Doi:10.1016/j.jsv.2005.02.013

Journal of Sound and Vibration 284 (2005) 1253–1254 A.P. Seyranian, A.A. Mailybaev, Multiparameter Stability Theory with Mechanical Applications,World Scientific, Singapore, ISBN981-238-406-5, 2004 (420 pp., US$86.00, £64.00).
Perturbation Theory has a long tradition. You take a problem that differs from a well- understood problem only through terms involving a small parameter, you expand everything youcan as a power series in the parameter, and then you try to learn as much as possible about thebehaviour of the problem from the low order terms in the expansion. One difficulty is that thisbehaviour itself can be sensitive to other arbitrarily small perturbations, and so it makes moresense to consider several parameters simultaneously. Formal multiparameter expansions becomevery complicated, so that it may be hard to see where things are going, but the major insights ofRene´ Thom and of Vladimir Arnold in the 1960s led to the recognition that a given problemtypically requires only a finite number of parameters (its codimension) in order to describe all itsperturbations, up to appropriate coordinate changes. Moreover, certain simple perturbationexpressions (universal unfoldings or versal deformations) suffice to capture the totality of thisperturbed behaviour.
These ideas have been slow to transfer from the mathematical to the engineering literature in the West, but in the Russian academic tradition of a more unified scientific culture the transitionhas been faster. The notion of versal deformation implies a shift in outlook toward the wholetheory of perturbations: rather than study directly the behaviour of a perturbed system, we dobetter first to understand the behaviour of the entire multiparameter class of systems to which itnaturally belongs (which may already have been classified and simply needs to be looked up), andthen set about determining exactly where the system being studied fits into the wider story. Manyare the reinvented wheels that can be avoided by taking this approach.
The book reviewed is an excellent representative both of the mathematical outlook just described and of the close Russian-style interaction between abstract geometrical thinking andspecific engineering applications. The stability of an equilibrium (or periodic) state of a systemdepends in the first instance on the eigenvalues of the system linearized about that state.
Therefore, the first step in a general study of stability of equilibria is a study of the typicalbehaviour of the eigenvalues of a matrix varying with several parameters. Of particular interestwill be the stability boundary, which is the frontier of the region in parameter space thatcorresponds to matrices all of whose eigenvalues have negative real parts. This becomesincreasingly complicated as the number of parameters increases, the structures occurringunavoidably in typical k-parameter families being precisely those whose codimension is no morethan k. The purpose of this book is to take the classification of stability boundaries due to Arnold Book Review / Journal of Sound and Vibration 284 (2005) 1253–1254 and to make it more quantitative, showing how certain important aspects of the geometry of thestability boundary can be calculated explicitly from the system at hand.
After an introductory chapter on stability theory, a long Chapter 2 deals with bifurcation analysis of eigenvalues of matrices depending on several parameters: this is the core of the bookon which most of the rest depends. Chapter 3 studies generic structure and local characterizationof the stability boundary up to codimension 3, using key ideas of general position that pervade thewhole approach. The singularities or non-smooth points that typically occur are important tounderstand, not least for the major influence they exert on the efficacy of numerical methods.
Motivated by ordinary differential equations of order m, the next chapter looks closely at the behaviour of the roots of an mth order polynomial as parameters vary, using classical techniquesof the Newton polygon and the Weierstrass preparation theorem. The next few chapters thenfocus on conservative systems, gyroscopic systems and linear Hamiltonian systems (all closelyrelated) and on mechanical interpretations of some of the theoretical results—including thecurious phenomenon of destabilization by damping. Chapter 9 turns to stability of parameter-dependent periodic systems, where it is the bifurcation of Floquet multipliers that is important,with Chapter 10 looking at stability boundaries in this context, and Chapters 11 and 12 studyinginteractions of parametric excitation and damping in several different specific contexts. The text isclearly written and the mathematics attractively set out with plenty of clear and instructivediagrams: an enjoyable book to read. Much of the work is based on the authors’ ownpublications, and the references naturally give prominence to Russian literature while recentWestern papers are less well represented. The English is almost flawless, with very occasionalnear-misses such as the use of enforced for strengthened and confusion of (in)definite articlesadding a little spice.
The book concludes with some remarks on extending the ideas to nonlinear aspects of bifurcation and to partial differential equations, although literature using methods of singularitytheory certainly does exist in both those fields. The increasing mathematical complications thatarise can often be tamed through exploitation of symmetries that are a natural componentof many real-world problems: for an overview of these matters see Ref. Incorporatingsymmetries of systems into the general study of stability boundaries would be a valuable extensionof the methods in this book.
[1] M. Golubitsky, I. Stewart, The Symmetry Perspective, Birkha¨user, Basel, 2003.
Department of Mathematics, University of Southampton, E-mail address: d.r.j.chillingworth@maths.soton.ac.uk

Source: http://mailybaev.imec.msu.ru/reviews/Chillingworth.pdf