Singularities and groups in bifurcation theory. Volume I.

*(English)*Zbl 0607.35004
Applied Mathematical Sciences, 51. New York etc.: Springer-Verlag. XVII, 463 p. DM 148.00 (1985).

Here is the first sentence in the Preface: ”This book has been written in a frankly partisan style - we believe that singularity theory offers an extremely useful approach to bifurcation problems and we hope to convert the reader to this view.” Later, the authors precise carefully their role: ”Let us emphasize that our principal contribution in this area has been to apply pre-existing techniques from singularity theory...to bifurcation problems.” On the other hand, the book will be followed by a companion where group theory and symmetry will play a more important role.

Indeed, the aim is to provide a rather systematic approach to the above topics. The recognition problem, perturbations and unfoldings, and classification and codimension, are strongly emphasized. They are developed at length, but this is balanced by the attention paid to applications. This balance is, in our opinion, one of the best qualities of the book. A second one is its introductory character, together with the importance given to motivation and guiding ideas.

First, let us give a survey of the contents. Chapter I is an introduction, where some interesting examples are considered, together with a first version of the Lyapunov-Schmidt reduction and the calculation of the derivatives of the reduced function.

Chapters II-IV are the core of the book. Chapter II is devoted to the recognition problem, i.e. to characterize mappings equivalent to a given map. This leads to the splitting of derivatives into low-order, higher- order and intermediate-order terms, and to the study of the restricted tangent space. Unfolding theory is the main subject of Chapter III. This means to find, up to equivalence, all perturbations for a given bifurcation problem. The Universal Unfolding Theorem is stated and used, but its proof is deferred until Vo. II. Codimension is characterized as the number of parameters entering in a universal unfolding. The second part deals with nonpersistent bifurcation diagrams and the three sources of nonpersistence: bifurcation, hysteresis and double limit points. The theorem saying that the transition set is a semi-algebraic variety is not proved. Classification of all bifurcation of all bifurcation problems with codimension \(\leq 3\) is the core of Chapter IV. The theorem asserting the existence of 11 elementary bifurcation problems is proved and tables providing a complete information about normal forms, universal unfoldings, transition varieties, etc., are included.

The important topic of moduli is considered in Chapter V by means of an interesting example. Moduli arise when problems are topologically but not \(C^{\infty}\) equivalent. The notion of codimension constant variety leads to the introduction of modal parameters, and the complicated behaviour related to moduli is nicely described by diagrams with distinguished points, connector points, etc.

Symmetry is introduced in Chapter VI in the simple case of a function g(x,\(\lambda)\) odd in x, i.e. with the group \(Z_ 2=\{I,R\}\), where \(Rx=-x\), and the influence of symmetry on codimension is studied. The basic theory of Chapters II-IV is then extended to this symmetric case. The Lyapunov-Schmidt reduction is extended in Chapter VII in three different directions, namely infinite-dimensional spaces, multidimensional \((>1)\) kernels, and the symmetric case. Applications, including a celebrated reaction-diffusion system (the Brusselator) are included.

Chapter VIII is devoted to Hopf Bifurcation, by emphasizing that Singularity Theory is well-adapted to the degenerate cases. The problem is reduced to an application of the Lyapunov-Schmidt reduction, and symmetry is exploited. The two first Hopf Theorems are proved. Floquet’s Theory and stability are carefully studied, including the degenerate case, and the results are applied to the glycolisis system. Another very nice application is the Clamped Hodgkin-Huxley Equation.

Problems in several variables are introduced in Chapter IX. After some heuristic considerations on codimension (which is not even defined) only the codimension 3 case is considered, completing the Classification Theorem. Chapter X deals with problems conmuting with the group \(Z_ 2\oplus Z_ 2\) and some relations between symmetry and stability. The book ends with a last nice application, the mode jumping in the buckling of a rectangular plate. Given the difficulties of the subject, the book is very pleasant to read. The authors try to put the statements and discussion of all results at the beginning of the section and the proofs at the end, and this is quite helpful. Some heuristic arguments are especially welcome, as is the case with emphasis on ”organizing centers” or the interest devoted to the global implications of the local results. The section dedicated to the Path Formulation and the comparison between Catastrophe Theory, Singularity Theory and Bifurcation Theory is also suggestive.

Finally, let us say that there are some misprints in the text. More important, some of the reference in the text are not included in the bibliography at the end (cf., e.g., Whitney (1943) in p. 248, but there are more).

Indeed, the aim is to provide a rather systematic approach to the above topics. The recognition problem, perturbations and unfoldings, and classification and codimension, are strongly emphasized. They are developed at length, but this is balanced by the attention paid to applications. This balance is, in our opinion, one of the best qualities of the book. A second one is its introductory character, together with the importance given to motivation and guiding ideas.

First, let us give a survey of the contents. Chapter I is an introduction, where some interesting examples are considered, together with a first version of the Lyapunov-Schmidt reduction and the calculation of the derivatives of the reduced function.

Chapters II-IV are the core of the book. Chapter II is devoted to the recognition problem, i.e. to characterize mappings equivalent to a given map. This leads to the splitting of derivatives into low-order, higher- order and intermediate-order terms, and to the study of the restricted tangent space. Unfolding theory is the main subject of Chapter III. This means to find, up to equivalence, all perturbations for a given bifurcation problem. The Universal Unfolding Theorem is stated and used, but its proof is deferred until Vo. II. Codimension is characterized as the number of parameters entering in a universal unfolding. The second part deals with nonpersistent bifurcation diagrams and the three sources of nonpersistence: bifurcation, hysteresis and double limit points. The theorem saying that the transition set is a semi-algebraic variety is not proved. Classification of all bifurcation of all bifurcation problems with codimension \(\leq 3\) is the core of Chapter IV. The theorem asserting the existence of 11 elementary bifurcation problems is proved and tables providing a complete information about normal forms, universal unfoldings, transition varieties, etc., are included.

The important topic of moduli is considered in Chapter V by means of an interesting example. Moduli arise when problems are topologically but not \(C^{\infty}\) equivalent. The notion of codimension constant variety leads to the introduction of modal parameters, and the complicated behaviour related to moduli is nicely described by diagrams with distinguished points, connector points, etc.

Symmetry is introduced in Chapter VI in the simple case of a function g(x,\(\lambda)\) odd in x, i.e. with the group \(Z_ 2=\{I,R\}\), where \(Rx=-x\), and the influence of symmetry on codimension is studied. The basic theory of Chapters II-IV is then extended to this symmetric case. The Lyapunov-Schmidt reduction is extended in Chapter VII in three different directions, namely infinite-dimensional spaces, multidimensional \((>1)\) kernels, and the symmetric case. Applications, including a celebrated reaction-diffusion system (the Brusselator) are included.

Chapter VIII is devoted to Hopf Bifurcation, by emphasizing that Singularity Theory is well-adapted to the degenerate cases. The problem is reduced to an application of the Lyapunov-Schmidt reduction, and symmetry is exploited. The two first Hopf Theorems are proved. Floquet’s Theory and stability are carefully studied, including the degenerate case, and the results are applied to the glycolisis system. Another very nice application is the Clamped Hodgkin-Huxley Equation.

Problems in several variables are introduced in Chapter IX. After some heuristic considerations on codimension (which is not even defined) only the codimension 3 case is considered, completing the Classification Theorem. Chapter X deals with problems conmuting with the group \(Z_ 2\oplus Z_ 2\) and some relations between symmetry and stability. The book ends with a last nice application, the mode jumping in the buckling of a rectangular plate. Given the difficulties of the subject, the book is very pleasant to read. The authors try to put the statements and discussion of all results at the beginning of the section and the proofs at the end, and this is quite helpful. Some heuristic arguments are especially welcome, as is the case with emphasis on ”organizing centers” or the interest devoted to the global implications of the local results. The section dedicated to the Path Formulation and the comparison between Catastrophe Theory, Singularity Theory and Bifurcation Theory is also suggestive.

Finally, let us say that there are some misprints in the text. More important, some of the reference in the text are not included in the bibliography at the end (cf., e.g., Whitney (1943) in p. 248, but there are more).

Reviewer: J.Hernandez

##### MSC:

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

35B32 | Bifurcations in context of PDEs |

37G99 | Local and nonlocal bifurcation theory for dynamical systems |

57R45 | Singularities of differentiable mappings in differential topology |

34D20 | Stability of solutions to ordinary differential equations |

74M20 | Impact in solid mechanics |

76L05 | Shock waves and blast waves in fluid mechanics |

34D05 | Asymptotic properties of solutions to ordinary differential equations |

34D30 | Structural stability and analogous concepts of solutions to ordinary differential equations |

35L67 | Shocks and singularities for hyperbolic equations |