MULTIPARAMETER STABILITY THEORY WITH MECHANICAL APPLICATIONS
Alexander P. Seyranian & Alexei A. Mailybaev(Moscow State Lomonosov University, Russia)
World Scientiﬁc, Singapore, 2003
xvi+403 pp. ISBN 981-238-406-5 (US$86.00,£64.00)
Mathematical Reviews MR2056325
What makes this new book an outstanding contribution to stability theory? First of all, the book succeeds in bringing qualitative results of the famous Russian school of applied mathematics to stability theory, making these results quantitative and applicable. The authors can take credit for a new multi-parameter bifurcation analysis of eigenvalues, which is a development of the perturbation theory created by Vishik, Lyusternik and Lidskii in the 1960's. Due to Seyranian's and Mailybaev's research in the last 15 years, documented by the references, it is shown how interaction and bifurcation of eigenvalues opens up a new understanding of many stability problems in mechanics like gyroscopic stabilization, flutter and divergence instabilities, transference of instability between eigenvalue branches, destabilization and stabilization by small damping, parametric resonance in periodically excited systems, etc. Another interesting issue in the book is the quantitative advancement of the qualitative singularity classification of differential equations, given by Arnold in the 1980's. A significant part is devoted to stability problems of periodic systems depending on multiple parameters. These problems go back to Mathieu, Floquet, Hill and others in the last part of the 19th century. Also, here new results by the authors have contributed to a better understanding of the subject. Last but not least we have to recognize that applications play a major role in this book. This feature makes it of great value, especially for graduate students and engineers. The theory is illustrated by many examples of pipes, beams, columns, rotating shafts, systems of connected bodies, panels, wings, etc. Beginners who want to take their first steps in stability theory should understand that there exist books for this purpose which are more elementary. Let me finally list some of the contents of the 12 chapters, which are important in addition to the above-mentioned topics: an excellent introduction to stability in the sense of Lyapunov, discussion of stability boundaries with singularities in the parameter space, classical gyroscopic and circulatory systems and Hamiltonian systems, sensitivity analysis of eigenfrequencies. The authors could have mentioned more results from classical stability theory, but they had to choose from an enormous amount of information on the subject. So it is natural and actually a good point that they gave priority to those issues where they have contributed with their own scientific results. Linear systems are the main subject, although a few connections to nonlinear systems are drawn. Those who are looking for numerical methods and computational vibration analysis will be disappointed. This book gives what the title promises: theory with applications. Without hesitation I can warmly recommend the book. It demands some serious work to study it. I have no doubt that it will fulfil what the authors hope at the end of their preface: "... to promote studies of new problems, effects, and phenomena associated with instabilities and catastrophes, and give a fresh view to classical problems".
Reviewed by Professor Wolfhard Kliem
(Technical University of Denmark)