨機(jī)力與非線性系統(tǒng)[M].上海:上?？萍冀逃霭嫔?1994.

HU Gang.Stochastic forces and nonlinear systems[M].Shanghai:Shanghai Scientific and Technological Education Publishing House，1994.

[4] ZHAO Yan,XU Wei,ZOU Shaocun.The steady state probability distribution and mean first passage time of FHN neural system driven by non-Gaussian noise[J].Acta Physica Sinica,2009,58(3):1396-1402.

[5] WANG Canjun.Colored noise induced switch in the gene transcriptional regulatory system[J].Acta Physica Sinica,2012,61(1): 010503.

[6] WU Dan,LUO Xiaoqin,ZHU Shiqun.Flutuations of single-mode laser driven by two different kinds of colored noise[J].Communications in Theoretical Physics,2006,45(4): 630-636.

[7] JIA Ya,LI Jiarong.Stead-state analysis of a bistable system with addibive and multiplicative noises [J].Physical Review Letters,1996,53(6): 5786-5792.

[8] XU Yong,FENG Jing,LI Juanjuan,et al.Stochastic bifurcation for tumor-immune system with symmetric Levy noise[J].Physical Letters A,2013,392(20): 4739-4748.

[9] WU Dajin,CAO Li,KE Shengzhi.Bistable kinetic model driven by correlated noises:Steady-state analysis [J].Physical Review Letters,1994,50(4): 2496-2502.

[10] CHEN Jun,CAO Li,WU Dajin.Bistable kinetic model driven by correlation noises:Unified colored-noise approxiamtion[J].Physical Review Letters,1995,52(3): 3228-3231.

[11] JIN Yanfei,XU Wei,LI Wei,et al.Stochastic resonance in an asymmetric bistable system driven by multiplicative and additive noise[J].Physics Reports B,2005,14(6):1077-1081.

[12] 胡麗萍，李鑫.高斯白噪聲對(duì)神經(jīng)元映射模型隨機(jī)共振的影響[J].紡織高校基礎(chǔ)科學(xué)學(xué)報(bào)，2014，27(4):492-495.

HU Liping,LI Xin.Impact of Gaussian white noise on stochastic resonance of map-based neural model[J].Basic Sciences Journal of Textile Universities,2014,27(4): 492-495.

[13] JIA Zhenglin.Time-delay induced reentrance phenomenon in a triple-well potential system driven by cross-corrslated noises[J].International Journal of Theoretical Physics,2009,48(1): 226-231.

[14] SHI Peiming,LI Pei,HAN Dongying.Steady-state analysis of a tristable system driven by a correlated multiplicative and and additive colored noises[J].Acta Physica Sinica,2014,63(17): 170504.

[15] ZHANG Huiqing,YANG Tingting,XU Wei.Effects of non-Gaussian noise on logical stochastic resonance in a triple-well potential system[J].Nonlinear Dynamics 2014,76(1): 649-656.

[16] BENZI R,SUTERA A,VULPINAI A.The mechanism of stochastic resonance[J].Journal of Physics A:Mathematical and Geneal,1981,14(11):L453-L457.

[17] LIANG Guiyun,CAO Lin,WU Dajin.Approximate Fokker-Planck equation of system driven by multiplicative colored noises with colored cross-correlation[J].Physical Letters A,2004,335(3/4): 371-384.

[18] KNUTH D E.The art of computer programming Vol.2[M].MA:Addiso-Wesley,Reading,1969.

編輯、校對(duì)：師 瑯

Steady-state analysis of a triple-well potential system with correlations between multiplicative and additive noise

ZHANGBu,NINGLijuan

(College of Mathematics and Information Science,Shaanxi Normal University,Xi′an 710062,China)

By virtue of the Liouville equation and Novikov theorem,the expressions of the stationary probability distribution function of one-dimensional triple-well potential system with correlations between multiplicative and additive noises are investigated. The results show that the intensity of cross-correlationλand the intensity of multiplicative noisePcan induce transition; the intensity of additive noiseQcan not induce transition.The accuracy of the conclusion is verified by using numerical simulation stationary probability distribution function.

triple-well system; noise; transition; stationary probability distribution function

1006-8341(2016)04-0496-05

10.13338/j.issn.1006-8341.2016.04.014

2016-09-28

國(guó)家自然科學(xué)基金資助項(xiàng)目(11202120)

寧麗娟(1977—),女,陜西省西安市人,陜西師范大學(xué)副教授,博士，研究方向?yàn)殡S機(jī)動(dòng)力學(xué).

E-mail:ninglijuan@snnu.edu.cn

張卜,寧麗娟.關(guān)聯(lián)的乘性和加性驅(qū)動(dòng)的三穩(wěn)系統(tǒng)穩(wěn)態(tài)分析[J].紡織高?；A(chǔ)科學(xué)學(xué)報(bào),2016,29(4)：496-500.

ZHANG Bu,NING Lijuan.Steady-state analysis of a triple-well potential system with correlations between multiplicative and additive noise[J].Basic Sciences Journal of Textile Universities,2016,29(4):496-500.

O 242.1

A