随机平均法是研究非线性随机动力学*有效且应用*广泛的近似 解析方法之一。《Stochastic Averaging Methods and Applications，Volume 1（随机平均法及其应用 上册）》是专门论述随机平均法的著作，介绍了随机平均 法的基本原理，给出了多种随机激励（高斯白噪声、高斯和泊松白噪 声、分数高斯噪声、色噪声、谐和与宽带噪声等）下多种类型非线性 系统（拟哈密顿系统、拟广义哈密顿系统、含遗传效应力系统等）的 随机平均法以及在自然科学和技术科学中的若干应用，主要是近30 年 来浙江大学朱位秋院士团队与美国佛罗里达大西洋大学Y.K. Lin 院士 和蔡国强教授关于随机平均法的研究成果的系统总结。《Stochastic Averaging Methods and Applications，Volume 1（随机平均法及其应用 上册）》论述深入 浅出，同时提供了必要的预备知识与众多算例，以利读者理解与掌握 《Stochastic Averaging Methods and Applications，Volume 1（随机平均法及其应用 上册）》内容。
目錄：
Contents1 Introduction 1References 82 Stochastic Processes 92.1 Fundamentals 92.1.1 Descriptions of Stochastic Processes 112.1.2 Stationarity and Ergodicity 132.1.3 Spectral Analysis 172.2 Gaussian Stochastic Processes 232.3 Markov Processes 242.3.1 Markov Processes and Chapman-Kolmogorov-Smoluwski Equation 242.3.2 Markov Diffusion Processes and Fokker–Planck-Kolmogorov (FPK) Equation 262.3.3 Wiener Processes and Gaussian White Noise 282.3.4 It？ Stochastic Differential Equations 312.3.5 Responses of Systems Under Gaussian White-Noise Excitations 342.4 PoissonWhite Noise Processes 382.4.1 Poisson Processes 382.4.2 PoissonWhite Noise 392.4.3 Stochastic Differential-Integral Equation and FPK Equation 422.5 Fractional Gaussian Processes 492.5.1 Fractional Calculus 492.5.2 Fractional Brownian Motion 502.5.3 Fractional Gaussian Noises 522.5.4 Stochastic Integration with Respect to Fractional Brownian Motion and Fractional Stochastic Differential Equations 542.5.5 Response of Linear Systems Excited by Fractional Gaussian Noises 572.6 Colored Noises 612.6.1 Noises Generated from Linear Filters 622.6.2 Noises Generated from Nonlinear Filters 642.6.3 Randomized Harmonic Process71References 743 Nonlinear Stochastic Dynamical Systems 773.1 Modeling of Nonlinear Stochastic Dynamical Systems 773.2 Hamiltonian Systems and Their Classification 803.2.1 Hamilton Equation 803.2.2 Poisson Bracket 843.2.3 Phase Flow 863.2.4 Canonical Transformation 873.2.5 Completely Integrable Hamiltonian System 883.2.6 Non-Integrable Hamiltonian System 933.2.7 Partially Integrable Hamiltonian System 943.2.8 Ergodicity of Hamiltonian Systems 953.2.9 Stochastically Excited and Dissipated Hamiltonian Systems 963.3 The Generalized Hamiltonian System and its Classification 983.4 Forces with Genetic Effects 1043.4.1 Hysteretic Forces 1043.4.2 Visco-Elastic Force 1143.4.3 Damping Force with Fractional Derivative 118References 1204 Stochastic Averaging Methods of Single-Degree-Of-Freedom Systems 1234.1 Stochastic Averaging Principles 1244.2 Stochastic Averaging Methods of SDOF Systems 1304.2.1 Stochastic Averaging of Amplitude Envelope 1314.2.2 Stochastic Averaging of Energy Envelope 1344.3 Systems Under Gaussian White Noise Excitations 1384.3.1 Linear Restoring Force 1384.3.2 Nonlinear Restoring Force 1424.4 Systems Under Broad-Band Random Excitations 1454.4.1 Linear Restoring Force 1464.4.2 A Primary-Secondary System 1484.4.3 Energy-Dependent White-Noise Approximation 1534.4.4 Fourier-Expansion Scheme 1554.4.5 Residual Phase Procedure 1594.5 Viscoelastic Systems Under Broad-Band Excitations 1674.5.1 Linear Restoring Force 1684.5.2 Nonlinear Restoring Force 1734.6 A System with Double-Well Potential 1804.6.1 Deterministic System with Double-Well Potential 1814.6.2 Stochastic Averaging 1844.7 Systems Under Combined Random and Harmonic Excitations 1904.8 Systems Under Poisson White Noise Excitations 2004.8.1 Amplitude Envelope 2014.8.2 Energy Envelope 2074.9 Systems Excited by Fractional Gaussian Noises 210References 2165 Stochastic Averaging Methods of Quasi-Hamiltonian Systems Under Gaussian White Noise Excitations. 2195.1 Quasi-Non-Integrable Hamiltonian Systems 2205.2 Quasi-Integrable Hamiltonian Systems 2325.2.1 Non-Internal Resonant Case 2345.2.2 Internal Resonant Case 2425.3 Quasi-Partially Integrable Hamiltonian Systems 2495.3.1 Noninternal Resonance Case 2515.3.2 Internal Resonant Case 2565.4 Stationary Response of 2-DOF Vibration-Impact System 2665.4.1 Exact Stationary Solution 2685.4.2 Application of Stochastic Averaging Method of Quasi-Non-Integrable Hamiltonian Systems 2695.4.3 Application of Stochastic Averaging Method of Quasi-Integrable Hamiltonian Systems 2745.4.4 Combined Application of Both Stochastic Averaging Methods of Quasi-Non-Integrable and Quasi-Integrable Hamiltonian Systems 2815.5 Quasi-Non-Integrable Hamiltonian Systems with Markov Jump Parameters 2845.5.1 Single-DOF Systems 2865.5.2 Multi-DOF Systems 294References 3026 Stochastic Averaging Methods of Quasi-Hamiltonian Systems Excited by Gaussian and PoissonWhite Noises 3036.1 Quasi-Hamiltonian Systems Excited by Gaussian and Poisson White Noises 3036.2 Quasi-Non-Integrable Hamiltonian Systems 3066.2.1 Combined Gaussian and Poisson White Noise Excitations3066.2.2 PoissonWhite Noise Excitation 3186.3 Quasi-Integrable Hamiltonian Systems 3306.3.1 Non-Internal Resonant Case 3326.3.2 Internal Resonant Case 3406.4 Quasi-Partially Integrable Hamiltonian Systems 3576.4.1 Non-Internal Resonant Case 3616.4.2 Internal Resonant Case 367References 3877 Stochastic Averaging Methods
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