Stochastic Dynamics

Focusing on the mathematical description of stochastic dynamics in discrete as well as in continuous time, this book investigates such dynamical phenomena as perturbations, bifurcations and chaos. It also introduces new ideas for the exploration of infinite dimensional systems, in particular stochastic partial differential equations. Example applications are presented from biology, chemistry and engineering, while describing numerical treatments of stochastic systems.

The aim of this book is to provide a well-structured and coherent overview of existing mathematical modeling approaches for biochemical reaction systems, investigating relations between both the conventional models and several types of deterministic-stochastic hybrid model recombinations. Another main objective is to illustrate and compare diverse numerical simulation schemes and their computational effort. Unlike related works, this book presents a broad scope in its applications, from offering a detailed introduction to hybrid approaches for the case of multiple population scales to discussing the setting of time-scale separation resulting from widely varying firing rates of reaction channels. Additionally, it also addresses modeling approaches for non well-mixed reaction-diffusion dynamics, including deterministic and stochastic PDEs and spatiotemporal master equations. Finally, by translating and incorporating complex theory to a level accessible to non-mathematicians, this book effectively bridges the gap between mathematical research in computational biology and its practical use in biological, biochemical, and biomedical systems.

This book is a result of many years of author’s research and teaching on random vibration and control. It was used as lecture notes for a graduate course. It provides a systematic review of theory of probability, stochastic processes, and stochastic calculus. The feedback control is also reviewed in the book. Random vibration analyses of SDOF, MDOF and continuous structural systems are presented in a pedagogical order. The application of the random vibration theory to reliability and fatigue analysis is also discussed. Recent research results on fatigue analysis of non-Gaussian stress processes are also presented. Classical feedback control, active damping, covariance control, optimal control, sliding control of stochastic systems, feedback control of stochastic time-delayed systems, and probability density tracking control are studied. Many control results are new in the literature and included in this book for the first time. The book serves as a reference to the engineers who design and maintain structures subject to harsh random excitations including earthquakes, sea waves, wind gusts, and aerodynamic forces, and would like to reduce the damages of structural systems due to random excitations. · Comprehensive review of probability theory, and stochastic processes · Random vibrations · Structural reliability and fatigue, Non-Gaussian fatigue · Monte Carlo methods · Stochastic calculus and engineering applications · Stochastic feedback controls and optimal controls · Stochastic sliding mode controls · Feedback control of stochastic time-delayed systems · Probability density tracking control

This textbook presents an exposition of stochastic dynamics and irreversibility. It comprises the principles of probability theory and the stochastic dynamics in continuous spaces, described by Langevin and Fokker-Planck equations, and in discrete spaces, described by Markov chains and master equations. Special concern is given to the study of irreversibility, both in systems that evolve to equilibrium and in nonequilibrium stationary states. Attention is also given to the study of models displaying phase transitions and critical phenomena both in thermodynamic equilibrium and out of equilibrium. These models include the linear Glauber model, the Glauber-Ising model, lattice models with absorbing states such as the contact process and those used in population dynamic and spreading of epidemic, probabilistic cellular automata, reaction-diffusion processes, random sequential adsorption and dynamic percolation. A stochastic approach to chemical reaction is also presented.The textbook is intended for students of physics and chemistry and for those interested in stochastic dynamics. It provides, by means of examples and problems, a comprehensive and detailed explanation of the theory and its applications.

At the dawn of the 21st century, computational stochastic dynamics is an emerging research frontier. This book focuses on advanced computational methods and software tools which can highly assist in tackling complex problems in stochastic dynamic/seismic analysis and design of structures. The book is primarily intended for researchers and post-graduate students in the fields of computational mechanics and stochastic structural dynamics. Nevertheless, practice engineers as well could benefit from it as most code provisions tend to incorporate probabilistic concepts in the analysis and design of structures. The book addresses mathematical and numerical issues in stochastic structural dynamics and connects them to real-world applications. It consists of 16 chapters dealing with recent advances in a wide range of related topics (dynamic response variability and reliability of stochastic systems, risk assessment, stochastic simulation of earthquake ground motions, efficient solvers for the analysis of stochastic systems, dynamic stability, stochastic modelling of heterogeneous materials). Numerical examples demonstrating the significance of the proposed methods are presented in each chapter.

In Stochastic Dynamics of Structures, Li and Chen present a unified view of the theory and techniques for stochastic dynamics analysis, prediction of reliability, and system control of structures within the innovative theoretical framework of physical stochastic systems. The authors outline the fundamental concepts of random variables, stochastic process and random field, and orthogonal expansion of random functions. Readers will gain insight into core concepts such as stochastic process models for typical dynamic excitations of structures, stochastic finite element, and random vibration analysis. Li and Chen also cover advanced topics, including the theory of and elaborate numerical methods for probability density evolution analysis of stochastic dynamical systems, reliability-based design, and performance control of structures. Stochastic Dynamics of Structures presents techniques for researchers and graduate students in a wide variety of engineering fields: civil engineering, mechanical engineering, aerospace and aeronautics, marine and offshore engineering, ship engineering, and applied mechanics. Practicing engineers will benefit from the concise review of random vibration theory and the new methods introduced in the later chapters. "The book is a valuable contribution to the continuing development of the field of stochastic structural dynamics, including the recent discoveries and developments by the authors of the probability density evolution method (PDEM) and its applications to the assessment of the dynamic reliability and control of complex structures through the equivalent extreme-value distribution." —A. H-S. Ang, NAE, Hon. Mem. ASCE, Research Professor, University of California, Irvine, USA "The authors have made a concerted effort to present a responsible and even holistic account of modern stochastic dynamics. Beyond the traditional concepts, they also discuss theoretical tools of recent currency such as the Karhunen-Loeve expansion, evolutionary power spectra, etc. The theoretical developments are properly supplemented by examples from earthquake, wind, and ocean engineering. The book is integrated by also comprising several useful appendices, and an exhaustive list of references; it will be an indispensable tool for students, researchers, and practitioners endeavoring in its thematic field." —Pol Spanos, NAE, Ryon Chair in Engineering, Rice University, Houston, USA

Stochastic dynamics has been a subject of interest since the early 20th Century. Since then, much progress has been made in this field of study, and many modern applications for it have been found in fields such as physics, chemistry, biology, ecology, economy, finance, and many branches of engineering including Mechanical, Ocean, Civil, Bio, and Earthquake Engineering.Elements of Stochastic Dynamics aims to meet the growing need to understand and master the subject by introducing fundamentals to researchers who want to explore stochastic dynamics in their fields and serving as a textbook for graduate students in various areas involving stochastic uncertainties. All topics within are presented from an application approach, and may thus be more appealing to users without a background in pure Mathematics. The book describes the basic concepts and theories of random variables and stochastic processes in detail; provides various solution procedures for systems subjected to stochastic excitations; introduces stochastic stability and bifurcation; and explores failures of stochastic systems. The book also incorporates some latest research results in modeling stochastic processes; in reducing the system degrees of freedom; and in solving nonlinear problems. The book also provides numerical simulation procedures of widely-used random variables and stochastic processes.A large number of exercise problems are included in the book to aid the understanding of the concepts and theories, and may be used for as course homework.

Stochastic Dynamics for Systems Biology is one of the first books to provide a systematic study of the many stochastic models used in systems biology. The book shows how the mathematical models are used as technical tools for simulating biological processes and how the models lead to conceptual insights on the functioning of the cellular processing