两类随机非线性动力系统和经济应用的研究
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摘要
随机非线性动力系统的随机响应、随机稳定性、随机分岔、随机混沌、首次穿越及其最优控制问题，无论是理论上还是应用上都是值得研究和非常重要的问题。近些年，对于这些问题的研究，在理论上已取得一些初步的研究成果，但是对于一些较为复杂的典型动力学系统在不同随机激励下的动力学行为的研究仍处于起步阶段。另外，利用随机非线性动力学方程来研究经济、金融领域的问题还很不成熟。鉴于此，本文将主要研究两类典型的非线性动力系统—Mathieu-Duffing振子和Rayleigh振子在随机噪声的激励下的动力学行为及其一类系统在物价模型中的应用。
第一部分较为系统的研究了随机Mathieu-Duffing系统的动力学行为。利用多尺度法研究了Gaussian白噪声激励下该系统的最大Lyapunov指数、零解和非零解的稳定性，研究结果表明：在参数主共振Ω=2ω_0附近邻域，最大Lyapunov指数λ在调谐参数σ=0时取得最大值，而随着σ的绝对值的增大，最大Lvapunov指数λ将趋于零。这说明，零解的稳定性随着调谐参数σ的绝对值的增大而趋于稳定；而对于系统的非零稳态响应，研究结果表明：非零解稳定的充要条件是-3βαa_0 cosη_0／2ω_0＞0。
第二部分研究了有界噪声参激下带有Van der Pol阻尼项的Mathieu-Duffing系统的混沌运动。首先，利用随机Melnikov方法推导了随机Van der PolMathieu-Duffing方程的随机Melnikov过程，由广义过程在均方意义上出现简单零点给出了可能出现混沌运动的临界激励幅值，从而预测该系统可能存在混沌运动的参数域。然后，用数值方法计算该系统的最大Lyapunov指数、Poincare截面、时间历程图和相轨图，并用最大Lyapunov指数和Poincare截面验证解析结果及研究有界噪声的强度对该系统产生随机混沌的临界幅值的影响。结果表明：在ω_0：Ω_1：Ω_2=1：2：2和ω_0：Ω_1：Ω_2=1：1：2两种情形下，随着噪声强度的增大，随机系统的混沌吸引子不断扩散，随机激励的强度δ越大混沌吸引子的区域越大。说明混沌响应加强，即随机噪声会诱导混沌的产生。
第三部分主要对确定性物价模型引入了噪声，建立了非线性随机动力学模型—随机Rayleigh方程，在此基础上利用动力学方法对该模型进行了系统研究。首先，对于非线性随机物价模型进行了最优化控制，通过对系统采用合理的控制策略，使得物价系统能在尽可能长的时间周期内处于平稳状态，避免出现通货膨胀或通货紧缩，这对于国家的经济良性发展也具有重要的意义。接着，对物价模型——Rayleigh方程在不同噪声激励下的稳态响应、首次穿越和随机Hopf分岔问题进行了进一步的讨论。具体的讲，研究了Rayleigh方程在Gaussian白噪声外激下的平稳响应首次穿越，利用随机平均法得出系统的随机平均It(?)微分方程，对平均方程建立条件可靠性函数的后向Kolmogorov方程及首次穿越时间条件矩的Pontragin方程，通过这两个方程研究该系统的非线性阻尼项、随机激励项对系统平稳响应与首次穿越的影响；利用拟Hamilton系统随机平均法及扩散过程的奇异点理论研究了Gaussian白噪声参激下Rayleigh方程的随机Hopf分岔，并利用最大Lyapunov指数方法和数值方法对结论进行的验证。
第四部分是全文的总结和展望。主要介绍文章的创新点和存在的不足，同时，提出在本研究工作的基础上，还有待进一步研究的问题以及所研究的方法和有可能碰到的困难。
It is very important and worthwhile to study random responses, random stabilities,random bifurcations, first-passage failures and random optimal control of stochasticnonlinear dynamical system in theory and its applications. In recent years, there aremany preliminary research results in theory about stochastic nonlinear science.However, the results of some complicated classical dynamical systems are notcompletely, and some results can not agree with others. In addition, there is sufficientwork to do as how to applying stochastic nonlinear dynamical methods to solve realproblems of economics and finance. Considering that, in the present paper, the authorhas studied the behaviors of two classical nonlinear dynamical systems, which areMathieu-Duffing system and Rayleigh oscillator, under different kinds of stochasticnoise excitations. At the same time, the author has studied a price model by usingstochastic nonlinear dynamical equations.
In the first section, the dynamical behaviors of stochastic Mathieu-Duffing systemhave been studied. Firstly, the author has studied the maximal Lyapunov exponentsand almost-sure stability of Mathieu-Duffing equation under Gaussian white noiseexcitation by multiple scales method. The results show that near the parametericresonance at excitation frequencyΩ=2ω_0, the system become more unstable with theincreasing of the maximal Lyapunov exponentλ, and the maximal Lyapunovexponents reache its maximum value whenσ=0. That is to say, the stability of trivialsolution of stochastic Mathieu-Duffing equation is more stable whileσtends tozero. But for the stability of nontrivial solution, the author has got the sufficient andnecessary condition is-3βαa_0 cosη_0/2ω_0＞0.
In the second section, the author has studied the onset of chaotic motion of Van der Pol-Mathieu-Duffing system under bounded noise excitation. By using randomMelnikov technique, a mean square criterion is used to detect the necessary conditionfor chaotic motion of this stochastic system. Then, the Poincare map and Lyapunovexponents have been calculated by numerical method. The results show that when thenoise is presented in the system, the shapes of Poincare maps begin to diffuse intolarger area and the threshold of bounded noise amplitude for the onset of chaos in thissystem increases as the intensity of the bounder noise increases, which is furtherverified by the maximal Lyapunov exponents of the system, i.e.,whenω_0:Ω_1:Ω_2=1:2:2 orω_0:Ω_1:Ω_2=1:1:2, the chaotic motions of thesystem can be enhanced by the bound noise.
In the third section, considering the government macro-control for price system asexternal control force, a dynamically controlled nonlinear stochastic price model isproposed based on deterministic model. By using stochastic averaging method andnonlinear stochastic control strategy, the stochastic model can been stabilized. Thecontrolling aim is to enable the system becoming more stable. Through comparingthe controlled Lyapunov exponents with uncontrolled Lyapunov exponents, it isshown that the controlled strategy is effective. This is very important to ourgovernment. Then, the author has not only studied the steady-state response andfirst-passage failure but also the applications in price modeling of Rayleigh oscillatorunder external and parametric excitations. Firstly, steady-state response and the firstpassage failure of Rayleigh oscillator under external Gaussian white noise excitationhave been studied. The equation of motion of the system is first reduced to averagedIto stochastic differential equations by using the stochastic averaging method. Abackward Kolmogorov equation governing the conditional reliability function and ageneralized Pontrayagin equation governing the conditional monments offirst-passage time are established. Statistical properties of the stationary response andfirst-passage failure are analyzed by using numerical results in three groups of varying parameters. Secondly, the stochastic Hopf bifurcation of Rayleigh oscillator subject toGaussian white noise parametric excitation is studied. The equation of motion of thesystem is reduced to an averaged It(?) stochastic differential equations by using thestochastic averaging method. Then, the critical D-bifurcation parameterc_D=-D/2μis determined approximately by using lyapunov exponents of theinvariable measureδ_c is trivial, and the critical P-bifurcation parameter valuec_P=0 is determined approximately by using the theory of singular points ofdiffusion processes. Finally, the procedure has been verified with the numericalsolution of the joint probability density of the stochastic system.
In the last section, the author has introduced the creations and the problems thatwill be further studied.
第一部分较为系统的研究了随机Mathieu-Duffing系统的动力学行为。利用多尺度法研究了Gaussian白噪声激励下该系统的最大Lyapunov指数、零解和非零解的稳定性，研究结果表明：在参数主共振Ω=2ω_0附近邻域，最大Lyapunov指数λ在调谐参数σ=0时取得最大值，而随着σ的绝对值的增大，最大Lvapunov指数λ将趋于零。这说明，零解的稳定性随着调谐参数σ的绝对值的增大而趋于稳定；而对于系统的非零稳态响应，研究结果表明：非零解稳定的充要条件是-3βαa_0 cosη_0／2ω_0＞0。
第二部分研究了有界噪声参激下带有Van der Pol阻尼项的Mathieu-Duffing系统的混沌运动。首先，利用随机Melnikov方法推导了随机Van der PolMathieu-Duffing方程的随机Melnikov过程，由广义过程在均方意义上出现简单零点给出了可能出现混沌运动的临界激励幅值，从而预测该系统可能存在混沌运动的参数域。然后，用数值方法计算该系统的最大Lyapunov指数、Poincare截面、时间历程图和相轨图，并用最大Lyapunov指数和Poincare截面验证解析结果及研究有界噪声的强度对该系统产生随机混沌的临界幅值的影响。结果表明：在ω_0：Ω_1：Ω_2=1：2：2和ω_0：Ω_1：Ω_2=1：1：2两种情形下，随着噪声强度的增大，随机系统的混沌吸引子不断扩散，随机激励的强度δ越大混沌吸引子的区域越大。说明混沌响应加强，即随机噪声会诱导混沌的产生。
第三部分主要对确定性物价模型引入了噪声，建立了非线性随机动力学模型—随机Rayleigh方程，在此基础上利用动力学方法对该模型进行了系统研究。首先，对于非线性随机物价模型进行了最优化控制，通过对系统采用合理的控制策略，使得物价系统能在尽可能长的时间周期内处于平稳状态，避免出现通货膨胀或通货紧缩，这对于国家的经济良性发展也具有重要的意义。接着，对物价模型——Rayleigh方程在不同噪声激励下的稳态响应、首次穿越和随机Hopf分岔问题进行了进一步的讨论。具体的讲，研究了Rayleigh方程在Gaussian白噪声外激下的平稳响应首次穿越，利用随机平均法得出系统的随机平均It(?)微分方程，对平均方程建立条件可靠性函数的后向Kolmogorov方程及首次穿越时间条件矩的Pontragin方程，通过这两个方程研究该系统的非线性阻尼项、随机激励项对系统平稳响应与首次穿越的影响；利用拟Hamilton系统随机平均法及扩散过程的奇异点理论研究了Gaussian白噪声参激下Rayleigh方程的随机Hopf分岔，并利用最大Lyapunov指数方法和数值方法对结论进行的验证。
第四部分是全文的总结和展望。主要介绍文章的创新点和存在的不足，同时，提出在本研究工作的基础上，还有待进一步研究的问题以及所研究的方法和有可能碰到的困难。
It is very important and worthwhile to study random responses, random stabilities,random bifurcations, first-passage failures and random optimal control of stochasticnonlinear dynamical system in theory and its applications. In recent years, there aremany preliminary research results in theory about stochastic nonlinear science.However, the results of some complicated classical dynamical systems are notcompletely, and some results can not agree with others. In addition, there is sufficientwork to do as how to applying stochastic nonlinear dynamical methods to solve realproblems of economics and finance. Considering that, in the present paper, the authorhas studied the behaviors of two classical nonlinear dynamical systems, which areMathieu-Duffing system and Rayleigh oscillator, under different kinds of stochasticnoise excitations. At the same time, the author has studied a price model by usingstochastic nonlinear dynamical equations.
In the first section, the dynamical behaviors of stochastic Mathieu-Duffing systemhave been studied. Firstly, the author has studied the maximal Lyapunov exponentsand almost-sure stability of Mathieu-Duffing equation under Gaussian white noiseexcitation by multiple scales method. The results show that near the parametericresonance at excitation frequencyΩ=2ω_0, the system become more unstable with theincreasing of the maximal Lyapunov exponentλ, and the maximal Lyapunovexponents reache its maximum value whenσ=0. That is to say, the stability of trivialsolution of stochastic Mathieu-Duffing equation is more stable whileσtends tozero. But for the stability of nontrivial solution, the author has got the sufficient andnecessary condition is-3βαa_0 cosη_0/2ω_0＞0.
In the second section, the author has studied the onset of chaotic motion of Van der Pol-Mathieu-Duffing system under bounded noise excitation. By using randomMelnikov technique, a mean square criterion is used to detect the necessary conditionfor chaotic motion of this stochastic system. Then, the Poincare map and Lyapunovexponents have been calculated by numerical method. The results show that when thenoise is presented in the system, the shapes of Poincare maps begin to diffuse intolarger area and the threshold of bounded noise amplitude for the onset of chaos in thissystem increases as the intensity of the bounder noise increases, which is furtherverified by the maximal Lyapunov exponents of the system, i.e.,whenω_0:Ω_1:Ω_2=1:2:2 orω_0:Ω_1:Ω_2=1:1:2, the chaotic motions of thesystem can be enhanced by the bound noise.
In the third section, considering the government macro-control for price system asexternal control force, a dynamically controlled nonlinear stochastic price model isproposed based on deterministic model. By using stochastic averaging method andnonlinear stochastic control strategy, the stochastic model can been stabilized. Thecontrolling aim is to enable the system becoming more stable. Through comparingthe controlled Lyapunov exponents with uncontrolled Lyapunov exponents, it isshown that the controlled strategy is effective. This is very important to ourgovernment. Then, the author has not only studied the steady-state response andfirst-passage failure but also the applications in price modeling of Rayleigh oscillatorunder external and parametric excitations. Firstly, steady-state response and the firstpassage failure of Rayleigh oscillator under external Gaussian white noise excitationhave been studied. The equation of motion of the system is first reduced to averagedIto stochastic differential equations by using the stochastic averaging method. Abackward Kolmogorov equation governing the conditional reliability function and ageneralized Pontrayagin equation governing the conditional monments offirst-passage time are established. Statistical properties of the stationary response andfirst-passage failure are analyzed by using numerical results in three groups of varying parameters. Secondly, the stochastic Hopf bifurcation of Rayleigh oscillator subject toGaussian white noise parametric excitation is studied. The equation of motion of thesystem is reduced to an averaged It(?) stochastic differential equations by using thestochastic averaging method. Then, the critical D-bifurcation parameterc_D=-D/2μis determined approximately by using lyapunov exponents of theinvariable measureδ_c is trivial, and the critical P-bifurcation parameter valuec_P=0 is determined approximately by using the theory of singular points ofdiffusion processes. Finally, the procedure has been verified with the numericalsolution of the joint probability density of the stochastic system.
In the last section, the author has introduced the creations and the problems thatwill be further studied.
引文
[1] 陈予恕，唐云等．非线性动力学中的现代分析方法论[M]．北京：科学出版社，2000：221-262
[2] Spanos, PD. Stochastic Linearization in Structural Dynamics [J]. Applied Mechanics Review, 1981, 34(1): 1-8
[3] Robert JB and Spanos PD. Random Vibration and Statistical Linearization [M]. New York: Wily, 1990
[4] 陆启韶．非线性复杂系统的动力学问题[J]．新疆大学学报(理工版)，2001，18(2)：141-146
[5] Stratonovich RL. Topics in the Theory of Random Noise [M]. New York: Gordon and Breach, 1967, Vol.2
[6] Khasminskii RZ. Principle of averaging of parabolic and elliptic differential equations for Markov process with small diffusion [J]. Theory Probability and Applications, 1963, 8:1-24
[7] Khasminskii RZ. A limit theorem for the solution of differential equations with random right-hand side [J]. Theory Probability and Applications, 1963, 11:309-405
[8] Zhu W Q. Stochastic averaging methods in random vibration [J]. ASME Applied Mechanics Review, 1988, 41(5): 189-199
[9] Zhu W Q. Recent developments and applications of the stochastic averaging method in random vibration [J]. ASME Applied Mechanics Review, 1996, 49(10):s72-s80
[10] Robert JB and Spanos PD. Stochastic averaging: an approximate method of solving random vibration problems [J]. International Journal of Non-linear mechanics., 1986, 21:111-134
[11] Risken H. The Fokker-Planck Equation [M]. New York: Springer, 1983
[12] Lin Y K and Cai G. Q. Probability Structural Dynamics: Advanced Theory and Applications. Singapore: McGraw-Hill, 1995
[13] Utz V W and Wedig W V. On the Calculation of Stationary Solutions of Multi-Dimensional Fokker-Planck Equations by Orthogonal Functions [J]. Nonlinear Dynamics, 2000, 21:289-306
[14] 朱位秋．非线性随机动力学与控制—Hamilton理论体系框架[M]．北京：科学出版社，2003
[15] 朱位秋．随机振动[M]．北京：科学出版社，1998：61-68，223-311
[16] Arnold L. Random Dynamical System [M]. Berlin-Heidelberg: Springer, 1998
[17] Zhu W Q., Huang Z L. Stochastic Hopf bifurcation of quasi-nonintegrable-Hamiltionian systems, Int. J. Non-linear Mechanics [J]. 1999, 34:437-447
[18] Cox D R, Miller H D. The Theory of Stochastic Processes [M], New York: Chapman and Hall, 1965.
[19] Bharucha-Reid A T. Elements of Markov Processes and Their Applications [M]. Now York: McGraw-Hill, 1960
[20] Zhu W Q., Deng M L., Huang Z. L. First-passage failure of quasi-integrable Hamiltonian systems [J]. ASME Journal of Applied Mechanics, 2002, 69: 274-282.
[21] Pontryagin L S, Boltyanski V G, Gamkrelidze RV, Mischenko E F. Mathematical Theory of Optimal Processes [M]. New York: Wily, 1962
[22] Kushner H J. Stochastic Stability and control [M]. New York: Academic Press, 1967
[23] Fleming WH, Rishel R W. Deterministic and Sttochastic Optimal Control [M]. New York: Springer-Verlag, 1975
[24] Fleming W H, Soner H M. Controlled Markov Processes and Viscosity Solutions [M]. Now York: Springer-Verlag, 1992
[25] Kushner H J, Dupuis P. Numerical Methods for Stochastic Control Problems in Continuous Time [M]. Now York: Springer-Verlag,1993
[26] Barnett, W. A., et al, (eds.), Economic Complexity[M]. Cambridge University Press, 1989
[27] Boldrin, M., and Montrucchio, L. On the Indeterminacy of Capital Accumulation Paths[J]. Journal of Economic Theory, 1986, 40, 26-39
[28] Brock, W. A., and C. Sayers" Is the Business Cycle Characterized by Deterministic Chaos[J]? Journal of Monetary Economics, 1988, 22, 71-90
[29] 徐龙炳，陆蓉，R／S分析探索中国股票市场的非线性[J]．预测，1999，2
[30] 徐寅峰，经济模型及经济混沌[J]．西安交通大学学报，1994，28(3)，83-86．
[31] 张立洪．经济数据中存在混沌吗[J]?自然杂志，1993，16(9-10)，12-16
[32] 徐前方．上证指数中的奇异吸引子[J]．数量经济技术经济研究，1994年第2期．
[33] 黄登仕、李后强．非线性经济学的理论与方法[M]．四川大学出版社，1993年。
[34] H. W. Lorenz. Multiple Attractors, Complex Basin Boundaries, and Transient Motion in Deterministic Economic Systems[M]. Dynamic Economic Models and Optimal Control, 1992, p411-415.
[35] 刘凯 湛垦华．非线性经济学研究的回顾与前瞻[J]．西安电子科技大学学报，1999年，第3期。
[36] 刘延柱，陈文良，陈立群．振动力学[M]．北京：高等教育出版社，1998：1-6，262-295
[37] 周纪卿，朱因远．非线性振动[M]．西安：西安交通大学出版社，1998：97-153
[38] Nayfeh A. Nonlinear Oscilations [M]. New York: John Wiley, 1973
[39] Nayfeh A. Perturbation method [M]. New York: John Wiley, 1973
[40] Oseledets V I. A multiplicative ergodic theorem: Lyapunov characteristic number for dynamical systems [J]. Trans. Moscow Math. Soc., 1968, 19: 197-231 (English translation)
[41] Nayfeh A H and Serhan S J. Response statistics of nonlinear systems to combined deterministic and random excitations [J]. International Journal of Non-linear mechanics, 1990, 25(5): 493-509
[42] Rajan S, and Davies HG. Multiple time scaling of the response of a Duffing oscillator to narrow-band excitations [J]. Journal of Sound and Vibration, 1988, 123: 497-506
[43] Rong H, Xu W, Fang T. Principal response of Duffing oscillator to combined deterministic and narrow-band random parametric excitation [J]. Journal of Sound and Vibration, 1998, 210(4): 483-515
[44] Xu W, He Q, Rong H, Fang T. One-to two intemal resonance in two degree-of-freedom nonlinear system with narrow-band excitation [J]. Nonlinear Dyn. 2002, 27: 385-395
[45] 戎海武，徐伟，方同．谐和与窄带随机噪声联合作用下Duffing系统的参数主共振[J]．力学学报，1998，30(2)：178-185
[46] Khasminskii RZ. On the averaging principle for It(?) stochastic differential equations [M]. Kibernetka, 1968, 3(4): 260-279
[47] Sobczyk K. Stochastic Differential Equations, With Applications to Physics and Engineering [M]. Dordrecht: Kluwer Academic Publishers, 1991
[48] Grigoriu M. White noise processes in random vibration. In: Nonlinear Dynamics and Stochastic Mechanics, Kliemann W, Sir Namachchivaya N. (Eds.)[M], Boca Raton: CRC Press, 1996
[49] Wedig W V. Invariant measures and Lyapunov exponents for generalized parameter fluctuations [J]. Structural Safety, 1990, 8: 13-25
[50] W. Q. Zhu, M. L. Deng, and Z. L. Huang, Optimal Bounded Control of First-Passage Failure of Quasi-Integrable Hamiltonian Systems with Wide-Band Random Excitation [J]. Nonlinear Dynamics 33: 189-207, 2003
[51] W. Q. Zhu and M. L. Deng, Optimal Bounded Control for Minimizing the Response of QuasiNon-Integrable Hamiltonian Systems [J]. Nonlinear. Dynamics 35: 81-100, 2004.
[52] W. Q. Zhu and Z. L. Huang, Stochastic Stabilization of Quasi-Partially Integrable Hamiltonian Systems by Using Lyapunov Exponent [J]. Nonlinear Dynamics 33: 209-224, 2003
[53] W. Q. Zhu and Z. G. Ying, Optimal nonlinear feedback control of quasi-Hamiltonian system [J]. Science in China, Series A, 42:1213-1219, 1999
[54] Yamaguchi Y. Structure of the stochastic layer of a perturbed double-well potentialsystem [J]. Physics Letter A, 1985,109:191-195.
[55] Esmailzadeh E, Nakhaie-Jazar G. Periodic Solution of A Mathieu-Duffing Type Equation[J]. International Journal of Non-linear Mechanics, 1997, 32: 905-912.
[56]Natsiavas S, Theodossiades S. Piecewise linear systems with time varying coefficient, ASME Design Engineering Technical Conf.[C]. Sacramento, CA,1997, 78-85.
[57] Natsiavas S,.Theodossiades S, Goudas I. Dynamic analysis of piecewise linear oscillators with time periodic coefficient[J]. International Journal of Non-linear Mechanics, 2000, 35:53-68.
[58] Albert C.J. Lou. Chaotic motion in the generic separatrix bands of a Mathieu-Duffing Oscillator with a twin-well potential [J]. Journal of Sound and Vibration, 2001,248:521-532.
[59] Albert C.J. Lou. Chaotic motion in the resonant separatrix bands of a Mathieu-Duffing Oscillator with a twin-well potential [J]. Journal of Sound and Vibration, 2004,473:653-666.
[60] Leslie Ng. Bifurcations in a Mathieu equation with cubic nonlinearities [J].. Chaos, Solitons and Fractals, 2002,14:173-181.
[61] Leslie Ng, Richard Band. Bifurcations in a Mathieu equation with cubic nonlinearities: Part II [J]. Communications in Nonlinear Science and Numerical Simulation, 2002,7: 107-121.
[62] Rong Haiwu, Xu Wei, Wang Xiangdong, Meng Guang, Fang Tong. Maximal Lyapunov exponent and almost-sure sample stability for second-order linear stochastic system [J]. International Journal of Non-linear Mechanics, 2003,38: 609-614.
[63] J. Warminski. Influence of external forcing on the parametric and self-excited system [J]. Journal of Technical Physics, 2001,42: 349-366
[64] A. Tan, B. S. Kang. Chaotic motions of a Duffing oscillator subjected to combined parametric and quasiperiodic excitation [J]. International Journal of nonlinear Science and Numerical Simulation, 2001, 2: 353-364
[65] G. M. Mahmoud. Periodic solutions of strongly non-linear Mathieu oscillators [J]. International Journal of Non-linear Mechanics, 1997, 32: 1177-1185
[66] D. V. Ramani, W. L. Keith, R. H. Rand. Perturbation solution for secondary bifurcation in the quadratically-damped Mathieu equation [J]. International Journal of Non-linear Mechanics, 2004, 39: 491-502
[67] A. Naess. Chaos and nonlinear stochastic dynamics [J]. Probabilistic Engineering Mechanics, 2000, 15: 37-47
[68] 唐驾时，贺新柱．参数振动系统的共振分析[J]．岳阳师范学院学报(自然科学版) 2001，14(1)，1-4．
[69] J. S. Yu, G. Q. Cai, Y. K. Lin. A new path integration procedure based on Gauss-Legendre scheme [J]. International Journal of Non-linear Mechanics, 1997, 32: 759-768
[70] J. S. Yu, Y. K. Lin. Numerical path integration of a non-homogeneous Markov process [J]. International Journal of Non-linear Mechanics, 2004, 39: 1493-1500
[71] 彭解华，唐驾时，于德介，李克安．Van der Pol-Duffing系统的非共振Hopf分岔[J]．国防科技大学学报，2001，23(2)，107-110．
[72] Holmes PJ. A nonlinear oscillator with a strange attractor [J]. Philos Trans R Soc 1979, A292: 419-48.
[73] Frey M, Simiu E. Noise-induced chaos and phase space flux [J]. Physica D 1993, 63: 321-40.
[74] S T Ariaratnam, W C Xie, E R Vrscay. Chaotic motion under parametric excitation [J]. Dynamics and Stability of Systems, 1989, 4(2): 111-130.
[75] W C Xie. Effect of noise on chaotic motion of buckled column under periodic excitation [J]. Nonlinear and Stochastic Dynamics, ASME, 1994, AMD-192 (DE-73): 215-225.
[76] H Lin, S C S Yim. Analysis of a nonlinear system exhibiting chaotic, and random behaviors [J]. Journal of Applied Mechanics, ASME, 1996, 63: 509-516.
[77] 刘雯彦，朱位秋，黄志龙，肖忠来．有界噪声激励下Duffing振子的混沌运动[J]．工程力学，1999，第16卷第6期，133-136．
[78] W. Y. Liu, W. Q. Zhu and Z. L. Huang, Effect of bounded noise on chaotic motion of duffing oscillator under parametric excitation[J]. Chaos, Solitons & Fractals, 12 (2001) 527-537
[79] Chunbiao Gan, Noise-induced chaos and basin erosion in softening Duffing oscillator [J]. Chaos, Solitons and Fractals 25 (2005) 1069-1081
[80] Xiaoli Yang, Wei Xu, Zhongkui Sun, Tong Fang. Effect of bounded noise on chaotic motion of a triple-well potential system [J]. Chaos, Solitons & Fractals, 25(2005)415-424.
[81] Xiaoli Yang, Wei Xu and Zhongkui Sun. Effect of bounded noise on the chaotic motion of a Duffing Van der pol oscillator in a φ~6 potential [J]. Chaos, Solitons & Fractals, 27(2006) 778-788.
[82] Zhongkui Sun, Wei Xu and Xiaoli Yang. Effect of random noise on chaotic motion of a particle in a φ~6 potential [J]. Chaos, Solitons & Fractals, 27(2006) 127-138
[83] Shaohua Li, Shaopu Yang and Wenwu Guo. Investigation on chaotic motion in hysteretic non-linear suspension system with multi-frequency excitations [J]. Mechanics Research Communications, 31 (2004) 229-236
[84] Erik M. Bollt, Lora Billings and Ira B. Schwartz. A manifold independent approach to understanding transport in stochastic dynamical systems [J]. Physica D: Nonlinear Phenomena, 173(2002)153-177
[85] Yannis Kominis and Kyriakos Hizanidis. The Hamiltonian perturbation approach of two interacting nonlinear waves or solitary pulses in an optical coupler [J]. Physica D: Nonlinear Phenomena, 173(2002)204-225
[86] R. Chacon, F. Sanchez-Bajo and J. A. Martinez Robustness in the suppression of bidirectional chaotic escape from a potential well by weak parametric excitations [J]. Physics Letters A, 303(2002) 190-196
[87]M. Belhaq and M. Houssni Suppression of chaos in averaged oscillator driven by external and parametric excitations [J]. Chaos, Solitons, & Fractals, 11(2000)1237-1246
[88] Wiggins S. Global bifurcations and chaos: Analytical Methods [M]. Springer-Verlag: New York; 1988.
[89] Wolf A, Swift J, Swinney H, Vastano A. Determining Lyapunov exponents from a time series [J]. Physica D 1985;16:285 - 317.
[90]Cox D R, Miller H D. The Theory of Stochastic Processes [M]. New York: Chapman and Hall, 1965.
[91]Bharucha-Reid A T.Elements of Markov Pocesses and Their Applications [M]. New York: McGraw-Hill, 1960.
[92]Zhu W.Q., Deng M.L., Huang Z.L., First-passage failure of quasi-integrable Hamiltonian systems [J]. ASME Journal of Applied Mechanics, 2002, 69: 274-282.
[93]Zhu W.Q., First-passage time of strongly nonlinear oscillators under combined harmonic and white noise excitations [J]. Nonlinear Dynamics 2003, 32: 291-305.
[93]Roberts J B, Spanos P D, Stochastic averaging: an approximate method of solving random vibration problems [J]. International Journal of Nonlinear Mechanics, 1986, 21:111-134
[94]Zhu W. Q., Stochastic averaging of quasi-Hamiltonian system [J]. Science in China, Series A, 1996,39:97-107.
[95]Zhu W. Q., Z. L. Huang, Y. Q. Yang. Stochastic averaging of quasi-integrable Hamiltonian systems [J]. ASME Journal of Applied Mechanics, 1997, 64: 975-984
[96]Zhu W.Q.,Yang Y.Q. Stochastic averaging of quasi-non-integrable -Hamiltonian systems[J]. J. Appl. Mech. ASME 64 (1997) 157-164.
[97]J. S. Yu, G. Q. Cai,Y. K. Lin. A new path integration procedure based on Gauss-Legendre scheme [J]. International Journal of Nonlinear Mechanics, 1997, 32: 159-768.
[98] Naess A, Moe V. New techniques for path integral solution of the random vibration of nonlinear oscillators. Structural Safety and Reliability[M], Rotterdam: Balkema.
[99] 李荣华，微分方程的数值解法[M]．高等教育出版社，2000
[100] Khasminskii. R Z., Necessary and Sufficient Conditions for the asymptotic stability of linear stochastic systems [J]. Theory Probability & Applications. 12(1) (1967): 144-147
[101] Sri Namachchivaya. N. Stochastic bifurcation[J]. Applied Mathematics. and Compute. 38 (1990) 101-159
[102] L. Arnold, N. Sri Namachchivaya, K. R. Schenk-Hoppe, Toward an understanding of stochastic Hopf bifurcation: a case study [J]. Int. J. Bifurcation Chaos 6 (11) (1996) 1947-1975
[103] L. Arnold, R. Lefever, Stochastic nonlinear systems in phsics, Chemistry and Biology[M]. Springer Verlag, Berlin (1981).
[104] Ariaratnam S.T, Xie W.C. Lyapunov exponents and stochastic bifurcations. In: Schiehlen Wed. Nonlinear Dynamics in Energy Systems [M]. Berlin, New York: Springer Verlag, 1989 1-8
[105] N. Sri Namachchivaya, S. Talwar, Lyapunov exponents and rotation number for stochastically perturbed co-dimes ion two bifurcation [J]. Journal of Sound and Vibration, 169(3) (1993): 349-372
[106] 刘先斌，陈虬．实噪声参激Hopf分叉系统研究[J]．力学学报，Vol．29，No．2，(1997)：158-166．
[107] 刘先斌，陈虬．非线性随机系统的稳定性和分岔研究[J]．力学进展，1996，26：437-452．
[108] Zhu W.Q., Huang Z.L. Huang, Stochastic Hopf bifurcation of quasi nonintegrable-Hamiltionian systems [J]. International Journal of Non-linear Mechanics 34 (1999) 437-447.
[109] Zhu W. Q., Lu M. Q, Wu Q. T. Stochastic jump and bifurcation of a Duffing oscillator under narrow-band excitation [J]. Journal of Sound and Vibration, 1993, 165: 185-304.
[110] Feller W. Diffusion process in one dimension [M]. Transaction of American Mathematical Society, 1954, 77: 1-31
[111] Zhang ZY. New developments in almost sure sample stability of nonlinear stochastic dynamical systems[D]. Ph.D dissertation, Polytechnic University, New York, 1991.
[112] 王树禾．微分方程模型与混沌[M]．中国科技大学出版社，2002
[113] 罗定军，张祥，董梅芳．动力系统的定性与分支理论[M]．pp103，科学出版社，2001
[114] Khasminskii R Z. Stochastic Stability of Differential Equation [M]. Alphen aan den Rijn Sijthoff & Noordhoff, 1980
[115] Florchinger P. Auniversal formula for the stabilization of control stochastic differential equations [J]. Stochastic Analysis and Applications, 1993, 11: 155-162.
[116] Florchinger P. Lyapunov-like techniques for stability [J]. SIAM Journal of Control and Optimization, 1995, 33: 1151-1169.
[117] Florchinger P. Feedback stabilization of affine in the control stochastic differential systems by the control Lyapunov function method [J]. SIAM Journal of Control and Optimization, 1997, 35: 500-511.
[118] Kristic M, Deng H. Stabilization of Nonlinear Uncertain System [M]. London: Springer, 1998.
[119] Bellman R. Dynamic Programming [M]. Princeton: Princeton University Press, 1957.
[120] B. Z. Kaplan and D. Guetta. Switching mode limit-cycle resonant oscillator as a relaxati on oscillator [J]. Journal of the Franklin Institute, 1996, 333: 833-847.
[121] J. P. Meijaard. A mechanism for the onset of chaos in mechanical systems with motion-limiting stops [J]. Chaos, Solitons & Fractals, 1996, 7: 1649-1658.
[122] S.V. Bulanov, E.Y. Echkina and I.N. Inovenkov. The nonlinear dynamics of the business center in Beckmann's model [J]. Physica A: Statistical Mechanics and its Applications, 2004, 344: 104-107.
[123] Teresa Carvalho and Fernando Durao. Control of a flotation column using fuzzy logic inference[J]. Fuzzy Sets and Systems, 2002, 125: 121-133.
[124] E. Ahmed, H. N. Agiza and S. Z. Hassan . On modifications of Puu's dynamical duopoly [J]. Chaos, Solitons & Fractals, 2000, 11: 1025-1028.
[125] Patrick Pintus, Duncan Sands and Robin de Vilder. On the transition from local regular to global irregular fluctuations [J]. Journal of Economic Dynamics and Control, 2000,24: 247-272.
[126] James W. Friedman and Claudio Mezzetti. Games with Partially Enforceable Agreements'[J]. Games and Economic Behavior, 1998,23: 176-200.
[127] Richard Cyert, Sok-Hyon Kang and Praveen Kumar. Managerial objectives and firm dividend policy: A behavioral theory and empirical evidence [J]. Journal of Economic Behavior & Organization, 1996, 31: 157-174.
[128] R. Kato and Y. Enomoto. Simulation of narrow-band noise induced by vortex dynamics in type-II superconducting films with periodic pinning arrays [J]. Physica C: Superconductivity, 2005, 426-431: Part 1,118-121.
[129] Mitsuhiro Nakazawa and Tomoyuki Sekine. Narrow band noise in the SDW state of (TMTSF)2PF6 [J]. Physica E: Low-dimensional Systems and Nanostructures, 2003,18: 192-193.
[130] Jing-Dong Bao. Rectification of different-color noise [J]. Physics Letters A, 1999,256:356-361.
[131] Mark W. Cannon, Greg J. Reese and Steven C. Fullenkamp. Target detection against narrow band noise backgrounds [J]. Vision Research, 1999, 39: 2191-2203.
[132] Pradip Majee, Gurupada Goswami and Bidhan Chandra Bag Colored non-Gaussian noise induced resonant activation [J]. Chemical Physics Letters, 2005, 416:256-260.
[133] Horacio S. Wio and Raul Toral. Effect of non-Gaussian noise sources in a noise-induced transition [J]. Physica D: Nonlinear Phenomena, 2004, 193: 161-168.
[134] Jorge A. Revelli, Alejandro D. Sanchez and Horacio S. Wio. Effect of non-Gaussian noises on the stochastic resonance-like phenomenon in gated traps [J]. Physica D: Nonlinear Phenomena, 2002,168-169: 165-170.
[135] M. A. Fuentes, Horacio S. Wio and Raul Toral. Effective Markovian approximation for non-Gaussian noises: a path integral approach [J]. Physica A: Statistical Mechanics and its Applications,2002, 303: 91-104.
[136] M. A. Fuentes, Raul Toral and Horacio S. Wio. Enhancement of stochastic resonance: the role of non Gaussian noises [J]. Physica A: Statistical Mechanics and its Applications, 2001, 295:114-122.
[2] Spanos, PD. Stochastic Linearization in Structural Dynamics [J]. Applied Mechanics Review, 1981, 34(1): 1-8
[3] Robert JB and Spanos PD. Random Vibration and Statistical Linearization [M]. New York: Wily, 1990
[4] 陆启韶．非线性复杂系统的动力学问题[J]．新疆大学学报(理工版)，2001，18(2)：141-146
[5] Stratonovich RL. Topics in the Theory of Random Noise [M]. New York: Gordon and Breach, 1967, Vol.2
[6] Khasminskii RZ. Principle of averaging of parabolic and elliptic differential equations for Markov process with small diffusion [J]. Theory Probability and Applications, 1963, 8:1-24
[7] Khasminskii RZ. A limit theorem for the solution of differential equations with random right-hand side [J]. Theory Probability and Applications, 1963, 11:309-405
[8] Zhu W Q. Stochastic averaging methods in random vibration [J]. ASME Applied Mechanics Review, 1988, 41(5): 189-199
[9] Zhu W Q. Recent developments and applications of the stochastic averaging method in random vibration [J]. ASME Applied Mechanics Review, 1996, 49(10):s72-s80
[10] Robert JB and Spanos PD. Stochastic averaging: an approximate method of solving random vibration problems [J]. International Journal of Non-linear mechanics., 1986, 21:111-134
[11] Risken H. The Fokker-Planck Equation [M]. New York: Springer, 1983
[12] Lin Y K and Cai G. Q. Probability Structural Dynamics: Advanced Theory and Applications. Singapore: McGraw-Hill, 1995
[13] Utz V W and Wedig W V. On the Calculation of Stationary Solutions of Multi-Dimensional Fokker-Planck Equations by Orthogonal Functions [J]. Nonlinear Dynamics, 2000, 21:289-306
[14] 朱位秋．非线性随机动力学与控制—Hamilton理论体系框架[M]．北京：科学出版社，2003
[15] 朱位秋．随机振动[M]．北京：科学出版社，1998：61-68，223-311
[16] Arnold L. Random Dynamical System [M]. Berlin-Heidelberg: Springer, 1998
[17] Zhu W Q., Huang Z L. Stochastic Hopf bifurcation of quasi-nonintegrable-Hamiltionian systems, Int. J. Non-linear Mechanics [J]. 1999, 34:437-447
[18] Cox D R, Miller H D. The Theory of Stochastic Processes [M], New York: Chapman and Hall, 1965.
[19] Bharucha-Reid A T. Elements of Markov Processes and Their Applications [M]. Now York: McGraw-Hill, 1960
[20] Zhu W Q., Deng M L., Huang Z. L. First-passage failure of quasi-integrable Hamiltonian systems [J]. ASME Journal of Applied Mechanics, 2002, 69: 274-282.
[21] Pontryagin L S, Boltyanski V G, Gamkrelidze RV, Mischenko E F. Mathematical Theory of Optimal Processes [M]. New York: Wily, 1962
[22] Kushner H J. Stochastic Stability and control [M]. New York: Academic Press, 1967
[23] Fleming WH, Rishel R W. Deterministic and Sttochastic Optimal Control [M]. New York: Springer-Verlag, 1975
[24] Fleming W H, Soner H M. Controlled Markov Processes and Viscosity Solutions [M]. Now York: Springer-Verlag, 1992
[25] Kushner H J, Dupuis P. Numerical Methods for Stochastic Control Problems in Continuous Time [M]. Now York: Springer-Verlag,1993
[26] Barnett, W. A., et al, (eds.), Economic Complexity[M]. Cambridge University Press, 1989
[27] Boldrin, M., and Montrucchio, L. On the Indeterminacy of Capital Accumulation Paths[J]. Journal of Economic Theory, 1986, 40, 26-39
[28] Brock, W. A., and C. Sayers" Is the Business Cycle Characterized by Deterministic Chaos[J]? Journal of Monetary Economics, 1988, 22, 71-90
[29] 徐龙炳，陆蓉，R／S分析探索中国股票市场的非线性[J]．预测，1999，2
[30] 徐寅峰，经济模型及经济混沌[J]．西安交通大学学报，1994，28(3)，83-86．
[31] 张立洪．经济数据中存在混沌吗[J]?自然杂志，1993，16(9-10)，12-16
[32] 徐前方．上证指数中的奇异吸引子[J]．数量经济技术经济研究，1994年第2期．
[33] 黄登仕、李后强．非线性经济学的理论与方法[M]．四川大学出版社，1993年。
[34] H. W. Lorenz. Multiple Attractors, Complex Basin Boundaries, and Transient Motion in Deterministic Economic Systems[M]. Dynamic Economic Models and Optimal Control, 1992, p411-415.
[35] 刘凯 湛垦华．非线性经济学研究的回顾与前瞻[J]．西安电子科技大学学报，1999年，第3期。
[36] 刘延柱，陈文良，陈立群．振动力学[M]．北京：高等教育出版社，1998：1-6，262-295
[37] 周纪卿，朱因远．非线性振动[M]．西安：西安交通大学出版社，1998：97-153
[38] Nayfeh A. Nonlinear Oscilations [M]. New York: John Wiley, 1973
[39] Nayfeh A. Perturbation method [M]. New York: John Wiley, 1973
[40] Oseledets V I. A multiplicative ergodic theorem: Lyapunov characteristic number for dynamical systems [J]. Trans. Moscow Math. Soc., 1968, 19: 197-231 (English translation)
[41] Nayfeh A H and Serhan S J. Response statistics of nonlinear systems to combined deterministic and random excitations [J]. International Journal of Non-linear mechanics, 1990, 25(5): 493-509
[42] Rajan S, and Davies HG. Multiple time scaling of the response of a Duffing oscillator to narrow-band excitations [J]. Journal of Sound and Vibration, 1988, 123: 497-506
[43] Rong H, Xu W, Fang T. Principal response of Duffing oscillator to combined deterministic and narrow-band random parametric excitation [J]. Journal of Sound and Vibration, 1998, 210(4): 483-515
[44] Xu W, He Q, Rong H, Fang T. One-to two intemal resonance in two degree-of-freedom nonlinear system with narrow-band excitation [J]. Nonlinear Dyn. 2002, 27: 385-395
[45] 戎海武，徐伟，方同．谐和与窄带随机噪声联合作用下Duffing系统的参数主共振[J]．力学学报，1998，30(2)：178-185
[46] Khasminskii RZ. On the averaging principle for It(?) stochastic differential equations [M]. Kibernetka, 1968, 3(4): 260-279
[47] Sobczyk K. Stochastic Differential Equations, With Applications to Physics and Engineering [M]. Dordrecht: Kluwer Academic Publishers, 1991
[48] Grigoriu M. White noise processes in random vibration. In: Nonlinear Dynamics and Stochastic Mechanics, Kliemann W, Sir Namachchivaya N. (Eds.)[M], Boca Raton: CRC Press, 1996
[49] Wedig W V. Invariant measures and Lyapunov exponents for generalized parameter fluctuations [J]. Structural Safety, 1990, 8: 13-25
[50] W. Q. Zhu, M. L. Deng, and Z. L. Huang, Optimal Bounded Control of First-Passage Failure of Quasi-Integrable Hamiltonian Systems with Wide-Band Random Excitation [J]. Nonlinear Dynamics 33: 189-207, 2003
[51] W. Q. Zhu and M. L. Deng, Optimal Bounded Control for Minimizing the Response of QuasiNon-Integrable Hamiltonian Systems [J]. Nonlinear. Dynamics 35: 81-100, 2004.
[52] W. Q. Zhu and Z. L. Huang, Stochastic Stabilization of Quasi-Partially Integrable Hamiltonian Systems by Using Lyapunov Exponent [J]. Nonlinear Dynamics 33: 209-224, 2003
[53] W. Q. Zhu and Z. G. Ying, Optimal nonlinear feedback control of quasi-Hamiltonian system [J]. Science in China, Series A, 42:1213-1219, 1999
[54] Yamaguchi Y. Structure of the stochastic layer of a perturbed double-well potentialsystem [J]. Physics Letter A, 1985,109:191-195.
[55] Esmailzadeh E, Nakhaie-Jazar G. Periodic Solution of A Mathieu-Duffing Type Equation[J]. International Journal of Non-linear Mechanics, 1997, 32: 905-912.
[56]Natsiavas S, Theodossiades S. Piecewise linear systems with time varying coefficient, ASME Design Engineering Technical Conf.[C]. Sacramento, CA,1997, 78-85.
[57] Natsiavas S,.Theodossiades S, Goudas I. Dynamic analysis of piecewise linear oscillators with time periodic coefficient[J]. International Journal of Non-linear Mechanics, 2000, 35:53-68.
[58] Albert C.J. Lou. Chaotic motion in the generic separatrix bands of a Mathieu-Duffing Oscillator with a twin-well potential [J]. Journal of Sound and Vibration, 2001,248:521-532.
[59] Albert C.J. Lou. Chaotic motion in the resonant separatrix bands of a Mathieu-Duffing Oscillator with a twin-well potential [J]. Journal of Sound and Vibration, 2004,473:653-666.
[60] Leslie Ng. Bifurcations in a Mathieu equation with cubic nonlinearities [J].. Chaos, Solitons and Fractals, 2002,14:173-181.
[61] Leslie Ng, Richard Band. Bifurcations in a Mathieu equation with cubic nonlinearities: Part II [J]. Communications in Nonlinear Science and Numerical Simulation, 2002,7: 107-121.
[62] Rong Haiwu, Xu Wei, Wang Xiangdong, Meng Guang, Fang Tong. Maximal Lyapunov exponent and almost-sure sample stability for second-order linear stochastic system [J]. International Journal of Non-linear Mechanics, 2003,38: 609-614.
[63] J. Warminski. Influence of external forcing on the parametric and self-excited system [J]. Journal of Technical Physics, 2001,42: 349-366
[64] A. Tan, B. S. Kang. Chaotic motions of a Duffing oscillator subjected to combined parametric and quasiperiodic excitation [J]. International Journal of nonlinear Science and Numerical Simulation, 2001, 2: 353-364
[65] G. M. Mahmoud. Periodic solutions of strongly non-linear Mathieu oscillators [J]. International Journal of Non-linear Mechanics, 1997, 32: 1177-1185
[66] D. V. Ramani, W. L. Keith, R. H. Rand. Perturbation solution for secondary bifurcation in the quadratically-damped Mathieu equation [J]. International Journal of Non-linear Mechanics, 2004, 39: 491-502
[67] A. Naess. Chaos and nonlinear stochastic dynamics [J]. Probabilistic Engineering Mechanics, 2000, 15: 37-47
[68] 唐驾时，贺新柱．参数振动系统的共振分析[J]．岳阳师范学院学报(自然科学版) 2001，14(1)，1-4．
[69] J. S. Yu, G. Q. Cai, Y. K. Lin. A new path integration procedure based on Gauss-Legendre scheme [J]. International Journal of Non-linear Mechanics, 1997, 32: 759-768
[70] J. S. Yu, Y. K. Lin. Numerical path integration of a non-homogeneous Markov process [J]. International Journal of Non-linear Mechanics, 2004, 39: 1493-1500
[71] 彭解华，唐驾时，于德介，李克安．Van der Pol-Duffing系统的非共振Hopf分岔[J]．国防科技大学学报，2001，23(2)，107-110．
[72] Holmes PJ. A nonlinear oscillator with a strange attractor [J]. Philos Trans R Soc 1979, A292: 419-48.
[73] Frey M, Simiu E. Noise-induced chaos and phase space flux [J]. Physica D 1993, 63: 321-40.
[74] S T Ariaratnam, W C Xie, E R Vrscay. Chaotic motion under parametric excitation [J]. Dynamics and Stability of Systems, 1989, 4(2): 111-130.
[75] W C Xie. Effect of noise on chaotic motion of buckled column under periodic excitation [J]. Nonlinear and Stochastic Dynamics, ASME, 1994, AMD-192 (DE-73): 215-225.
[76] H Lin, S C S Yim. Analysis of a nonlinear system exhibiting chaotic, and random behaviors [J]. Journal of Applied Mechanics, ASME, 1996, 63: 509-516.
[77] 刘雯彦，朱位秋，黄志龙，肖忠来．有界噪声激励下Duffing振子的混沌运动[J]．工程力学，1999，第16卷第6期，133-136．
[78] W. Y. Liu, W. Q. Zhu and Z. L. Huang, Effect of bounded noise on chaotic motion of duffing oscillator under parametric excitation[J]. Chaos, Solitons & Fractals, 12 (2001) 527-537
[79] Chunbiao Gan, Noise-induced chaos and basin erosion in softening Duffing oscillator [J]. Chaos, Solitons and Fractals 25 (2005) 1069-1081
[80] Xiaoli Yang, Wei Xu, Zhongkui Sun, Tong Fang. Effect of bounded noise on chaotic motion of a triple-well potential system [J]. Chaos, Solitons & Fractals, 25(2005)415-424.
[81] Xiaoli Yang, Wei Xu and Zhongkui Sun. Effect of bounded noise on the chaotic motion of a Duffing Van der pol oscillator in a φ~6 potential [J]. Chaos, Solitons & Fractals, 27(2006) 778-788.
[82] Zhongkui Sun, Wei Xu and Xiaoli Yang. Effect of random noise on chaotic motion of a particle in a φ~6 potential [J]. Chaos, Solitons & Fractals, 27(2006) 127-138
[83] Shaohua Li, Shaopu Yang and Wenwu Guo. Investigation on chaotic motion in hysteretic non-linear suspension system with multi-frequency excitations [J]. Mechanics Research Communications, 31 (2004) 229-236
[84] Erik M. Bollt, Lora Billings and Ira B. Schwartz. A manifold independent approach to understanding transport in stochastic dynamical systems [J]. Physica D: Nonlinear Phenomena, 173(2002)153-177
[85] Yannis Kominis and Kyriakos Hizanidis. The Hamiltonian perturbation approach of two interacting nonlinear waves or solitary pulses in an optical coupler [J]. Physica D: Nonlinear Phenomena, 173(2002)204-225
[86] R. Chacon, F. Sanchez-Bajo and J. A. Martinez Robustness in the suppression of bidirectional chaotic escape from a potential well by weak parametric excitations [J]. Physics Letters A, 303(2002) 190-196
[87]M. Belhaq and M. Houssni Suppression of chaos in averaged oscillator driven by external and parametric excitations [J]. Chaos, Solitons, & Fractals, 11(2000)1237-1246
[88] Wiggins S. Global bifurcations and chaos: Analytical Methods [M]. Springer-Verlag: New York; 1988.
[89] Wolf A, Swift J, Swinney H, Vastano A. Determining Lyapunov exponents from a time series [J]. Physica D 1985;16:285 - 317.
[90]Cox D R, Miller H D. The Theory of Stochastic Processes [M]. New York: Chapman and Hall, 1965.
[91]Bharucha-Reid A T.Elements of Markov Pocesses and Their Applications [M]. New York: McGraw-Hill, 1960.
[92]Zhu W.Q., Deng M.L., Huang Z.L., First-passage failure of quasi-integrable Hamiltonian systems [J]. ASME Journal of Applied Mechanics, 2002, 69: 274-282.
[93]Zhu W.Q., First-passage time of strongly nonlinear oscillators under combined harmonic and white noise excitations [J]. Nonlinear Dynamics 2003, 32: 291-305.
[93]Roberts J B, Spanos P D, Stochastic averaging: an approximate method of solving random vibration problems [J]. International Journal of Nonlinear Mechanics, 1986, 21:111-134
[94]Zhu W. Q., Stochastic averaging of quasi-Hamiltonian system [J]. Science in China, Series A, 1996,39:97-107.
[95]Zhu W. Q., Z. L. Huang, Y. Q. Yang. Stochastic averaging of quasi-integrable Hamiltonian systems [J]. ASME Journal of Applied Mechanics, 1997, 64: 975-984
[96]Zhu W.Q.,Yang Y.Q. Stochastic averaging of quasi-non-integrable -Hamiltonian systems[J]. J. Appl. Mech. ASME 64 (1997) 157-164.
[97]J. S. Yu, G. Q. Cai,Y. K. Lin. A new path integration procedure based on Gauss-Legendre scheme [J]. International Journal of Nonlinear Mechanics, 1997, 32: 159-768.
[98] Naess A, Moe V. New techniques for path integral solution of the random vibration of nonlinear oscillators. Structural Safety and Reliability[M], Rotterdam: Balkema.
[99] 李荣华，微分方程的数值解法[M]．高等教育出版社，2000
[100] Khasminskii. R Z., Necessary and Sufficient Conditions for the asymptotic stability of linear stochastic systems [J]. Theory Probability & Applications. 12(1) (1967): 144-147
[101] Sri Namachchivaya. N. Stochastic bifurcation[J]. Applied Mathematics. and Compute. 38 (1990) 101-159
[102] L. Arnold, N. Sri Namachchivaya, K. R. Schenk-Hoppe, Toward an understanding of stochastic Hopf bifurcation: a case study [J]. Int. J. Bifurcation Chaos 6 (11) (1996) 1947-1975
[103] L. Arnold, R. Lefever, Stochastic nonlinear systems in phsics, Chemistry and Biology[M]. Springer Verlag, Berlin (1981).
[104] Ariaratnam S.T, Xie W.C. Lyapunov exponents and stochastic bifurcations. In: Schiehlen Wed. Nonlinear Dynamics in Energy Systems [M]. Berlin, New York: Springer Verlag, 1989 1-8
[105] N. Sri Namachchivaya, S. Talwar, Lyapunov exponents and rotation number for stochastically perturbed co-dimes ion two bifurcation [J]. Journal of Sound and Vibration, 169(3) (1993): 349-372
[106] 刘先斌，陈虬．实噪声参激Hopf分叉系统研究[J]．力学学报，Vol．29，No．2，(1997)：158-166．
[107] 刘先斌，陈虬．非线性随机系统的稳定性和分岔研究[J]．力学进展，1996，26：437-452．
[108] Zhu W.Q., Huang Z.L. Huang, Stochastic Hopf bifurcation of quasi nonintegrable-Hamiltionian systems [J]. International Journal of Non-linear Mechanics 34 (1999) 437-447.
[109] Zhu W. Q., Lu M. Q, Wu Q. T. Stochastic jump and bifurcation of a Duffing oscillator under narrow-band excitation [J]. Journal of Sound and Vibration, 1993, 165: 185-304.
[110] Feller W. Diffusion process in one dimension [M]. Transaction of American Mathematical Society, 1954, 77: 1-31
[111] Zhang ZY. New developments in almost sure sample stability of nonlinear stochastic dynamical systems[D]. Ph.D dissertation, Polytechnic University, New York, 1991.
[112] 王树禾．微分方程模型与混沌[M]．中国科技大学出版社，2002
[113] 罗定军，张祥，董梅芳．动力系统的定性与分支理论[M]．pp103，科学出版社，2001
[114] Khasminskii R Z. Stochastic Stability of Differential Equation [M]. Alphen aan den Rijn Sijthoff & Noordhoff, 1980
[115] Florchinger P. Auniversal formula for the stabilization of control stochastic differential equations [J]. Stochastic Analysis and Applications, 1993, 11: 155-162.
[116] Florchinger P. Lyapunov-like techniques for stability [J]. SIAM Journal of Control and Optimization, 1995, 33: 1151-1169.
[117] Florchinger P. Feedback stabilization of affine in the control stochastic differential systems by the control Lyapunov function method [J]. SIAM Journal of Control and Optimization, 1997, 35: 500-511.
[118] Kristic M, Deng H. Stabilization of Nonlinear Uncertain System [M]. London: Springer, 1998.
[119] Bellman R. Dynamic Programming [M]. Princeton: Princeton University Press, 1957.
[120] B. Z. Kaplan and D. Guetta. Switching mode limit-cycle resonant oscillator as a relaxati on oscillator [J]. Journal of the Franklin Institute, 1996, 333: 833-847.
[121] J. P. Meijaard. A mechanism for the onset of chaos in mechanical systems with motion-limiting stops [J]. Chaos, Solitons & Fractals, 1996, 7: 1649-1658.
[122] S.V. Bulanov, E.Y. Echkina and I.N. Inovenkov. The nonlinear dynamics of the business center in Beckmann's model [J]. Physica A: Statistical Mechanics and its Applications, 2004, 344: 104-107.
[123] Teresa Carvalho and Fernando Durao. Control of a flotation column using fuzzy logic inference[J]. Fuzzy Sets and Systems, 2002, 125: 121-133.
[124] E. Ahmed, H. N. Agiza and S. Z. Hassan . On modifications of Puu's dynamical duopoly [J]. Chaos, Solitons & Fractals, 2000, 11: 1025-1028.
[125] Patrick Pintus, Duncan Sands and Robin de Vilder. On the transition from local regular to global irregular fluctuations [J]. Journal of Economic Dynamics and Control, 2000,24: 247-272.
[126] James W. Friedman and Claudio Mezzetti. Games with Partially Enforceable Agreements'[J]. Games and Economic Behavior, 1998,23: 176-200.
[127] Richard Cyert, Sok-Hyon Kang and Praveen Kumar. Managerial objectives and firm dividend policy: A behavioral theory and empirical evidence [J]. Journal of Economic Behavior & Organization, 1996, 31: 157-174.
[128] R. Kato and Y. Enomoto. Simulation of narrow-band noise induced by vortex dynamics in type-II superconducting films with periodic pinning arrays [J]. Physica C: Superconductivity, 2005, 426-431: Part 1,118-121.
[129] Mitsuhiro Nakazawa and Tomoyuki Sekine. Narrow band noise in the SDW state of (TMTSF)2PF6 [J]. Physica E: Low-dimensional Systems and Nanostructures, 2003,18: 192-193.
[130] Jing-Dong Bao. Rectification of different-color noise [J]. Physics Letters A, 1999,256:356-361.
[131] Mark W. Cannon, Greg J. Reese and Steven C. Fullenkamp. Target detection against narrow band noise backgrounds [J]. Vision Research, 1999, 39: 2191-2203.
[132] Pradip Majee, Gurupada Goswami and Bidhan Chandra Bag Colored non-Gaussian noise induced resonant activation [J]. Chemical Physics Letters, 2005, 416:256-260.
[133] Horacio S. Wio and Raul Toral. Effect of non-Gaussian noise sources in a noise-induced transition [J]. Physica D: Nonlinear Phenomena, 2004, 193: 161-168.
[134] Jorge A. Revelli, Alejandro D. Sanchez and Horacio S. Wio. Effect of non-Gaussian noises on the stochastic resonance-like phenomenon in gated traps [J]. Physica D: Nonlinear Phenomena, 2002,168-169: 165-170.
[135] M. A. Fuentes, Horacio S. Wio and Raul Toral. Effective Markovian approximation for non-Gaussian noises: a path integral approach [J]. Physica A: Statistical Mechanics and its Applications,2002, 303: 91-104.
[136] M. A. Fuentes, Raul Toral and Horacio S. Wio. Enhancement of stochastic resonance: the role of non Gaussian noises [J]. Physica A: Statistical Mechanics and its Applications, 2001, 295:114-122.