Normal forms and unfoldings for local dynamical systems.

*(English)*Zbl 1014.37001
Springer Monographs in Mathematics. New York, NY: Springer. xix, 494 p. (2003).

This book is a treatise on normal forms and unfoldings of a dynamical systems near a singular point. The goal is to lay down basic principles and this is claimed to be the main originality of this work. Moreover it is selfcontained.

Chapter one presents two classical introductory examples leading to the Hopf and the Takens-Bogdanov bifurcations. Chapter two introduces to linear algebraic tools involving the splitting of linear operators (typically into semisimple and nilpotent part). Several methods are developed for finding the complement of the image of a linear operator. Chapters three and four are devoted to linear and nonlinear normal forms respectively. One uses methods of Lie theory. Some generalizations are included and concerning Chapter four, the author states in his preface that it “is the central chapter of the book”. The purpose is “to explain the structure of normal forms using the language of a module of equivariants (of some one-parameter group) over a ring of invariants and (as a secondary goal) to give algorithms suitable for use in symbolic computation systems”.

The last two chapters present selected topics (geometrical structures in truncated systems in relation to the issue of \(k\)-determinacy, some examples involving bifurcation theory and the author’s theory of asymptotic unfolding).

The author insists on the distinction between various notions of simplicity characterizing normal forms and various ways of computing them, all independent (a priori). He also distinguishes the dogmatic approach for unfolding in the sense that one looks for universal ones but they may not exist. The pragmatic way consists in unfolding truncated systems up to a given order \((k)\), but then the description may be incomplete. He also emphasizes the role of nilpotent normal forms. The volume contains new results including some of the author. There is a bibliography (115 entries) and an index at the end.

This conceptually attractive and clearly written book is recommended.

Chapter one presents two classical introductory examples leading to the Hopf and the Takens-Bogdanov bifurcations. Chapter two introduces to linear algebraic tools involving the splitting of linear operators (typically into semisimple and nilpotent part). Several methods are developed for finding the complement of the image of a linear operator. Chapters three and four are devoted to linear and nonlinear normal forms respectively. One uses methods of Lie theory. Some generalizations are included and concerning Chapter four, the author states in his preface that it “is the central chapter of the book”. The purpose is “to explain the structure of normal forms using the language of a module of equivariants (of some one-parameter group) over a ring of invariants and (as a secondary goal) to give algorithms suitable for use in symbolic computation systems”.

The last two chapters present selected topics (geometrical structures in truncated systems in relation to the issue of \(k\)-determinacy, some examples involving bifurcation theory and the author’s theory of asymptotic unfolding).

The author insists on the distinction between various notions of simplicity characterizing normal forms and various ways of computing them, all independent (a priori). He also distinguishes the dogmatic approach for unfolding in the sense that one looks for universal ones but they may not exist. The pragmatic way consists in unfolding truncated systems up to a given order \((k)\), but then the description may be incomplete. He also emphasizes the role of nilpotent normal forms. The volume contains new results including some of the author. There is a bibliography (115 entries) and an index at the end.

This conceptually attractive and clearly written book is recommended.

Reviewer: A.Akutowicz (Berlin)

##### MSC:

37-02 | Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory |

37G05 | Normal forms for dynamical systems |

37G10 | Bifurcations of singular points in dynamical systems |