Description 
xv, 321 pages ; 23 cm 
Series 
Series on advances in mathematics for applied sciences ; v. 17 

Series on advances in mathematics for applied sciences ; v. 17

Contents 
Preface; Contents; General Notations; CHAPTER I Stochastic Canonical Equation of Multidimensional Nonlinear Dissipative Hamiltonian Dynamical Systems; 1. Introduction; 2. General Equation of Stochastic Nonlinear Dissipative Hamiltonian Dynamical Systems; 3. General Considerations on the Kinetic Energy; 4. Modeling of the Kinetic Energy and Mass Matrix; 5. Generalized Potential, Potential Energy Function and Lagrangian; 6. Generalized Momentum and Hamiltonian; 7. Energy Function; 8. Total Energy of the Dynamical System; 9. Model of the Hamiltonian 

1. Introduction 

10. Generalized Momentum and Generalized Velocity11. Modeling of the Nonconservative Generalized Force; 12. QuasiLinear Term in the Generalized Velocity; 13. QuasiLinear Damping Term; 14. QuasiLinear Gyroscopic Term; 15. External Stochastic Excitation; 16. Stochastic Parametric Excitation Term; CHAPTER II Fundamental Examples of Nonlinear Dynamical Systems and Associated SecondOrder Equation; 1. Introduction; 2. SecondOrder Equation Associated with the Dissipative Hamiltonian Dynamical System; 3. Examples for Multidimensional Nonlinear Oscillators Under External Random Excitations 

4. Examples for Multidimensional Nonlinear Oscillators Under Parametric and External Random Excitations5. Remark Concerning Modeling of Hysteresis Loops; 6. Complements on the Potential Energy Function; Generalized Restoring ForceDisplacement Relationship; Potential Energy Function and Generalized Restoring ForceDisplacement Relationship; CHAPTER III Brief Review of Probability and Random Variables; 1. Introduction; 2. Principles of Probability and Random Variables; Principles of Probability Theory and Probability Space in Stochastic Modeling; Conditional Probability and Independence 

CrossCovariance Matrix5. Classical Examples of Probability Laws; 6. Transformations of Random Variables with Values in Rn; General Result for a Nonlinear Bijective Mapping; The Use of Characteristic Functions; SecondOrder Calculation; 7. Convergence of a Sequence of Random Variables with Values in Rn; MeanSquare Convergence or Convergence in L2(.4,Rn); Convergence in Probability or Stochastic Convergence; Almost Sure Convergence; Convergence in Law or in Distribution; Relationships Between the Four Modes of Convergence; CHAPTER IV Probabilistic Tools I. Classical Stochastic Processes 

Negligible Subsets, Properties Almost Surely TrueCausality Principle and Random Variable; 3. Random Variable with Values in Rn; Probability Law of a Random Variable with Values in Rn; Almost Sure Equality of Random Variables with Values in Rn; Integration of Random Variables with Values in Rn; Characteristic Function of a Random Variable with Values in Rn; Moments of a Random Variable with Values in Rn; Tchebychev Inequality; 4. SecondOrder Random Variable with Values in Rnand L2(A,Rn)Space; Mean Vector; Correlation Matrix; Covariance Matrix; CrossCorrelation Matrix 
Summary 
This is an analysis of multidimensional nonlinear dissipative Hamiltonian dynamical systems subjected to parametric and external stochastic excitations by the FokkerPlanck equation method.The author answers three types of questions concerning this area. First, what probabilistic tools are necessary for constructing a stochastic model and deriving the FKP equation for nonlinear stochastic dynamical systems? Secondly, what are the main results concerning the existence and uniqueness of an invariant measure and its associated stationary response? Finally, what is the class of multidimensional dy 
Notes 
Description based upon print version of record 
Bibliography 
Includes bibliographical references (pages 301314) and index 
Notes 
English 
Subject 
Diffusion processes.


FokkerPlanck equation  Numerical solutions.


Mathematical physics.


Hamiltonian systems.


Stochastic differential equations  Numerical solutions.

LC no. 
94006348 
ISBN 
9810217552 
