1 Stochastic Averaging Methods of Quasi-integrable Hamiltonian Systems Excited by Colored Noises 1.1 Stationary Wideband Noise Excitation 1.1.1 SDOF System 1.1.2 MDOF System 1.2 Fractional Gaussian Noise Excitation 1.2.1 Non-internal Resonant Case 1.2.2 Internal Resonant Case 1.3 Combined Harmonic and Stationary Wideband Noise Excitations 1.3.1 Single-DOF System 1.3.2 MDOF System 1.4 Narrowband Randomized Harmonic Noise Excitation 1.4.1 SDOF System 1.4.2 MDOF System References 2 Stochastic Averaging Methods of Quasi-integrable Hamiltonian Systems with Genetic Effective Forces 2.1 Quasi-integrable Hamiltonian System with Hysteretic Forces 2.1.1 Equalization of Hysteretic Forces 2.1.2 Stochastic Averaging for the Equivalent Quasi-integrable Hamiltonian Systems 2.2 Quasi-integrable Hamiltonian Systems with Viscoelastic Forces 2.3 Quasi-integrable Hamiltonian Systems with Fractional Derivative Damping Forces 2.4 Quasi-integrable Hamiltonian Systems with Time-Delay Forces References 3 Stochastic Averaging Methods of Quasi-generalized Hamiltonian Systems Excited by Gaussian White Noises 3.1 Quasi-nonintegrable Generalized Hamiltonian Systems 3.2 Quasi-integrable Generalized Hamiltonian Systems 3.2.1 Non-internal Resonance 3.2.2 Internal Resonant Case 3.3 Quasi-partially Integrable Generalized Hamiltonian Systems 3.3.1 Non-resonant Case 3.3.2 Internal Resonant Case References 4 Stochastic Averaging Method of Predator-Prey Ecosystems 4.1 Classical Lotka-Volterra Predator-Prey Ecosystem 4.1.1 Deterministic Models 4.1.2 Stochastic Model 4.1.3 Stochastic Averaging 4.1.4 Stationary Probability Density 4.2 Ecosystem with Predator-Saturation and Predator-Competition 4.2.1 Deterministic Model 4.2.2 Stochastic Model 4.2.3 Stochastic Averaging 4.3 Ecosystem Under Colored Noise Excitations 4.3.1 Low-Pass Filtered Stochastic Excitation 4.3.2 Excitation of Randomized Harmonic Process 4.4 Time-Delayed Ecosystem 4.4.1 Deterministic Model 4.4.2 Stochastic Model 4.4.3 Stochastic Averaging 4.5 Ecosystem with Habitat Complexity 4.5.1 Deterministic Model
4.5.2 Equilibriums and Stability 4.5.3 Modified Lotka-Volterra Model 4.5.4 Stochastic Model and Stochastic Averaging References 5 Several Applications of the Stochastic Averaging Methods in Natural Sciences 5.1 Motion of Active Brownian Particles 5.1.1 Deterministic Motion of Active Brownian Particle 5.1.2 Stochastic Motion of Active Brownian Particle 5.1.3 Random Swarm Motion of Active Brownian Particles 5.2 Reaction Rate Theory 5.2.1 Kramers Reaction Rate Theory 5.2.2 Reaction Rate Dominated by Energy Diffusion 5.2.3 Reaction Rate on Multi-dimensional Potential Energy Landscape 5.2.4 Reaction Rate Under Colored Noise Excitation 5.2.5 Prediction of Reaction Rate Under Colored Noise Excitation Using the Stochastic Averaging Method in Sect. 1.1 5.3 Fermi Resonance 5.3.1 Pippard Model of Fermi Resonance 5.3.2 First-Passage Time of Pippard System Under Stochastic Excitation 5.3.3 Reaction Rate of Fermi Resonance Under Stochastic Excitation 5.4 Thermal Motion of DNA Molecule 5.4.1 PBD Model of DNA Molecule 5.4.2 Stationary Motion of DNA Molecules 5.5 Conformational Transformation of Biomacromolecule 5.5.1 Model and Motion of Conformational Transformation 5.5.2 Stochastic Dynamics of Conformational Transformation 5.5.3 Denaturation of DNA Molecule References 6 Several Applications of the Stochastic Averaging Methods in Technical Sciences 6.1 Vortex-Induced Random Vibration 6.1.1 Hartlen-Currie Wake Oscillator Model 6.1.2 Hartlen-Currie Model with Fluctuating Wind Excitation—Resonance Case 6.1.3 Hartlen-Currie Model Under Fluctuating Wind Excitation—Non-resonance Case 6.1.4 Nonlinear Structural Oscillator 6.2 Multi-machine Power Systems with Stochastic Excitations 6.2.1 Model of Single/Multi-machine Power Systems Subjected to Stochastic Excitations 6.2.2 Stochastic Averaging 6.2.3 Reliability of Multi-machine Power Systems 6.3 Ship Rolling Motion 6.3.1 Rolling Motion Equation of Ship Under Irregular Wave Excitation 6.3.2 Averaged It? Stochastic Differential Equation 6.3.3 Ship Capsize Probability 6.4 Asymptotic Lyapunov Stability with Probability 1 of Quasi-Hamiltonian Systems 6.4.1 Asymptotic Lyapunov Stability with Probability 1 of Stochastic Systems 6.4.2 Maximum Lyapunov Exponent 6.4.3 Lyapunov Asymptotic Stability with Probability 1 for Quasi-non-integrable Hamiltonian Systems 6.4.4 Lyapunov Asymptotic Stability with Probability 1 of Quasi-integrable Hamiltonian Systems 6.5 Nonlinear Sto Index
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