Contents
1 Stochastic Averaging Methods of Quasi-integrable Hamiltonian Systems Excited by Colored Noises 1
1.1 Stationary Wideband Noise Excitation 1 1.1.1 SDOF System 2
1.1.2 MDOF System 13
1.2 Fractional Gaussian Noise Excitation 28
1.2.1 Non-internal Resonant Case 30
1.2.2 Internal Resonant Case 38
1.3 Combined Harmonic and Stationary Wideband Noise Excitations 47
1.3.1 Single-DOF System 47
1.3.2 MDOF System 68
1.4 Narrowband Randomized Harmonic Noise Excitation 87
1.4.1 SDOF System 88
1.4.2 MDOF System 92
References 102
2 Stochastic Averaging Methods of Quasi-integrable Hamiltonian Systems with Genetic Effective Forces 105
2.1 Quasi-integrable Hamiltonian System with Hysteretic Forces 105
2.1.1 Equalization of Hysteretic Forces 105
2.1.2 Stochastic Averaging for the Equivalent Quasi-Integrable Hamiltonian Systems 110
2.2 Quasi-integrable Hamiltonian Systems with Viscoelastic Forces 123
2.3 Quasi-integrable Hamiltonian Systems with Fractional Derivative Damping Forces 139
2.4 Quasi-integrable Hamiltonian Systems with Time-Delay Forces 160
References 174
3 Stochastic Averaging Methods of Quasi-generalized Hamiltonian Systems Excited by Gaussian White Noises 177
3.1 Quasi-nonintegrable Generalized Hamiltonian Systems 179
3.2 Quasi-integrable Generalized Hamiltonian Systems 190
3.2.1 Non-internal Resonance 191
3.2.2 Internal Resonant Case 198
3.3 Quasi-partially Integrable Generalized Hamiltonian Systems 209
3.3.1 Non-resonant Case 210
3.3.2 Internal Resonant Case 217
References 236
4 Stochastic Averaging Method of Predator–Prey Ecosystems 237
4.1 Classical Lotka-Volterra Predator–Prey Ecosystem 237
4.1.1 Deterministic Models 237
4.1.2 Stochastic Model 240
4.1.3 Stochastic Averaging 241
4.1.4 Stationary Probability Density 243
4.2 Ecosystem with Predator-Saturation and Predator-Competition 246
4.2.1 Deterministic Model 246
4.2.2 Stochastic Model 248
4.2.3 Stochastic Averaging 249
4.3 Ecosystem Under Colored Noise Excitations 250
4.3.1 Low-Pass Filtered Stochastic Excitation 255
4.3.2 Excitation of Randomized Harmonic Process 257
4.4 Time-Delayed Ecosystem 261
4.4.1 Deterministic Model 261
4.4.2 Stochastic Model 264
4.4.3 Stochastic Averaging 266
4.5 Ecosystem with Habitat Complexity 268
4.5.1 Deterministic Model 269
4.5.2 Equilibriums and Stability 271
4.5.3 Modified Lotka-Volterra Model 274
4.5.4 Stochastic Model and Stochastic Averaging 276
References 282
5 Several Applications of the Stochastic Averaging Methods in Natural Sciences 285
5.1 Motion of Active Brownian Particles 285
5.1.1 Deterministic Motion of Active Brownian Particle 286
5.1.2 Stochastic Motion of Active Brownian Particle 289
5.1.3 Random Swarm Motion of Active Brownian Particles 304
5.2 Reaction Rate Theory 311
5.2.1 Kramers Reaction Rate Theory 312
5.2.2 Reaction Rate Dominated by Energy Diffusion 315
5.2.3 Reaction Rate on Multi-dimensional Potential Energy Landscape 318
5.2.4 Reaction Rate Under Colored Noise Excitation 321
5.2.5 Prediction of Reaction Rate Under Colored Noise Excitation Using the Stochastic Averaging Method in Sect1.1 324
5.3 Fermi Resonance 330
5.3.1 Pippard Model of Fermi Resonance 330
5.3.2 First-Passage Time of Pippard System Under Stochastic Excitation 332
5.3.3 Reaction Rate of Fermi Resonance Under Stochastic Excitation 342
5.4 Thermal Motion of DNA Molecule 346
5.4.1 PBD Model of DNA Molecule 346
5.4.2 Stationary Motion of DNA Molecules 349
5.5 Conformational Transformation of Biomacromolecule 355
5.5.1 Model and Motion of Conformational Transformation 355
5.5.2 Stochastic Dynamics of Conformational Transformation 358
5.5.3 Denaturation of DNA Molecule 363
References 366
6 Several Applications of the Stochastic Averaging Methods in Technical Sciences 371
6.1 Vortex-Induced Random Vibration 371
6.1.1 Hartlen-Currie Wake Oscillator Model 372
6.1.2 Hartlen-Currie Model with Fluctuating Wind Excitation—Resonance Case 373
6.1.3 Hartlen-Currie Model Under Fluctuating Wind Excitation—Non-resonance Case 381
6.1.4 Nonlinear Structural Oscillator 383
6.2 Multi-machine Power Systems with Stochastic Excitations 389
6.2.1 Model of Single/Multi-machine Power Systems Subjected to Stochastic Excitations 389
6.2.2 Stochastic Averaging 392
6.2.3 Reliability of Multi-machine Power Systems 394
6.3 Ship Rolling Motion 398
6.3.1 Rolling Motion Equation of Ship Under Irregular Wave Excitation 398
6.3.2 Averaged It？ Stochastic Differential Equation 400
6.3.3 Ship Capsize Probability 404
6.4 Asymptotic Lyapunov Stability with Probability 1 of Quasi-Hamiltonian Systems 407
6.4.1 Asymptotic Lyapunov Stability with Probability 1 of Stochastic Systems 408
6.4.2 Maximum Lyapunov Exponent 410
6.4.3 Lyapunov Asymptotic Stability with Probability 1 for Quasi-non-integrable Hamiltonian Systems 415
6.4.4 Lyapunov Asymptotic Stability with Probability 1 of Quasi-integrable Hamiltonian Systems 420
6.5 Nonlinear Stochastic Optimal Control of Quasi-Hamiltonian Systems 434
6.5.1 Controlled Quasi-Hamiltonian Systems 434
6.5.2 Optimal Control of Quasi-non-integrable Hamiltonian Systems 436
References 448
Index 451