1R11. Qualitative Methods in Nonlinear Dynamics: Novel Approaches to Liapunov’s Matrix Functions. Pure and Applied Mathematics, Vol 246. - AA Martynyuk (Stability of Processes Dept, Inst of Mech, Natl Acad of Sci, Kiev, Ukraine). Marcel Dekker, New York. 2002. 301 pp. ISBN 0-8247-0735-4. $150.00.
Reviewed by RA Ibrahim (Dept of Mech Eng, Wayne State Univ, 5050 Anthony Wayne Dr, Rm 2119 Engineering Bldg, Detroit MI 48202).
This addition to the series of pure and applied mathematics monographs deals with the modern theory of dynamics of continuous, discrete-time, and impulsive nonlinear systems using Liapunov matrix-valued functions. It is known that this theory is originally rooted in the developments of Poincare´’s and Liapunov’s ideas for treating nonlinear systems of differential equations. The book is devoted to introduce mathematical theorems for analyzing Liapunov matrix-valued functions in five chapters.
The first chapter introduces the mathematical statements of qualitative methods of the general equations of continuous nonlinear systems. The definitions of various types of stability are introduced for nonlinear non-autonomous systems. Scalar, vector, and matrix-valued Liapunov functions, and the comparison principle were introduced to allow the estimation of the distance from every point of the system integral curve to the origin when the time changes from the fixed value. Other stability theorems, based on the work of the author and others, are stated with their proofs.
Some methods for analyzing continuous nonlinear systems of hierarchical structure are presented in Chapter 2. These methods are supported by an example of third-order systems. Some stability theorems of systems with regular hierarchy subsystems, large systems, and their extension to overlapping decomposition are discussed. The problem of poly-stability of nonlinear systems with separable motion is analyzed as an application of the matrix-valued function. Chapter 2 includes the concepts of integral and Lipschitz stability based on the use of the principle of comparison with a matrix-valued Liapunov function. Chapter 3 presents the qualitative analysis of discrete-time systems that model mechanical systems with impulse control, digital computing devices, population dynamics, chaotic dynamics of economical systems, and many others. These systems are usually described in terms of difference equations whose stability conditions are defined in terms of the matrix-valued functions method.
Chapter 4 introduces the stability of nonlinear dynamical systems subjected to impulsive perturbations. The impulsive system of differential equations are stated for general class of dynamical systems. The stability definitions presented in Chapter 2 for ordinary differential equations are adapted for the impulsive systems. Conditions and definitions of uniqueness, continuity, boundedness, and stability of solutions of impulsive systems are presented.
Chapter 5 culminates the theorems and general results presented in the first four chapters by introducing some applications. They include numerical algorithms of constructing a point network supported by illustrative examples. The oscillations and stability of coupled mechanical systems are demonstrated for three pendulums through elastic springs and coupled two non-autonomous parametric oscillators.
Qualitative Methods in Nonlinear Dynamics: Novel Approaches to Liapunov’s Matrix Functions is recommended to researchers who are studying the mathematical stability theory of dynamical systems. The author is commended for introducing illustrative examples from different applications to support the idea of Liapunov’s matrix functions.