This book covers the impact of noise on models that are widely used in science and engineering, and applies perturbed methods which assume noise changes on a faster time or space scale than the system being studied. The book is written in two parts. The first part carefully develops mathematical methods of studying random perturbations of dynamical systems. The second part presents non-random problems, reformulated to account for both external and system random noise, and analyzed using results from Part I. Researchers and graduate students in mathematics and engineering will find this book useful.

Provides a good combination of a general theory and well-chosen interesting and important applications Most of the material presentes is not easily accessible in the existing literature Authors are one of the world-renowned leaders in probalilistic theory and one of the world-renowned leaders in applied mathematics respectively

Autorentext

Biography of I.I. Gikhman Iosif Ilyich Gikhman was born on the 26th of May 1918 in the city of Uman, Ukraine. He studied in Kiev, graduating in 1939, then remained there to teach and do research under the supervision of N. Bogolyubov, defending a "candidate" thesis on the influence of random processes on dynamical systems in 1942 and a doctoral dissertation on Markov processes and mathematical statistics in 1955. I.I. Gikhman is one of the founders of the theory of stochastic differential equations and also contributed significantly to mathematical statistics, limit theorems, multidimensional martingales, and stochastic control. He died in 1985, in Donetsk. Biography of A.V. Skorokhod Anatoli Vladimirovich Skorokhod was born on September 10th, 1930 in the city Nikopol, Ukraine. He graduated from Kiev University in 1953, after which his graduate studies at Moscow University, were directed by E.B. Dynkin. From 1956 to 1964 Anatoli Skorokhod was a professor of Kiev university. Threafter he worked at the Institute of Mathematics of the Ukrainian Academy of Science, but he has also, since 1993, been professor of Statistics and Probability at Michigan State University. Skorokhod was elected to the Ukrainian Academy of Sciences in 1985 and became a Fellow of American Academy of Arts and Sciences in 2000. His mathematical research interests are the theory of stochastic processes, stochastic differential equations, Markov processes, randomly perturbed dynamical systems.

Klappentext
As systems evolve, they are subjected to random operating environments. In addition, random errors occur in measurements of their outputs and in their design and fabrication where tolerances are not precisely met. This book develops methods for describing random dynamical systems, and it illustrates how the methods can be used in a variety of applications. The first half of the book concentrates on finding approximations to random processes using the methodologies of probability theory. The second half of the book derives approximations to solutions of various problems in mechanics, electronic circuits, population biology, and genetics. In each example, the underlying physical or biological phenomenon is described in terms of nonrandom models taken from the literature, and the impact of random noise on the solutions is investigated. The mathematical problems in these applicitons involve random pertubations of gradient systems, Hamiltonian systems, toroidal flows, Markov chains, difference equations, filters, and nonlinear renewal equations. The models are analyzed using the approximation methods described here and are visualized using MATLAB-based computer simulations.
This book will appeal to those researchers and graduate students in science and engineering who require tools to investigate stochastic systems.

Inhalt
Ergodic Theorems.- Convergence Properties of Stochastic Processes.- Averaging.- Normal Deviations.- Diffusion Approximation.- Stability.- Markov Chains with Random Transition Probabilities.- Randomly Perturbed Mechanical Systems.- Dynamical Systems on a Torus.- PhaseLocked Loops.- Models in Population Biology.- Genetics.