羅瑞芬，張建剛，杜文舉

(蘭州交通大學數(shù)理學院，甘肅蘭州 730070)

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一個帶有隨機參數(shù)的新的二維混沌系統(tǒng)的隨機Hopf分岔分析

羅瑞芬，張建剛，杜文舉

(蘭州交通大學數(shù)理學院，甘肅蘭州 730070)

研究了一個帶有隨機參數(shù)的新的二維混沌系統(tǒng)的隨機Hopf分岔.首先根據(jù)正交多項式逼近將此帶有隨機參數(shù)的二維混沌系統(tǒng)化為等價的確定性系統(tǒng)，然后通過第一Lyapunov系數(shù)法研究了等價的確定性系統(tǒng)的Hopf分岔.研究發(fā)現(xiàn)，隨機系統(tǒng)與確定性系統(tǒng)不同，隨機Hopf分岔的臨界值不僅是由隨機系統(tǒng)中的確定性參數(shù)決定，隨機參數(shù)的強度也是決定因素之一.最后，理論結(jié)果通過數(shù)值模擬被證實.

穩(wěn)定性；隨機混沌；隨機Hopf分岔；Chebyshev多項式逼近

Hopf分岔一直是非線性動力學領域中的一個研究熱點，特別是近幾年，許多學者對不同系統(tǒng)的Hopf分岔現(xiàn)象進行了較為深入的研究，涉及的領域也非常廣泛，有化學工廠[1]、機械系統(tǒng)[2]、生態(tài)系統(tǒng)[3]、金融系統(tǒng)[4-5]、生物系統(tǒng)[6-7]、計算機網(wǎng)絡[8-9]等.由于系統(tǒng)的復雜性，這些研究大體上僅限于定性階段，對于系統(tǒng)的定量研究還相當有限，特別是對隨機分岔的研究還處于初始階段.隨機系統(tǒng)在自然界中廣泛存在，越來越多地被用來刻畫事物間的動態(tài)關系，尤其是含有隨機參數(shù)的隨機系統(tǒng).目前，解決含有隨機參數(shù)的系統(tǒng)問題，有三個數(shù)學方法比較常用：蒙特卡羅方法[10]、隨機微擾法和正交多項式逼近法[11]，其中第三種方法后來由Li[12]做了改善，是個有效的分析方法[13].近幾年，利用Chebyshev多項式逼近法對一些經(jīng)典的動力學模型的隨機分岔和混沌現(xiàn)象[13-16]進行了成功的分析.Fang[17]運用Chebyshev多項式逼近法研究了有界隨機變量的隨機參數(shù)系統(tǒng)，李永坤等也運用此方法研究了隨機動力系統(tǒng)的分岔與混沌現(xiàn)象以及控制和同步問題[18-22]，結(jié)果表明，Chebyshev多項式逼近法是研究含有隨機參數(shù)的隨機動力學問題的有效方法.本文用相同的方法對一個新的二維混沌系統(tǒng)的隨機混沌和Hopf分岔進行了研究，并借助Maple軟件詳細討論了這個系統(tǒng)的Hopf分岔現(xiàn)象，給出了這個系統(tǒng)產(chǎn)生Hopf分岔的參數(shù)條件.該方法對分析其它系統(tǒng)的Hopf分岔現(xiàn)象具有一定的參考意義.

1 一個帶有隨機參數(shù)的新的二維混沌系統(tǒng)的正交多項式逼近

確定性的二維混沌系統(tǒng)：

相應的方程（3）選擇第二類切比雪夫多項式[16].ρU(u)表示第i個正交多項式，M表示選取多項式的最大階數(shù)，同時正交多項式的正交性能可表達成如下形式：

的正交性，可以得到等價的確定性系統(tǒng).當M →∞，方程（5）嚴格成立，否則方程（5）就是個近似方程.本文中取2M=，可得等價確定性的近似方程如下：

2 隨機Hopf分岔分析

2.1 Hopf分岔的存在性

借助Maple，獲得特征方程如下：

2.2 分岔的趨勢和穩(wěn)定性

為了進一步研究Hopf分岔，對系統(tǒng)（6）的第一Lyapunov系數(shù)進行研究.

當 L1＜0時，平衡點漸進穩(wěn)定，發(fā)生超臨界的Hopf分岔，并且在平衡點附近存在穩(wěn)定的極限環(huán)；當 L1＞0時，平衡點不穩(wěn)定，發(fā)生亞臨界的Hopf分岔，并在平衡點附近存在不穩(wěn)定的極限環(huán).

令Cn是一個定義在復數(shù)域C上的非線性空間，p, q∈Cn，其中q∈Cn是特征值 Ρ1所對的特征向量，p∈Cn是一個伴隨特征向量，滿足：

當a=a0時，方程（6）有平衡點 o(0， 0， 0， 0， 0， 0)，方程（6）能表達成：

這里的x=(x0,y0, x1, y1, x2,y2)，B(x, y)，C（x, y, z）分別是雙線性和三線性函數(shù)，可表示為：

這里的K見附錄.也可得到：

并且有：

3 結(jié) 語

本文通過正交多項式逼近來研究含有隨機參數(shù)的二維混沌系統(tǒng)的隨機Hopf分岔現(xiàn)象，利用正交多項式逼近將隨機二維混沌系統(tǒng)化為等價的確定性系統(tǒng)，然后運用Lyapunov系數(shù)法對等價的確定性系統(tǒng)的Hopf分岔進行研究.結(jié)果表明，隨機二維混沌系統(tǒng)中的隨機Hopf分岔與傳統(tǒng)的Hopf分岔不同，隨機Hopf分岔的臨界值不僅是由隨機系統(tǒng)中的確定性參數(shù)決定，隨機參數(shù)的強度也是決定因素之一.有關該系統(tǒng)在控制和同步方面的一些問題還需要進一步研究.

[1] Continillo G, Grabski A, Mancusi E, et al. Parallel tools for the bifurcation analysis of large-scale chemically reactive dynamical systems [J]. Computers Chemical Engineering, 2012, 38: 94-105.

[2] Walter V W. Stability and bifurcation in multi-scaled stochastic mechanic [J]. Procedia IUTAM, 2013, 6: 169-179.

[3] Huang D, Wang H. Hopf bifurcation of the stochastic model on HAB nonlinear stochastic dynamics [J]. Chaos, Solitions and Fractals, 2006, 27: 1072-1079.

[4] Zhang Q, Xu Z Z, Feng T J, et al. A dynamic stochastic frontier model to evaluate regional efficiency: Evidence fromChinese county-level panel data [J]. European Journal of Operational Research, 2015, 241: 907-916.

[5] Klaus B, Klijn F, Walzl M. Stochastic stability for roommate markets [J]. Journal of Economic Theory, 2010, 145: 2218-2240.

[6] Vinals J, Lepine F, Gaudreault M. Pitchfork and Hopf bifurcations in stochastic regulatory networks with delayed feedback [J]. Biophysical Journal, 2009, 96: 305.

[7] Hasegawa H. Stochastic bifurcation in FitzHugh-Nagumo ensembles subjected to additive and/or multiplicative noises [J]. Physica D, 2008, 237: 137-155.

[8] Li W, Su H, Wang K. Global stability analysis for stochastic coupled systems on networks [J]. Automatica, 2011, 47: 215-220.

[9] Huang Z T, Yang Q G, Can J F. The stochastic stability and bifurcation behavior of an Internet congestion control model [J]. Mathematical and Computer Modelling, 2011, 54: 1954-1965.

[10] Zhao H Y, Huang X X, Zhang X B. Hopf bifurcation and harvesting control of a bioeconomic plankton model with delay and diffusion terms [J]. Physica A, 2015, 421: 300-315.

[11] Zhang Y, Xu W, Fang T. Stochastic Hopf bifurcation and chaos of stochastic Bonhoeffer-vander Pol system via Chebyshev polynomial approximation [J]. Applied Mathematics and Computation, 2007, 190: 1225-1236.

[12] Li J. The expanded order system method of combined random vibration analysis [J]. Acta Mechanica Sinica, 1996, 28: 63-68.

[13] Ma S J. The stochastic Hopf bifurcation analysis in Brusselator system with random parameter [J]. Applied Mathematics and Computation, 2012, 219: 306-319.

[14] Pandey R K, Suman S, Singh K K, et al. An approximate method for Abel inversion using Chebyshev polynomials [J]. Applied Mathematics and Computation, 2014, 237: 120-132.

[15] Eslahchi M R, Dehghan M, Amani S. The third and fourth kinds of Chebyshev polynomials and best uniform Approximation [J]. Mathematical and Computer Modelling, 2012, 55: 1746-1762.

[16] Ma S J, Xu W, Li W, et al. Period-doubling bifurcation analysis of stochastic van der Pol system via Chebyshev polynomial approximation [J]. Acta Physica Sinica, 2005, 54: 3508-3515.

[17] Fang T, Leng X L, Song C Q. Chebyshev polynomials approximation for dynamical response problem of random system [J]. Journal of sound and vibration, 2003, 226:198-206.

[18] Li Y K, Li C Z. Stability and Hopf bifurcation analysis on a delayed Leslie-Gower Predator-prey system incorporating a prey refuge [J]. Applied Mathematics and Computation, 2013, 219: 4576-4589.

[19] Hu Z Y, Teng Z D , Zhang L. Stability and bifurcation analysis in a discrete SIR epidemic model [J]. Mathematics and Computers in Simulation, 2014, 97: 80-93.

[20] Anton C, Deng J, Wong Y S. Hopf bifurcation analysis of an aeroelastic model using stochastic normal form [J]. Journal of Sound and Vibration, 2012, 331: 3866-3886.

[21] Zhang G D, Shen Y, Chen B. Bifurcation analysis in a discrete differential-algebraic Predator-prey system [J]. Applied Mathematical Modelling, 2014, 38: 4835-4848.

[22] Wang B. Existence, stability and bifurcation of random complete and periodic solutions of stochastic parabolic equations [J]. Nonlinear Analysis, 2014, 103: 9-25.

[23] Hassard B, Kazarinoff N, Wan Y. Theory and Application of Hopf bifurcation [M]. Cambridge: CambridgeUniversity Press, 1981: 306-319.

附錄：

(編輯：王一芳)

Stochastic Hopf Bifurcation Analysis of a Novel Two-dimensional Chaotic System with Random Parameters

LUO Ruifen, ZHANG Jiangang, DU Wenju
(School of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou, China 730070)

It is probed in this paper that a novel two-dimensional chaotic system with random parameter is proposed. First of all, the system is trasformed from two-dimensional chaotic system with stochastic parameters to the equivalent deterministic system based on orthogonal polynomials approximation. Then the Hopf bifurcation of equivalent deterministic system is studied by the calculation of the first Lyapunov coefficient method. It is discovered from the research that the stochastic system differs from the deterministic system. The critical value of stochastic Hopf bifurcation is determined not only by deterministic parameters but also the intensity of random parameters in stochastic system which is one of the decisive factors. Finally, the numerical simulations results show the effectiveness of the method and the correctness of the theoretical results in the paper.

Stability; Stochastic Chaos; Stochastic Hopf Bifurcation; Chebyshev Polynomial Approximation

O175.12

A

1674-3563(2016)01-0026-10

10.3875/j.issn.1674-3563.2016.01.004 本文的PDF文件可以從xuebao.wzu.edu.cn獲得

2015-05-07

羅瑞芬(1990- )，女，甘肅臨夏人，碩士研究生，研究方向：隨機動力學