基于混沌理论的非线性声学特性研究
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摘要
随着科学技术的不断发展进步,非线性声学理论被越来越多的应用到工程实践当中,在国防、医疗、工业等领域都有对大功率声源的广泛应用,这也迫使对非线性声学理论的研究更加深入。非线性声学作为一门物理学领域中的非线性理论,其中的许多问题具有非线性科学问题的普遍特性,混沌理论为非线性问题的研究提供了有效手段,可以通过混沌理论方法对其进行研究。
本文首先研究了非线性时间序列的分析方法,以非线性时间序列为基础对非线性系统特性进行研究,利用非线性系统的混沌吸引子特征对不同非线性系统特性进行了判断,通过相空间重构方法对非线性时间序列进行了重构,并对相空间重构中延迟时间和嵌入维数的选取方法进行了研究,研究了非线性时间序列的Lyapunov (?)旨数计算方法,通过Lyapunov指数对非线性时间序列的特征进行了分析。
本文从非线性动力学系统振动特性出发,对非线性系统振动特性及其研究方法进行了探讨,通过系统相图、Poincare截面、功率谱、Lyapunov指数等研究方法,对不同状态的非线性系统特性进行了分析,讨论了外激励参数对非线性系统振动状态的影响,并重点研究了不同状态条件下系统的输出功率谱特性,得到系统在进入混沌状态后输出的激励频点功率明显下降的结论。为了对非线性系统振动状态进行有效控制,文中针对非线性系统参数对系统特性的影响进行了研究,讨论了非线性系数和阻尼系数改变时系统特性的变化规律,总结了外激励条件下非线性系统进入混沌状态的规律,以及系统进入混沌状态后对激励参数的敏感性,同时,讨论了系统非线性系数和阻尼系数对非线性系统输出功率的影响。研究了多频激励条件下非线性系统的振动状态,通过多尺度分析方法对多频激励条件下的非线性系统幅频响应特性进行了分析,对比研究了单频激励和多频激励条件下非线性系统的振动状态,对加入多频激励参数对系统振动状态的影响进行了总结。
针对非线性声学理论中的大振幅波传播问题进行了研究,通过非线性波动方程研究了单频和多频大振幅波的非线性传播问题,对单频大振幅平面波、柱面波和球面波的传播规律进行了对比研究,分析了在单频大振幅波传播过程中基频波能量和高阶谐波能量的变化情况,利用谱分解法对大振幅波的非线性相互作用问题进行了研究,讨论了两波非线性相互作用条件下,不同初始相位、频率及能量对低频波能量的影响,并将谱分解法的计算结果与Fenlon理论计算结果进行了比较。对时域有限差分计算方法进行了研究,并利用时域有限差分方法通过对非线性波动方程的数值计算研究了大振幅波的非线性传播问题,通过计算结果对单频和多频大振幅波的传播进行了时域和频域分析。对频散介质中的大振幅波非线性传播问题进行了研究,通过行波变换方法对非线性波动方程进行了变换,得到了非线性波动方程的等效非线性系统模型,通过相平面分析方法和Melnikov方法对该系统进行了研究,得到了系统进入混沌状态的理论阈值。通过试验研究了大振幅波非线性相互作用问题,得到了两波相互作用使低频波能量产生下降的试验结果,对两波非线性相互作用产生的和、差频波能量进行了理论计算和实验测量,得到两波能量较大条件下理论计算与实际结果较为符合的结论。采用锥形弹簧为非线性元件构建了非线性振动模型,对非线性波动方程行波变换得到的等效非线性系统进行了试验研究,通过调整外激励频率和幅度研究了非线性系统的振动状态,利用重构相空间、Lyapunov指数和功率谱方法对系统的周期、倍周期和混沌状态进行了分析。
对外激励条件下气泡的非线性振动特性进行了研究,讨论了不同外激励参数条件对气泡非线性振动的影响,并对不同状态条件下的气泡振动特性进行了分析,通过等效介质方法对含气泡水介质的非线性特性进行了研究,利用对不同含气泡水介质的等效非线性参数(B/A)n的计算研究了介质的非线性特性,得到含气泡水介质具有较强非线性特性的结论,通过水池试验对含气泡水介质的非线性系数进行了测量和计算,为实际问题中确定非线性系统参数,建立等效非线性系统模型提供了有效方法。
With the continuous development and progress of science and technology, nonlinear acoustic theory is more applications to engineering practice, in national defence, medical treatment, industrial and other fields have a wide range of high-power source applications, which also forced nonlinear acoustic theory more in depth research. Nonlinear acoustics as a nonlinear theory in physics, in which many of the problems have the common characteristics of the problem of nonlinear science, chaos theory provides an effective means for study nonlinear problems, we can be to study nonlinear acoustic theory through chaos theory approach.
This paper studies the nonlinear time series analysis method, based on nonlinear time series to study the characteristics of nonlinear systems, the different characteristics of nonlinear systems were judged through nonlinear systems chaotic attractor, by phase space reconstruction reconstructed phase space of nonlinear time series, and the selection methods of delay time and embedding dimension were studied, sdudied the method of Lyapunov exponent computation of nonlinear time series, by Lyapunov exponent characteristic of nonlinear time series were judged.
In this paper, the vibration characteristics of the nonlinear system and its research methods are discussed, through the system phase diagram, Poincare section, power spectrum, Lyapunov exponent methods analyzed the characteristics of the non-linear system different states, discussed influence of the external excitation parameters on the vibration state of nonlinear systems, and focus on the output power spectrum characteristics of the system states under the different conditions, get the conclusions that the output excitation frequency power of system decreased when system into chaos. To the state of the nonlinear vibration control, in this paper the influence of the nonlinear system parameters on the system characteristics were studied, discussed the influence of the nonlinear coefficient and damping change on characteristics of the system, summarized the law that nonlinear system into chaos under conditions of external excitation and the parameters sensitivity that the system enters chaotic state under the excitation, also discussed the influence of the nonlinear coefficient and damping coefficient on the output power of nonlinear systems. Sdudied the vibration state of nonlinear systems under multi-frequency excitation, amplitude frequency response characteristics of nonlinear systems under multi-frequency excitation were analyzed through multi-scale analysis, compare the vibration state nonlinear system of single-frequency excitation condition with multi-frequency excitation, summarized the influence of joining multi-frequency excitation parameters on the vibration state of the system.
The problem of large amplitude acoustic wave propagation in nonlinear acoustic theory was studied, studied the nonlinear propagation of single frequency and multi-frequency large amplitude wave through a nonlinear wave equation, compared propagation law of single frequency plane wave, cylindrical wave and spherical wave, analysed the energy changes of the basis frequency and high frequency wave in the single-frequency large amplitude wave propagation, by the spectral decomposition method studied the nonlinear interaction of large amplitude waves, discussed the influence of different initial phase, frequency and energy on the low-frequency wave energy under two waves nonlinear interaction conditions, and the calculated results of spectral decomposition and Fenlon theoretical results are compared. Studied Finite difference time domain method, and by numerical calculation of nonlinear wave equation studied the problem of large amplitude wave nonlinear propagation, studied the propagations of single-frequency and multi-frequency large amplitude wave through the time domain and frequency domain analysis.Nonlinear propagation problem of large amplitude wave is studied in the dispersion medium, by traveling wave transformation of nonlinear wave equation obtained the equivalent nonlinear system model, through the phase plane analysis method and Melnikov method studied this system, and get the theoretical threshold of the system into chaotic state. By experiment studied the large amplitude wave nonlinear interaction, get the experiment results that low frequency wave energy generated drop in two waves interaction, the sum and difference frequency wave energy were calculated and experimental measured under nonlinear interaction of two waves, obtained the conclusions that the calculation with actual results are more accordant under two waves interaction of large energy. Using the conical spring constructed nonlinear vibration model, through experiment studied the equivalent nonlinear system of nonlinear wave equation using traveling wave transformation, by adjusting the excitation frequency and amplitude studied the vibration state of the nonlinear system, using phase space, Lyapunov exponent and power spectrum methods studied the periodic, doubling period and chaos states.
Under the conditions of external excitation the nonlinear vibration characteristics of bubbles were studied, discussed the influence of difference external excitation parameters on the nonlinear vibration of the bubble, and analyzed the vibration characteristics of the bubbles under different states, through the effective medium method nonlinear characteristics of water medium containing the bubbles were studied, using equivalent nonlinear parameter (B/A)n of different water media containing bubbles studied the nonlinear characteristics of the media, get the conclusions that the water medium with bubbles has strong nonlinear characteristics, through experiment the nonlinear coefficients of water medium containing the bubble were measured and calculatedto determine nonlinear system parameters of the actual problem, constructed the equivalent nonlinear system model provides effective way.
本文首先研究了非线性时间序列的分析方法,以非线性时间序列为基础对非线性系统特性进行研究,利用非线性系统的混沌吸引子特征对不同非线性系统特性进行了判断,通过相空间重构方法对非线性时间序列进行了重构,并对相空间重构中延迟时间和嵌入维数的选取方法进行了研究,研究了非线性时间序列的Lyapunov (?)旨数计算方法,通过Lyapunov指数对非线性时间序列的特征进行了分析。
本文从非线性动力学系统振动特性出发,对非线性系统振动特性及其研究方法进行了探讨,通过系统相图、Poincare截面、功率谱、Lyapunov指数等研究方法,对不同状态的非线性系统特性进行了分析,讨论了外激励参数对非线性系统振动状态的影响,并重点研究了不同状态条件下系统的输出功率谱特性,得到系统在进入混沌状态后输出的激励频点功率明显下降的结论。为了对非线性系统振动状态进行有效控制,文中针对非线性系统参数对系统特性的影响进行了研究,讨论了非线性系数和阻尼系数改变时系统特性的变化规律,总结了外激励条件下非线性系统进入混沌状态的规律,以及系统进入混沌状态后对激励参数的敏感性,同时,讨论了系统非线性系数和阻尼系数对非线性系统输出功率的影响。研究了多频激励条件下非线性系统的振动状态,通过多尺度分析方法对多频激励条件下的非线性系统幅频响应特性进行了分析,对比研究了单频激励和多频激励条件下非线性系统的振动状态,对加入多频激励参数对系统振动状态的影响进行了总结。
针对非线性声学理论中的大振幅波传播问题进行了研究,通过非线性波动方程研究了单频和多频大振幅波的非线性传播问题,对单频大振幅平面波、柱面波和球面波的传播规律进行了对比研究,分析了在单频大振幅波传播过程中基频波能量和高阶谐波能量的变化情况,利用谱分解法对大振幅波的非线性相互作用问题进行了研究,讨论了两波非线性相互作用条件下,不同初始相位、频率及能量对低频波能量的影响,并将谱分解法的计算结果与Fenlon理论计算结果进行了比较。对时域有限差分计算方法进行了研究,并利用时域有限差分方法通过对非线性波动方程的数值计算研究了大振幅波的非线性传播问题,通过计算结果对单频和多频大振幅波的传播进行了时域和频域分析。对频散介质中的大振幅波非线性传播问题进行了研究,通过行波变换方法对非线性波动方程进行了变换,得到了非线性波动方程的等效非线性系统模型,通过相平面分析方法和Melnikov方法对该系统进行了研究,得到了系统进入混沌状态的理论阈值。通过试验研究了大振幅波非线性相互作用问题,得到了两波相互作用使低频波能量产生下降的试验结果,对两波非线性相互作用产生的和、差频波能量进行了理论计算和实验测量,得到两波能量较大条件下理论计算与实际结果较为符合的结论。采用锥形弹簧为非线性元件构建了非线性振动模型,对非线性波动方程行波变换得到的等效非线性系统进行了试验研究,通过调整外激励频率和幅度研究了非线性系统的振动状态,利用重构相空间、Lyapunov指数和功率谱方法对系统的周期、倍周期和混沌状态进行了分析。
对外激励条件下气泡的非线性振动特性进行了研究,讨论了不同外激励参数条件对气泡非线性振动的影响,并对不同状态条件下的气泡振动特性进行了分析,通过等效介质方法对含气泡水介质的非线性特性进行了研究,利用对不同含气泡水介质的等效非线性参数(B/A)n的计算研究了介质的非线性特性,得到含气泡水介质具有较强非线性特性的结论,通过水池试验对含气泡水介质的非线性系数进行了测量和计算,为实际问题中确定非线性系统参数,建立等效非线性系统模型提供了有效方法。
With the continuous development and progress of science and technology, nonlinear acoustic theory is more applications to engineering practice, in national defence, medical treatment, industrial and other fields have a wide range of high-power source applications, which also forced nonlinear acoustic theory more in depth research. Nonlinear acoustics as a nonlinear theory in physics, in which many of the problems have the common characteristics of the problem of nonlinear science, chaos theory provides an effective means for study nonlinear problems, we can be to study nonlinear acoustic theory through chaos theory approach.
This paper studies the nonlinear time series analysis method, based on nonlinear time series to study the characteristics of nonlinear systems, the different characteristics of nonlinear systems were judged through nonlinear systems chaotic attractor, by phase space reconstruction reconstructed phase space of nonlinear time series, and the selection methods of delay time and embedding dimension were studied, sdudied the method of Lyapunov exponent computation of nonlinear time series, by Lyapunov exponent characteristic of nonlinear time series were judged.
In this paper, the vibration characteristics of the nonlinear system and its research methods are discussed, through the system phase diagram, Poincare section, power spectrum, Lyapunov exponent methods analyzed the characteristics of the non-linear system different states, discussed influence of the external excitation parameters on the vibration state of nonlinear systems, and focus on the output power spectrum characteristics of the system states under the different conditions, get the conclusions that the output excitation frequency power of system decreased when system into chaos. To the state of the nonlinear vibration control, in this paper the influence of the nonlinear system parameters on the system characteristics were studied, discussed the influence of the nonlinear coefficient and damping change on characteristics of the system, summarized the law that nonlinear system into chaos under conditions of external excitation and the parameters sensitivity that the system enters chaotic state under the excitation, also discussed the influence of the nonlinear coefficient and damping coefficient on the output power of nonlinear systems. Sdudied the vibration state of nonlinear systems under multi-frequency excitation, amplitude frequency response characteristics of nonlinear systems under multi-frequency excitation were analyzed through multi-scale analysis, compare the vibration state nonlinear system of single-frequency excitation condition with multi-frequency excitation, summarized the influence of joining multi-frequency excitation parameters on the vibration state of the system.
The problem of large amplitude acoustic wave propagation in nonlinear acoustic theory was studied, studied the nonlinear propagation of single frequency and multi-frequency large amplitude wave through a nonlinear wave equation, compared propagation law of single frequency plane wave, cylindrical wave and spherical wave, analysed the energy changes of the basis frequency and high frequency wave in the single-frequency large amplitude wave propagation, by the spectral decomposition method studied the nonlinear interaction of large amplitude waves, discussed the influence of different initial phase, frequency and energy on the low-frequency wave energy under two waves nonlinear interaction conditions, and the calculated results of spectral decomposition and Fenlon theoretical results are compared. Studied Finite difference time domain method, and by numerical calculation of nonlinear wave equation studied the problem of large amplitude wave nonlinear propagation, studied the propagations of single-frequency and multi-frequency large amplitude wave through the time domain and frequency domain analysis.Nonlinear propagation problem of large amplitude wave is studied in the dispersion medium, by traveling wave transformation of nonlinear wave equation obtained the equivalent nonlinear system model, through the phase plane analysis method and Melnikov method studied this system, and get the theoretical threshold of the system into chaotic state. By experiment studied the large amplitude wave nonlinear interaction, get the experiment results that low frequency wave energy generated drop in two waves interaction, the sum and difference frequency wave energy were calculated and experimental measured under nonlinear interaction of two waves, obtained the conclusions that the calculation with actual results are more accordant under two waves interaction of large energy. Using the conical spring constructed nonlinear vibration model, through experiment studied the equivalent nonlinear system of nonlinear wave equation using traveling wave transformation, by adjusting the excitation frequency and amplitude studied the vibration state of the nonlinear system, using phase space, Lyapunov exponent and power spectrum methods studied the periodic, doubling period and chaos states.
Under the conditions of external excitation the nonlinear vibration characteristics of bubbles were studied, discussed the influence of difference external excitation parameters on the nonlinear vibration of the bubble, and analyzed the vibration characteristics of the bubbles under different states, through the effective medium method nonlinear characteristics of water medium containing the bubbles were studied, using equivalent nonlinear parameter (B/A)n of different water media containing bubbles studied the nonlinear characteristics of the media, get the conclusions that the water medium with bubbles has strong nonlinear characteristics, through experiment the nonlinear coefficients of water medium containing the bubble were measured and calculatedto determine nonlinear system parameters of the actual problem, constructed the equivalent nonlinear system model provides effective way.
引文
[1]P.J.Westervelt. Parametric Acoustic Array. J.Acoust.Soc.Am.,1963,35(4):535-537P
[2]F.H.FENLON. An extension of the bessel-fubini series for a mutihle-frequency CW acoustic source of finite amplitude. J.Acoust.Soc.Am.,1972,51(1):284-289 P
[3]龚秀芬,朱哲民,杜功焕.有限振幅与小振幅平面声波的非线性相互作用研究.南京大学学报(自然科学版),1979,3：19-28页
[4]朱哲民,杜功焕,龚秀芬.有限振幅声波非线性传播畸变消除的研究.声学学报,1981,3：189-193页
[5]H.O.Berktay. Possible Exploitation of Non-Linear Acoustics in Underwater Transmitting Applications. J.Sound Vib.,1965,4:435-461 P
[6]王焕磊,朱晓峰,龚秀芬,章东.利用反射差频参量阵进行生物组织B/A值的层析成像.科学通报,2003,48(21)：2240-2243页
[7]马大猷.现代声学理论基础.科学出版社,2004
[8]李胜利,李劲,王泽文,姚宏霖,高秋华,尹小根.用高压脉冲放电等离子体处理印染废水的研究.中国环境科学,1996,16(1)：73-76页
[9]Rim, Geun Hie, Chu-Hyun Cho, Hong-Sik Lee, E.P. Pavlov. An electric-blast system for rock fragmentation. IEEE International Pulsed Power Conference,1999:165-168P
[10]潘小京,黄建国,陈军.基于液电效应的水下宽带干扰源研究.通信电声,2008,32(5)：66-68页
[11]潘正伟,刘平香.水下爆炸对鱼雷毁伤的实验研究.舰船科学技术,2003,25(6)：52-55页
[12]陆达人,刘森.水声对抗强干扰对弱信号的压制作用.舰船科学技术,2003,25(6)：42-45页
[13]孙鹏,刘平香.水下定向声能技术在水声对抗中的应用研究.舰船科学技术,2007,29(3)：65-67页
[14]楼京俊,朱石坚,何琳.混沌隔振方法研究.船舶力学,2006,10(5)：135-141页
[15]Liu Z, Zhang X, Mao Y, Zhu Y Y, Yang Z, Chan C T, Sheng P. Locally resonant sonic materials. Science,2000,289:1734-1736 P
[16]F.Cervera, L.Sanchis, J.V.Sanchez-Perez, R.Martinez-Sala, C.Rubio, F. Meseguer. Refractive acoustic devices for airborne sound. Phys.Rev.Lett,2002, 88:023902 P
[17]A.A.Krokhim, J.Arriaga, L.N.Gumen. Speed of sound in periodic elastic composites. Phys.Rev.Lett,2003,91:264302 P
[18]吴福根,刘正猷,刘有延.二维周期性复合介质中弹性波的能带结构.声学学报,2001,26(4)：319-323页
[19]温激鸿,王刚,刘耀宗,赵宏刚.金属/丁腈橡胶杆状结构声子晶体振动带隙研究.振动工程学报,2005,18(1)：1-7页
[20]Greer S.Garrett.. Nearfield of a large acoustic transducer. J.Acoustic.Soc.Am,1982, 72(3):1056-1061 P
[21]C.Campos-Pozuelo, B.Dubus, J.A.Gallego-Juraez. Finite-element analysis of the nonlinear propagation of high-intensity acoustic waves. J. Acoustic. Soc.Am.,1999, 106(1):91-101 P
[22]J. Hoffelner, H.Landes, M.Kaltenbaches and R.Lerch. Numerical simulation of nonlinear wave propagation in thermoviscous fluids including dissipation. IEEE Ultrasonics Symposium,1999:513-516 P
[23]J.Hoffelner, H.Landes, R.Lerch. Calculation of acoustic streaming velocity and radiation force based on finite element simulations of nonlinear wave propagation. IEEE Ultrasonics Symposium,2000:585-588 P
[24]J.Hoffelner, H.Landes, R.Lerch. Numerical simulation of nonlinear effects in high power ultrasound applications. IEEE Ultrasonics Symposium,2001:645-648 P
[25]Christopher J.Vecchio, Peter A.Lewin. Finite amplitude acoustic propagation modeling using the extended angular spectrum method. J.Acoustic.Soc.Am,1994,95(5), Pt.1: 2399-2407 P
[26]Yang-Sub Lee, Mark F.Hamilton. Time-domain modeling of pulsed finite-amplitude sound beams. J.Acoust.Soc.Am,1995,97(2):906-917 P
[27]G.Wojcik, B.Fornberg, R.Waag, L.Carcione. Pseudospectral method for large scale bioacoustic models. IEEE Ultrasonics Symposium,1998:1501-1506 P
[28]T.Nouri-Baranger, E.Closset, D.Cathignol. Numerical solution for nonlinear acoustic beam radiated from non axisymmetric plane transducer using the operator splitting method. IEEE Ultrasonics Symposium,2001:479-482 P
[29]M.E.Anderson, A 2-D Nonlinear Wave Propagation Solver Written in Open-Source, NIH grant R01-CA76059
[30]崔华义.利用非线性声学测量冰厚的方法研究.海洋技术,2005,24(3)：58-69页
[31]Sarvazyan A P. Acoustic nonlinearity parameter B/A of aqueous solutions of some amino acids and proteins. J. Acoust. Soc. Am.,1990,88:1555-1561 P
[32]刘杰惠,章东,龚秀芬,陈曦,邵力正,李肖蓉,张琴兰.超声造影剂SDA的巨非线性参量,声学技术,2000,19(3)：137-138页
[33]冯绍松,章瑞铨.非线性声学简介与在医学诊断中的应用.声学技术,2004,23(3)(I)：58-60页
[34]冯绍松,章瑞铨.超声造影术的声学原理.声学技术,1993,12(1)：42-45页
[35]冯绍松,章瑞铨.非线性声学与医学诊断.声学技术,2002,21(1)：34-37页
[36]Lauterborn W, Parlitz U. Methods of chaos physics and their application to acoustics. J.Acoust.Soc.Am,1988,84:1975-1993 P
[37]王本仁,缪国王,魏荣爵.在液氮及水中声次谐波的实验观察.物理学报,1984,33(3)：434-436页
[38]Maa D Y, Liu K. Nonlinear standing waves:Theory and experiment. J.Acoust.Soc.Am, 1995,98(5):2753-2763 P
[39]赵翔,徐健学.有限振幅驻波声场中分岔的数值研究.声学学报,1999,24(4)：424-428页
[40]陈赵江,张淑仪,郑凯高.功率超声脉冲激励下金属板的非线性振动现象研究.物理学报,2010,59(6)：4071-4083页
[41]Poincare H. Memoire Sur Les Courbes Definies par une Equation Differentiells. Jour.de Math,1881,7(3):375-422; 1882,8(3):251-296; 1885, 1(4):167-244; 1886,2(4):151-217 P
[42]Lyapunov A M. Problem e General de la Stabilite du Movement. Princeton University Press,1949
[43]Birkhoff G D. Dynamical Systems. A. M. S. Publications,1927
[44]Smale S. Differential Dynamical Systems. Bull. Amer. Math. Society,1967,73: 747-817 P
[45]Peixoto M. Structural Stability on Two-dimensional Manifolds. Topology,1962,1: 101-120 P
[46]廖山涛.紧致微积分流行上常微分方程的某类性质.北京大学学报,1963,9(3)：241-265页
[47]Grassberger P. Generalized Dimension of Strange Attractors. Physics Letter,1983, 97(5):227-230 P
[48]Grassberge P, Procaccia I. Characterization of Strange Attractors. Physics Review Letter,1983,50:346-349 P
[49]HuBler A. Adaptive Cotrol of Chaotic Systems. Helvetica Physica Acta,1989,62(1): 343-346 P
[50]Ott E, Grebogi C and Yorke J A. Controlling Chaos. Physical Review Letter,1990, 64(11):1196-1199 P
[51]王润田,章瑞铨,周艳等.非线性声学的进展与应用.声学技术,2007,4：348-357页
[52]邵道远,钱祖文,张彬铨,张金城.参量阵水下通信.应用声学,1982,1(1)：33-34页
[53]许振夏.非线性声学的水下应用.声学技术,1992,1：7-11页
[54]龚秀芬.医学超声中的声学非线性研究.物理学进展,1996,16(3)：286-298页
[55]陈予恕,唐云.非线性动力学中的现代分析方法.科学出版社,1992
[56]E.N.Lorenz. Deterministic non-periodic flows. Atmosphere Sci,1963,20:130-141P
[57]Henon.M.A. TWO-dmensionai system ping with a strang attractor. Commun MathPHys,1976,50(2):69-77 P
[58]O.E.Rossler. An equation for continuous chaos. Phys.Lett.A,1976,57:397-398P
[59]G.Chen, T.Ueta. Yet another chaotic attractor. Bifurcation and Chaos,1999,9: 1465-1466 P
[60]G.Chen, X.Dong. From Chaos to Order:Methodologies,Perspectives and Applications. World Scientific,1998
[61]Babin A V, Vishik M I. Attractor for the Navier-Stokes system and for parabolic equations and Estimates of their Dimension. J Soviet Math,1983,28:619-627 P
[62]Ladyhenskaya O A. Attractors for Seml-group and Evolutions Equations.Lezioni Lincee, Cambridge University Press,1991
[63]范剑青,姚琦伟.非线性时间序列——建模、预报和应用.高等教育出版社,2005
[64]李开泰,马逸尘.数理方程HILBERT空间方法(下).西安交通大学出版社,1992
[65]郝柏林.混沌与分形.上海科学技术出版社,2004
[66]章新华.舰船辐射噪声的混沌现象研究.声学学报,1998,23(2)：134-140页
[67]朱石坚.舰艇减振降噪系统中的混沌隔振技术研究.国防科技大学博士论文,2006
[68]孙克辉,谈国强,盛利元,张泰山.Lyapunov(?)旨数计算算法的设计与实现.计算机工程与应用,2004,35：12-14页
[69]Iwanski J, Bradley E. Recurrence plots of experimental data:to embed or not to embed. Chaos,1998,8(4):861-871 P
[70]Robert Cawley. Local-geometric-projection method for noise ruduction in chaotic maps and flows. Phys. Rev. A,1992,46:3057-3082 P
[71]Kapitaniak T. Combined Bifurcations and Transition to Chaos in a Nonlinear Oscillator with Two External Periodic Forces. Journal of Sound and Vibration,1988, 121(2):259-268 P
[72]毕勤胜,陈予恕,吴志强.多频激励Duffing系统的分岔和混沌.应用数学和力学,1998,19(2)：113-120页
[73]钱祖文.非线性声学.科学出版社,1992
[74]陈式刚.映象与混沌.国防工业出版社,1992
[75]葛晓慧,黄进.利用Duffing振子估计多频信号参数.浙江大学学报(工学版),2008,42(6)：954-959页
[76]戎海武,王向东,徐伟,方同.多频谐和与噪声作用下Duffing振子的安全盆侵蚀与混沌.应用力学学报,2009,26(2)：274-278页
[77]季颖,毕勤胜.参外联合激励复合非线性振子的分岔分析.物理学报,2009,58(7)：4431-4438页
[78]H.M.Abdelhafez. Resonance of a nonlinear forced system with two-frequency parametric and self-excitation. Mathematics and Computers in Simulation,2004,66: 69-83 P
[79]Jiang Rongjun, Zhu Shijian. He Lin, et al. Prospect of applying controlling chaotic vibration in the waterborne-noise confrontation.18th biennial conference on mechanical vibration and noise, ASME, Pitsburgh, USA,2001
[80]Z Xu, Y K Cheung. A non-linear scales method for strongly non-linear oscillators. Nonlinear Dynamics,1995,7(3):285-299 P
[81]Mary Beth, Blackstock. Parametric Array in Air. J.Acoust.Soc.Am.,1975,57(3): 562-568 P
[82]Laszlo Adler. Nonlinear acoustics and its impacts on nondestructive evaluation and acousto-optics. SPIE,26(43),2004:228-243 P
[83]John C.MouId, Gregory L.Wojcik, etc. Validation of FFT-based algorithms for large-scale modeling of wave propagation in tissue. IEEE Ultrasonics Symposium, 1999:1551-1556 P
[84]楼京俊.基于混沌理论的线谱控制技术研究.海军工程大学博士论文,2006
[85]J. D. Farmer, J. J. Sidorowich. Predicting chaotic time series. Phys. Rev. Lett.,1987, 59(8):845-848 P
[86]H.D.I. Ababrbanel, R. Brown, J.J. Sidorowich, L.S. Tsimring. The analysis of observed chaotic data in physical systems. Rev.Mod. Phys,1993,65(4):1331-1392P
[87]G..G. Malinetskii, A.B. Potapov, A.I. Rakhmanov, E.B. Rodichev. Limitations of delay reconstruction for chaotic systems with broad spectrum. Phys. Lett. A.,1993, 179(1):15-22 P
[88]Alexei Potopov. Distortions of reconstruction for chaotic attractors. Physica D,1997 101:207-226 P
[89]Leung H., Huang X.P.. Parameter estimation in chaotic noise. IEEE Trans.Sig.Proc, 1996,44(10):2456-2463 P
[90]Iwanski J, Bradley E. Recurrence plots of experimental data:to embed or not to embed. Chaos,1998,8(4):861-871 P
[91]Temam R. Infinite Dimensional Dynamieal System in Mechanics and Physies. Springer Verlag, New York,1988
[92]Aolstolos Serletis, Asghar Shahmoradi, Demitre Serletis. Effect of noise on estimation of Lyapunov exponents from a time series. Chaos solitons and Fractals,2007,32: 883-887 P
[93]杜功焕,朱哲民,龚秀芬.声学基础.南京大学出版社,2001
[94]李太宝.计算声学-声场的方程和计算方法.科学出版社,2005
[95]张振福,赵云.基于Westervelt方程的时域有限差分法模拟理想介质中非线性声场的传播.第十一届全国非线性振动学术会议,2007：299305页
[96]王润田.海底声学探测与底质识别技术的新发展.声学技术,2002,1：96-98页
[97]樊斌,章东,颜永生,王焕磊,龚秀芬.非线性声参量阵成像系统设计.仪器与仪表,2002,10：13-18页
[98]褚宏宪,赵铁虎等.参量阵浅地层剖面测量技术在近岸海洋工程的应用效果.物探与化探,2005,29(6)：526-532页
[103]C.Campos-Pozuelo, B.Dubus, J.A.Gallego-Juraez. Finite-element analysis of the nonlinear propagation of high-intensity acoustic waves. J.Acoustic.Soc.Am.,1999, 106(1):91-101 P
[104]J.Hoffelner, H.Landes, M.Kaltenbaches, R.Lerch. Numerical simulation of nonlinear wave propagation in thermoviscous fluids including dissipation. IEEE Ultrasonies Symposium,1999:513-516 P
[105]J.Hoffelner, H.Landes, R.Lerch. Calculation of acoustic streaming velocity and radiation force based on finite element simulations of nonlinear wave propagation. IEEE Ultrasonics Symposium,2000:585-588 P
[106]J.Hoffelner, H.Landes, R.Lerch. Numerical simulation of nonlinear effects in high power ultrasound applications. IEEE Ultrasonics Symposium,2001:645-648 P
[107]Christopher J. Vecchio, Peter A.Lewin. Finite amplitude acoustic propagation modeling using the extended angular speetrum method. J.Acoustic.Soc.Am,1994,95(5), Pt.l: 2399-2407 P
[108]Mark D.Cahill, Andrew C.Baker. Numerical simulation of the acoustic field of a phased-array medieal ultrasound seanner. J.Aeoust.Soc.Am,1998,104(3), SePtember:1274-1283 P
[109]Greg Wojeik, John Mould. A study of second harmonic generation by focused medical transdueer pulses. IEEE Ultrasonies Symposium,1998:1583-1588 P
[110]John B.Sehneider, Kurt L.Shlager. FDTD Simulations of TEM Horns and the Implications for Staircased Representations. IEEE Transactions on Antennas and Propagation,1997:1830-1838 P
[111]Michelle Ghrist, Bengt Fombrerg. Tobin A.Driscoll. Staggered Time Integrators For Wave Equations, SIAM J.NUMER.ANAL.2000,38(3):718-741 P
[112]CARY B B. Nonlinear losses induced in spherical waves. J Acoust Soc Am,1967, 42(1):88-92 P
[114]杨训仁.大气声学.科学出版社,1997
[115]郭柏灵.非线性演化方程.上海科学出版社,1995
[116]张振福.空气介质中非线性声场特性研究.国防科技大学硕士论文,2007
[117]叶建军,陈虬.一类非线性振动系统的混沌运动.西南交通大学学报,2001,36(6)：629-632页
[118]杜功焕,龚秀芬.在强无规噪声场中有规声能量的抑制.声学学报,1984,9(3)：129-133页
[119]龚秀芬,朱哲民,石涛,吴玉奎.行波管中声参量放大效应的研究.声学学报,1988,1：9-13页
[120]张正娣.非线性波动方程分岔中的若干问题分析.江苏大学博士论文,2008
[121]L Van Wijingaarden. On the equations of motion for mixtures of liquid and gas bubbles. Fluid Mech.,1968,33:465-474 P
[122]Commander K W, Prosperitti A. Linear pressure waves in bubbly liquids:Comparison between theory and experiments. J. Acoust. Soc.Am.,1989,85(2):732-746 P
[123]Apfel R E. The effective nonlinearity parameter for immisible liquid mixture. J. Acoust. Soc.Am,1983,74(6):1866-1868 P
[124]Junru Wu, Zhemin Zhu. Measurements of the effective nonlinearity parameter B/A of water containing trapped cylindrical bubbles. J. Acoust. Soc. Am.,1991,89(6): 2634-2639 P
[125]徐晓辰,毛峰,龚秀芬.液体媒质的三阶非线性参量C/A的实验研究.中国声学学会2001年青年学术会议,2001：36-38页
[126]钱梦騄,程茜,葛曹燕.单泡声致发光中气泡的动力学特性-振子模型.声学学报,2002,27(4)：289-294页
[127]王捷.单一超声空化气泡动力学过程的数值分析.陕西师范大学硕士论文,2006
[128]赵晓亮,朱哲民,周林,杜功焕.含气泡液体中声传播的解析解及其强非线性声特性.应用声学,1999,18(6)：18-23页
[129]聂邦胜,邱仁贵.气泡声学特性探潜方法研究.海洋技术,2007,26(4)：51-53页
[130]孙贵新.水中气泡群对水声信号传播的影响.辽宁师范大学学报(自然科学版),2006,29(3)：308-310页
[131]黄景泉,李福新.水中孤立气泡对声波的耗散作用.应用数学和力学,1994,15(6)：549-554页
[132]张凤华,廖振方,唐川林.空泡运动非线性动力学特性的数值研究.重庆大学学报(自然科学版),2001,24(4)：1-4页
[133]张凤华,廖振方,唐川林.振荡压力场中空泡混沌运动的数值研究.应用力学学报,2001,18(3)：14-19页
[134]龚燕君,章东,龚秀芬.基于超声造影剂中包膜微气泡非线性振动的次谐波和超谐波研究.科学通报,2005,50(17)：1820-1823页
[135]李福新,孙进才,黄景泉.声波激励下水中孤立气泡的多阶谐振.声学学报,1996,24(6)：475-482页
[136]陈曦,章东,龚秀芬.非线性声参量B/A的参量阵成像.声学学报,1999,24(5)：470-477页
[137]颜永生,章东,龚秀芬.反射式非线性声参量层析成像的研究.声学学报,2002, 27(2)：157-161页
[138]龚秀芬,章东.非线性声参量成像及其在医学诊断中的应用.应用声学,2005,24(4)：208-215页
[139]J.R.Wu, Z.M.Zhu, G.H.Du. Nonlinear behavior of a liquid containing uniform bubbles:comparison between theory and experiments. Ultrasound in Med.&Biol, 1995,21:545-552 P
[140]何祚镛,赵玉芳著.声学理论基础.国防工业出版社,1981
[141]R.J.尤立克.水声原理(第三版).哈尔滨：哈尔滨船舶工程学院出版社,1990
[142]郭玉翠.非线性偏微分方程引论.清华大学出版社,2008
[143]朱哲民,杜功焕.含气泡水的强非线性声学特性.声学学报,1995,20(6)：425-431页
[144]C.M.Sehgal, R.C.Bahn, J.F.Greenleaf. Measurement of the acoustic nonlinearity parameter B/A in human tissues by thermodynamic method. J.Acoust.Soc.Am,1983, 74:1518 P
[145]X.F.Gong, R.Feng, C.Y.Zhu, T.Shi. Ultrasonic investigation of the nonlinearity parameter B/A in biological media. J.Acoust.Soc.Am,1984,76(3):949 P
[2]F.H.FENLON. An extension of the bessel-fubini series for a mutihle-frequency CW acoustic source of finite amplitude. J.Acoust.Soc.Am.,1972,51(1):284-289 P
[3]龚秀芬,朱哲民,杜功焕.有限振幅与小振幅平面声波的非线性相互作用研究.南京大学学报(自然科学版),1979,3：19-28页
[4]朱哲民,杜功焕,龚秀芬.有限振幅声波非线性传播畸变消除的研究.声学学报,1981,3：189-193页
[5]H.O.Berktay. Possible Exploitation of Non-Linear Acoustics in Underwater Transmitting Applications. J.Sound Vib.,1965,4:435-461 P
[6]王焕磊,朱晓峰,龚秀芬,章东.利用反射差频参量阵进行生物组织B/A值的层析成像.科学通报,2003,48(21)：2240-2243页
[7]马大猷.现代声学理论基础.科学出版社,2004
[8]李胜利,李劲,王泽文,姚宏霖,高秋华,尹小根.用高压脉冲放电等离子体处理印染废水的研究.中国环境科学,1996,16(1)：73-76页
[9]Rim, Geun Hie, Chu-Hyun Cho, Hong-Sik Lee, E.P. Pavlov. An electric-blast system for rock fragmentation. IEEE International Pulsed Power Conference,1999:165-168P
[10]潘小京,黄建国,陈军.基于液电效应的水下宽带干扰源研究.通信电声,2008,32(5)：66-68页
[11]潘正伟,刘平香.水下爆炸对鱼雷毁伤的实验研究.舰船科学技术,2003,25(6)：52-55页
[12]陆达人,刘森.水声对抗强干扰对弱信号的压制作用.舰船科学技术,2003,25(6)：42-45页
[13]孙鹏,刘平香.水下定向声能技术在水声对抗中的应用研究.舰船科学技术,2007,29(3)：65-67页
[14]楼京俊,朱石坚,何琳.混沌隔振方法研究.船舶力学,2006,10(5)：135-141页
[15]Liu Z, Zhang X, Mao Y, Zhu Y Y, Yang Z, Chan C T, Sheng P. Locally resonant sonic materials. Science,2000,289:1734-1736 P
[16]F.Cervera, L.Sanchis, J.V.Sanchez-Perez, R.Martinez-Sala, C.Rubio, F. Meseguer. Refractive acoustic devices for airborne sound. Phys.Rev.Lett,2002, 88:023902 P
[17]A.A.Krokhim, J.Arriaga, L.N.Gumen. Speed of sound in periodic elastic composites. Phys.Rev.Lett,2003,91:264302 P
[18]吴福根,刘正猷,刘有延.二维周期性复合介质中弹性波的能带结构.声学学报,2001,26(4)：319-323页
[19]温激鸿,王刚,刘耀宗,赵宏刚.金属/丁腈橡胶杆状结构声子晶体振动带隙研究.振动工程学报,2005,18(1)：1-7页
[20]Greer S.Garrett.. Nearfield of a large acoustic transducer. J.Acoustic.Soc.Am,1982, 72(3):1056-1061 P
[21]C.Campos-Pozuelo, B.Dubus, J.A.Gallego-Juraez. Finite-element analysis of the nonlinear propagation of high-intensity acoustic waves. J. Acoustic. Soc.Am.,1999, 106(1):91-101 P
[22]J. Hoffelner, H.Landes, M.Kaltenbaches and R.Lerch. Numerical simulation of nonlinear wave propagation in thermoviscous fluids including dissipation. IEEE Ultrasonics Symposium,1999:513-516 P
[23]J.Hoffelner, H.Landes, R.Lerch. Calculation of acoustic streaming velocity and radiation force based on finite element simulations of nonlinear wave propagation. IEEE Ultrasonics Symposium,2000:585-588 P
[24]J.Hoffelner, H.Landes, R.Lerch. Numerical simulation of nonlinear effects in high power ultrasound applications. IEEE Ultrasonics Symposium,2001:645-648 P
[25]Christopher J.Vecchio, Peter A.Lewin. Finite amplitude acoustic propagation modeling using the extended angular spectrum method. J.Acoustic.Soc.Am,1994,95(5), Pt.1: 2399-2407 P
[26]Yang-Sub Lee, Mark F.Hamilton. Time-domain modeling of pulsed finite-amplitude sound beams. J.Acoust.Soc.Am,1995,97(2):906-917 P
[27]G.Wojcik, B.Fornberg, R.Waag, L.Carcione. Pseudospectral method for large scale bioacoustic models. IEEE Ultrasonics Symposium,1998:1501-1506 P
[28]T.Nouri-Baranger, E.Closset, D.Cathignol. Numerical solution for nonlinear acoustic beam radiated from non axisymmetric plane transducer using the operator splitting method. IEEE Ultrasonics Symposium,2001:479-482 P
[29]M.E.Anderson, A 2-D Nonlinear Wave Propagation Solver Written in Open-Source, NIH grant R01-CA76059
[30]崔华义.利用非线性声学测量冰厚的方法研究.海洋技术,2005,24(3)：58-69页
[31]Sarvazyan A P. Acoustic nonlinearity parameter B/A of aqueous solutions of some amino acids and proteins. J. Acoust. Soc. Am.,1990,88:1555-1561 P
[32]刘杰惠,章东,龚秀芬,陈曦,邵力正,李肖蓉,张琴兰.超声造影剂SDA的巨非线性参量,声学技术,2000,19(3)：137-138页
[33]冯绍松,章瑞铨.非线性声学简介与在医学诊断中的应用.声学技术,2004,23(3)(I)：58-60页
[34]冯绍松,章瑞铨.超声造影术的声学原理.声学技术,1993,12(1)：42-45页
[35]冯绍松,章瑞铨.非线性声学与医学诊断.声学技术,2002,21(1)：34-37页
[36]Lauterborn W, Parlitz U. Methods of chaos physics and their application to acoustics. J.Acoust.Soc.Am,1988,84:1975-1993 P
[37]王本仁,缪国王,魏荣爵.在液氮及水中声次谐波的实验观察.物理学报,1984,33(3)：434-436页
[38]Maa D Y, Liu K. Nonlinear standing waves:Theory and experiment. J.Acoust.Soc.Am, 1995,98(5):2753-2763 P
[39]赵翔,徐健学.有限振幅驻波声场中分岔的数值研究.声学学报,1999,24(4)：424-428页
[40]陈赵江,张淑仪,郑凯高.功率超声脉冲激励下金属板的非线性振动现象研究.物理学报,2010,59(6)：4071-4083页
[41]Poincare H. Memoire Sur Les Courbes Definies par une Equation Differentiells. Jour.de Math,1881,7(3):375-422; 1882,8(3):251-296; 1885, 1(4):167-244; 1886,2(4):151-217 P
[42]Lyapunov A M. Problem e General de la Stabilite du Movement. Princeton University Press,1949
[43]Birkhoff G D. Dynamical Systems. A. M. S. Publications,1927
[44]Smale S. Differential Dynamical Systems. Bull. Amer. Math. Society,1967,73: 747-817 P
[45]Peixoto M. Structural Stability on Two-dimensional Manifolds. Topology,1962,1: 101-120 P
[46]廖山涛.紧致微积分流行上常微分方程的某类性质.北京大学学报,1963,9(3)：241-265页
[47]Grassberger P. Generalized Dimension of Strange Attractors. Physics Letter,1983, 97(5):227-230 P
[48]Grassberge P, Procaccia I. Characterization of Strange Attractors. Physics Review Letter,1983,50:346-349 P
[49]HuBler A. Adaptive Cotrol of Chaotic Systems. Helvetica Physica Acta,1989,62(1): 343-346 P
[50]Ott E, Grebogi C and Yorke J A. Controlling Chaos. Physical Review Letter,1990, 64(11):1196-1199 P
[51]王润田,章瑞铨,周艳等.非线性声学的进展与应用.声学技术,2007,4：348-357页
[52]邵道远,钱祖文,张彬铨,张金城.参量阵水下通信.应用声学,1982,1(1)：33-34页
[53]许振夏.非线性声学的水下应用.声学技术,1992,1：7-11页
[54]龚秀芬.医学超声中的声学非线性研究.物理学进展,1996,16(3)：286-298页
[55]陈予恕,唐云.非线性动力学中的现代分析方法.科学出版社,1992
[56]E.N.Lorenz. Deterministic non-periodic flows. Atmosphere Sci,1963,20:130-141P
[57]Henon.M.A. TWO-dmensionai system ping with a strang attractor. Commun MathPHys,1976,50(2):69-77 P
[58]O.E.Rossler. An equation for continuous chaos. Phys.Lett.A,1976,57:397-398P
[59]G.Chen, T.Ueta. Yet another chaotic attractor. Bifurcation and Chaos,1999,9: 1465-1466 P
[60]G.Chen, X.Dong. From Chaos to Order:Methodologies,Perspectives and Applications. World Scientific,1998
[61]Babin A V, Vishik M I. Attractor for the Navier-Stokes system and for parabolic equations and Estimates of their Dimension. J Soviet Math,1983,28:619-627 P
[62]Ladyhenskaya O A. Attractors for Seml-group and Evolutions Equations.Lezioni Lincee, Cambridge University Press,1991
[63]范剑青,姚琦伟.非线性时间序列——建模、预报和应用.高等教育出版社,2005
[64]李开泰,马逸尘.数理方程HILBERT空间方法(下).西安交通大学出版社,1992
[65]郝柏林.混沌与分形.上海科学技术出版社,2004
[66]章新华.舰船辐射噪声的混沌现象研究.声学学报,1998,23(2)：134-140页
[67]朱石坚.舰艇减振降噪系统中的混沌隔振技术研究.国防科技大学博士论文,2006
[68]孙克辉,谈国强,盛利元,张泰山.Lyapunov(?)旨数计算算法的设计与实现.计算机工程与应用,2004,35：12-14页
[69]Iwanski J, Bradley E. Recurrence plots of experimental data:to embed or not to embed. Chaos,1998,8(4):861-871 P
[70]Robert Cawley. Local-geometric-projection method for noise ruduction in chaotic maps and flows. Phys. Rev. A,1992,46:3057-3082 P
[71]Kapitaniak T. Combined Bifurcations and Transition to Chaos in a Nonlinear Oscillator with Two External Periodic Forces. Journal of Sound and Vibration,1988, 121(2):259-268 P
[72]毕勤胜,陈予恕,吴志强.多频激励Duffing系统的分岔和混沌.应用数学和力学,1998,19(2)：113-120页
[73]钱祖文.非线性声学.科学出版社,1992
[74]陈式刚.映象与混沌.国防工业出版社,1992
[75]葛晓慧,黄进.利用Duffing振子估计多频信号参数.浙江大学学报(工学版),2008,42(6)：954-959页
[76]戎海武,王向东,徐伟,方同.多频谐和与噪声作用下Duffing振子的安全盆侵蚀与混沌.应用力学学报,2009,26(2)：274-278页
[77]季颖,毕勤胜.参外联合激励复合非线性振子的分岔分析.物理学报,2009,58(7)：4431-4438页
[78]H.M.Abdelhafez. Resonance of a nonlinear forced system with two-frequency parametric and self-excitation. Mathematics and Computers in Simulation,2004,66: 69-83 P
[79]Jiang Rongjun, Zhu Shijian. He Lin, et al. Prospect of applying controlling chaotic vibration in the waterborne-noise confrontation.18th biennial conference on mechanical vibration and noise, ASME, Pitsburgh, USA,2001
[80]Z Xu, Y K Cheung. A non-linear scales method for strongly non-linear oscillators. Nonlinear Dynamics,1995,7(3):285-299 P
[81]Mary Beth, Blackstock. Parametric Array in Air. J.Acoust.Soc.Am.,1975,57(3): 562-568 P
[82]Laszlo Adler. Nonlinear acoustics and its impacts on nondestructive evaluation and acousto-optics. SPIE,26(43),2004:228-243 P
[83]John C.MouId, Gregory L.Wojcik, etc. Validation of FFT-based algorithms for large-scale modeling of wave propagation in tissue. IEEE Ultrasonics Symposium, 1999:1551-1556 P
[84]楼京俊.基于混沌理论的线谱控制技术研究.海军工程大学博士论文,2006
[85]J. D. Farmer, J. J. Sidorowich. Predicting chaotic time series. Phys. Rev. Lett.,1987, 59(8):845-848 P
[86]H.D.I. Ababrbanel, R. Brown, J.J. Sidorowich, L.S. Tsimring. The analysis of observed chaotic data in physical systems. Rev.Mod. Phys,1993,65(4):1331-1392P
[87]G..G. Malinetskii, A.B. Potapov, A.I. Rakhmanov, E.B. Rodichev. Limitations of delay reconstruction for chaotic systems with broad spectrum. Phys. Lett. A.,1993, 179(1):15-22 P
[88]Alexei Potopov. Distortions of reconstruction for chaotic attractors. Physica D,1997 101:207-226 P
[89]Leung H., Huang X.P.. Parameter estimation in chaotic noise. IEEE Trans.Sig.Proc, 1996,44(10):2456-2463 P
[90]Iwanski J, Bradley E. Recurrence plots of experimental data:to embed or not to embed. Chaos,1998,8(4):861-871 P
[91]Temam R. Infinite Dimensional Dynamieal System in Mechanics and Physies. Springer Verlag, New York,1988
[92]Aolstolos Serletis, Asghar Shahmoradi, Demitre Serletis. Effect of noise on estimation of Lyapunov exponents from a time series. Chaos solitons and Fractals,2007,32: 883-887 P
[93]杜功焕,朱哲民,龚秀芬.声学基础.南京大学出版社,2001
[94]李太宝.计算声学-声场的方程和计算方法.科学出版社,2005
[95]张振福,赵云.基于Westervelt方程的时域有限差分法模拟理想介质中非线性声场的传播.第十一届全国非线性振动学术会议,2007：299305页
[96]王润田.海底声学探测与底质识别技术的新发展.声学技术,2002,1：96-98页
[97]樊斌,章东,颜永生,王焕磊,龚秀芬.非线性声参量阵成像系统设计.仪器与仪表,2002,10：13-18页
[98]褚宏宪,赵铁虎等.参量阵浅地层剖面测量技术在近岸海洋工程的应用效果.物探与化探,2005,29(6)：526-532页
[103]C.Campos-Pozuelo, B.Dubus, J.A.Gallego-Juraez. Finite-element analysis of the nonlinear propagation of high-intensity acoustic waves. J.Acoustic.Soc.Am.,1999, 106(1):91-101 P
[104]J.Hoffelner, H.Landes, M.Kaltenbaches, R.Lerch. Numerical simulation of nonlinear wave propagation in thermoviscous fluids including dissipation. IEEE Ultrasonies Symposium,1999:513-516 P
[105]J.Hoffelner, H.Landes, R.Lerch. Calculation of acoustic streaming velocity and radiation force based on finite element simulations of nonlinear wave propagation. IEEE Ultrasonics Symposium,2000:585-588 P
[106]J.Hoffelner, H.Landes, R.Lerch. Numerical simulation of nonlinear effects in high power ultrasound applications. IEEE Ultrasonics Symposium,2001:645-648 P
[107]Christopher J. Vecchio, Peter A.Lewin. Finite amplitude acoustic propagation modeling using the extended angular speetrum method. J.Acoustic.Soc.Am,1994,95(5), Pt.l: 2399-2407 P
[108]Mark D.Cahill, Andrew C.Baker. Numerical simulation of the acoustic field of a phased-array medieal ultrasound seanner. J.Aeoust.Soc.Am,1998,104(3), SePtember:1274-1283 P
[109]Greg Wojeik, John Mould. A study of second harmonic generation by focused medical transdueer pulses. IEEE Ultrasonies Symposium,1998:1583-1588 P
[110]John B.Sehneider, Kurt L.Shlager. FDTD Simulations of TEM Horns and the Implications for Staircased Representations. IEEE Transactions on Antennas and Propagation,1997:1830-1838 P
[111]Michelle Ghrist, Bengt Fombrerg. Tobin A.Driscoll. Staggered Time Integrators For Wave Equations, SIAM J.NUMER.ANAL.2000,38(3):718-741 P
[112]CARY B B. Nonlinear losses induced in spherical waves. J Acoust Soc Am,1967, 42(1):88-92 P
[114]杨训仁.大气声学.科学出版社,1997
[115]郭柏灵.非线性演化方程.上海科学出版社,1995
[116]张振福.空气介质中非线性声场特性研究.国防科技大学硕士论文,2007
[117]叶建军,陈虬.一类非线性振动系统的混沌运动.西南交通大学学报,2001,36(6)：629-632页
[118]杜功焕,龚秀芬.在强无规噪声场中有规声能量的抑制.声学学报,1984,9(3)：129-133页
[119]龚秀芬,朱哲民,石涛,吴玉奎.行波管中声参量放大效应的研究.声学学报,1988,1：9-13页
[120]张正娣.非线性波动方程分岔中的若干问题分析.江苏大学博士论文,2008
[121]L Van Wijingaarden. On the equations of motion for mixtures of liquid and gas bubbles. Fluid Mech.,1968,33:465-474 P
[122]Commander K W, Prosperitti A. Linear pressure waves in bubbly liquids:Comparison between theory and experiments. J. Acoust. Soc.Am.,1989,85(2):732-746 P
[123]Apfel R E. The effective nonlinearity parameter for immisible liquid mixture. J. Acoust. Soc.Am,1983,74(6):1866-1868 P
[124]Junru Wu, Zhemin Zhu. Measurements of the effective nonlinearity parameter B/A of water containing trapped cylindrical bubbles. J. Acoust. Soc. Am.,1991,89(6): 2634-2639 P
[125]徐晓辰,毛峰,龚秀芬.液体媒质的三阶非线性参量C/A的实验研究.中国声学学会2001年青年学术会议,2001：36-38页
[126]钱梦騄,程茜,葛曹燕.单泡声致发光中气泡的动力学特性-振子模型.声学学报,2002,27(4)：289-294页
[127]王捷.单一超声空化气泡动力学过程的数值分析.陕西师范大学硕士论文,2006
[128]赵晓亮,朱哲民,周林,杜功焕.含气泡液体中声传播的解析解及其强非线性声特性.应用声学,1999,18(6)：18-23页
[129]聂邦胜,邱仁贵.气泡声学特性探潜方法研究.海洋技术,2007,26(4)：51-53页
[130]孙贵新.水中气泡群对水声信号传播的影响.辽宁师范大学学报(自然科学版),2006,29(3)：308-310页
[131]黄景泉,李福新.水中孤立气泡对声波的耗散作用.应用数学和力学,1994,15(6)：549-554页
[132]张凤华,廖振方,唐川林.空泡运动非线性动力学特性的数值研究.重庆大学学报(自然科学版),2001,24(4)：1-4页
[133]张凤华,廖振方,唐川林.振荡压力场中空泡混沌运动的数值研究.应用力学学报,2001,18(3)：14-19页
[134]龚燕君,章东,龚秀芬.基于超声造影剂中包膜微气泡非线性振动的次谐波和超谐波研究.科学通报,2005,50(17)：1820-1823页
[135]李福新,孙进才,黄景泉.声波激励下水中孤立气泡的多阶谐振.声学学报,1996,24(6)：475-482页
[136]陈曦,章东,龚秀芬.非线性声参量B/A的参量阵成像.声学学报,1999,24(5)：470-477页
[137]颜永生,章东,龚秀芬.反射式非线性声参量层析成像的研究.声学学报,2002, 27(2)：157-161页
[138]龚秀芬,章东.非线性声参量成像及其在医学诊断中的应用.应用声学,2005,24(4)：208-215页
[139]J.R.Wu, Z.M.Zhu, G.H.Du. Nonlinear behavior of a liquid containing uniform bubbles:comparison between theory and experiments. Ultrasound in Med.&Biol, 1995,21:545-552 P
[140]何祚镛,赵玉芳著.声学理论基础.国防工业出版社,1981
[141]R.J.尤立克.水声原理(第三版).哈尔滨：哈尔滨船舶工程学院出版社,1990
[142]郭玉翠.非线性偏微分方程引论.清华大学出版社,2008
[143]朱哲民,杜功焕.含气泡水的强非线性声学特性.声学学报,1995,20(6)：425-431页
[144]C.M.Sehgal, R.C.Bahn, J.F.Greenleaf. Measurement of the acoustic nonlinearity parameter B/A in human tissues by thermodynamic method. J.Acoust.Soc.Am,1983, 74:1518 P
[145]X.F.Gong, R.Feng, C.Y.Zhu, T.Shi. Ultrasonic investigation of the nonlinearity parameter B/A in biological media. J.Acoust.Soc.Am,1984,76(3):949 P