Bifurcations in Hamiltonian systems. Computing singularities by Gröbner bases
``This book deals with nonlinear Hamiltonian systems, depending on parameters.\dots The general goal is to understand their dynamics in a qualitative, and if possible, also quantitative way. In many important cases, it is possible to reduce a skeleton of the dynamics to lower dimension, sometimes leading to a Hamiltonian system in one degree of freedom. Such reduced systems allow a singularity theory or catastrophe theory approach which gives raise to transparent, in a sense polynomial, normal forms. Moreover the whole process of arriving at these normal forms is algorithmic. The purpose of this book is to develop computer-algebraic tools for the implementation of these algorithms. This set-up allows for many applications concerning resonances in coupled or driven oscillators, the \(n\)-body problem, the dynamics of the rigid body \dots''. This description opens the book, and conveniently summarizes its scope. Given a Hamiltonian system (around an equilibrium point) described by a Hamiltonian \(H\) and the associated Hamiltonian vector field \(X\), the latter can be taken to Birkhoff-Gustavsson normal form up to some degree \(k\) by means of a sequence of canonical transformations; denoting the resulting canonical transformation as \(\Phi\), one has thus mapped \(X\) into the vector field \(\Phi_* (X) = N + R\), where \(N\) is the normalized part, and \(R\) a (small) remainder. Thus, the flow under the normalized part \(N\) provides an approximation of the ``true'' flow under \(\Phi_*(X)\), hence (upon use of \(\Phi^{-1}\)) also under \(X\). This is the ``skeleton'' of the dynamics mentioned above. In many interesting cases, the vector field \(N\) in normal form exhibits a large degree of symmetry, and can thus be reduced to a lower-dimensional (Hamiltonian) system \(N_0\) by more and less standard methods. As mentioned above, it is not so rare that the lower-dimensional system is actually a Hamiltonian system in one degree of freedom, thus amenable to complete analysis and displaying a regular behaviour. It should be stressed that the remainder part \(R\) of \(\Phi_* (X)\) does in general not share the symmetries of \(N\), so that the analysis of the effect of the (small) perturbation \(R\) on the ``skeleton'' provided by \(N\) is far from trivial; indeed taking \(R\) into account does often lead to the appearance of a chaotic dynamics. As the authors suggest, ``one could say that the approximation \(N\) contains the regular skeleton supporting the chaotic zones of instability''. The study of the dynamics is thus divided into two steps, i.e. (a) study of the dynamics of the normalized part \(N\), maybe via its symmetry reduction; and (b) study of the perturbation of this by the remainder \(R\). The book is essentially concerned with the first step (see e.g. \textit{H. W. Broer} and \textit{F. Takens} [Dynamics reported, vol. 2, 39-59 (1989; Zbl 0704.58047)] for an approach to the second step worked out by one of these authors). The study of the dynamics generated by \(N\) is, in this approach, split in substeps: (i) Symmetry reduction to an effective dynamics \(N_0\) in lower dimension; (ii) a singularity theory analysis to study the bifurcations of the reduced normalized dynamics \(N_0\). It should be stressed that here the final aim is to be able to map the results of this analysis back to the original system and ``physical'' coordinates: this imposes to keep complete track of all the changes of coordinates and projections involved in the procedure, and to be able to work them backwards. This is a nontrivial and computationally demanding task, which the authors tackle very effectively. Their theoretical discussion (which is presented in part II of this book, only after showing the results which can be obtained by their methods) is compact but deep; readers not at ease with this part should however find motivation in the results given in part I. A wide set of comparisons of exact results with ``experimental'' ones (i.e. outcomes of numerical simulations) is also presented, providing further evidence of the power of the general results and methods discussed here for dealing with concrete applications. The symmetry reduction of \(N\) to \(N_0\) can be performed in different ways; in particular the authors considers two methods. These are: (i) the ``planar reduction method'' fostered by two of them [\textit{H. W. Broer} and \textit{G. Vegter}, Dyn. Rep., Expo. Dyn. Syst., New Ser. 1, 1-53 (1992; Zbl 0893.58043); \textit{H. W. Broer, S. N. Chow, Y. Kim} and \textit{G. Vegter}, Z. Angew. Math. Phys. 44, 389-432 (1993; Zbl 0805.58047)]; and (ii) the ``energy-momentum map'' [\textit{J. J. Duistermaat}, Bifurcation theory and applications, Lect. Notes Math. 1057, 57-105 (1984; Zbl 0546.58036); \textit{J.-C. van der Meer},The Hamiltonian Hopf bifurcation, Lect. Notes Math. 1160. Berlin etc.: Springer-Verlag (1985; Zbl 0585.58019)], on which the authors also gave relevant contributions [\textit{H. W. Broer, I. Hoveijn, G. A. Lunter} and \textit{G. Vegter}, Nonlinearity 11, 1569-1605 (1998; Zbl 0932.37041)]. As mentioned above, with both methods one obtains a formal normal form \(N\) (corresponding to a normalized Hamiltonian \(H_n\)), after which one reduces to a one degree of freedom Hamiltonian system \(N_0\) (effective Hamiltonian \(H_0\)); singularity theory is then applied on this. The most interesting cases contain rather strong resonances, which yields discrete symmetries in the normalized vector field; to these should be added the discrete symmetries present due to physical constraints, such as time reversibility. The systems considered here depend on distinguished parameters; the latter are not generic, but actually correspond to (nonnegative) action variables. Thus the unfolding considered in the singularity theory analysis should not be generic, but reflect this specific nature; such an unfolding was developed in [\textit{H. W. Broer, S. N. Chow, Y. Kim} and \textit{G. Vegter}, Z. Angew. Math. Phys. 44, 389-432 (1993; Zbl 0805.58047)]. The two methods can be formalized in a unified way by the use of ``standard bases'' [\textit{H. Hironaka}, Ann. Math. (2) 79, 109-203, 326 (1964; Zbl 0122.38603)]; these can be considered as a generalization of Gröbner bases. The book consists of 7 chapters, organized into an Introduction (Chapter 1) and two parts; the first of these (Chapters 2-3) is devoted to Applications, or actually to methods and their applications to specific examples, while the second (Chapters 4-7) deals with the theory lying behind the methods exposed previously. An Appendix deals with classification of term orders, and with the proof of a result concerning equivalence of symmetric germs given in the main body of the book. Throughout the book, the spring pendulum is used as a case study. The content of different chapters is as follows. Chapter 1 contains an extensive Introduction, and some discussion of the results obtained. Chapter 2 is devoted to the planar reduction method, and applications of this, while Chapter 3 deals with the energy-momentum method, and applications. The short Chapters 4 and 5 deal, respectively with Birkhoff normalization and algorithms for actually performing it, and with singularity theory. Chapter 6 is devoted to Gröbner bases and standard bases, with applications to the ring of formal power series. Finally, Chapter 7 considers the computation of the normalizing transformations. The book considers a specific class of systems (identified by symmetry properties), and for these goes at the heart of what Poincaré deemed to be ``the fundamental problem of dynamics''. It can be read at different levels; it is directed, and should result of interest, to different groups of scientists (not only mathematicians) active in nonlinear dynamics, from the very theoretical to the very applied. Addressing such a varied audience is definitely a hard problem in itself, but this book is quite successful in solving it, as well as in solving the mathematical problems it tackles.
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