E-raamat: Weakly Connected Nonlinear Systems: Boundedness and Stability of Motion

"Preface The investigation of nonlinear systems with a small parameter is attributable by a lot of modern problems of mechanics, physics, hydrodynamics, electrodynamics of charge-particle beams, space technology, astrodynamics and many others. The key problem in solution of various applied problems is that of the stability of solutions of systems of equations in various senses. The methods of the classical stability theory, if appropriately adapted, may be applied to systems containing a small parameter.The progress in solving problems of the theory of stability and nonlinear perturbations is associated with finding way around significant difficulties connected with the growth of the number of variables characterizing the state of a system, which may include critical variables. In addition, the presence of critical variables may result in a situation when not only the first approximation cannot solve a stability problem, but also the further nonlinear approximations below some order cannot solve it. Newapproaches recently developed for systems with a small parameter may include the following. A. The development of the direct Lyapunov method for the study of the boundedness and stability of systems with a finite number of degrees of freedom with respectto two different measures. B. The analysis of stability on the basis of the combination of the concepts of the direct Lyapunov method and the averaging method of nonlinear mechanics for some classes of linear and nonlinear systems. C. The generalization of the direct Lyapunov method on the basis of the concepts of the comparison principle and the averaging method of nonlinear mechanics. D. The development of the method of matrix-valued Lyapunov functions and its application in the study of stability of"--

Weakly Connected Nonlinear Systems: Boundedness and Stability of Motion provides a systematic study on the boundedness and stability of weakly connected nonlinear systems, covering theory and applications previously unavailable in book form. It contains many essential results needed for carrying out research on nonlinear systems of weakly connected equations.

After supplying the necessary mathematical foundation, the book illustrates recent approaches to studying the boundedness of motion of weakly connected nonlinear systems. The authors consider conditions for asymptotic and uniform stability using the auxiliary vector Lyapunov functions and explore the polystability of the motion of a nonlinear system with a small parameter. Using the generalization of the direct Lyapunov method with the asymptotic method of nonlinear mechanics, they then study the stability of solutions for nonlinear systems with small perturbing forces. They also present fundamental results on the boundedness and stability of systems in Banach spaces with weakly connected subsystems through the generalization of the direct Lyapunov method, using both vector and matrix-valued auxiliary functions.

Designed for researchers and graduate students working on systems with a small parameter, this book will help readers get up to date on the knowledge required to start research in this area.