Bifurcations of planar vector fields. Nilpotent singularities and Abelian integrals.

*(English)*Zbl 0755.58002
Lecture Notes in Mathematics. 1480. Berlin etc.: Springer-Verlag. viii, 226 p. (1991).

This volume consists of two separate, closely related articles. In part I, F. Dumortier, R. Roussarie and J. Sotomayor study generic 3-parameter families of planar vector fields with nilpotent linear part, while in part II, H. Żoładek discusses Abelian integrals in unfoldings of codimension 3 singular planar vector fields. Let us go through both parts separately.

In part I the authors continue and complete a previous study. Here three unfoldings of codimension 3 singularities of planar vector fields are presented. Together with the unfolding in the earlier paper they constitute a complete list of all topological models for all possible \(k\) parameter families with \(k \leq 3\). Linear planar vector fields with nilpotent linear part have (up to conjugacy) the form \(y{\partial\over\partial x}\). Normal form theory allows to transform a smooth, parameter dependent vector field with this linear part into \[ y{\partial\over\partial x}+(F(x,\lambda)+yG(x,\lambda)){\partial\over\partial y}+Q_ 1{\partial\over\partial x}+Q_ 2{\partial\over\partial y}. \] Here \(F\), \(G\) can be assumed to be polynomials of order \(N\) and \(Q_ 1\), \(Q_ 2\) are of order \(o(\|(x,y)\|^ N)\) in the variables \((x,y)\). The 2- jet of such a vector field has the form \(y{\partial\over\partial x}+(\alpha x^ 2+\beta xy){\partial\over\partial y}\). The cusp case (\(\alpha\neq 0\), \(\beta=0\)) is dealt with in the earlier paper. The other codimension 3 case (\(\alpha=0\), \(\beta\neq 0\)) is the subject of the present work. The goal is to study 3-parameter local families which generically unfold such a germ. By normal form theory the 4-jet of a germ with the above 2-jet is conjugate to \[ y{\partial\over\partial x}+(\varepsilon_ 1x^ 3+dx^ 4+bxy+ax^ 2y+ex^ 3y){\partial\over\partial y} \] with \(b > 0\), \(\varepsilon_ 1 = 0, \pm 1\). By a previous result of Dumortier the 3-jet determines the topological type if \(\varepsilon_ 1\neq 0\) or \(\varepsilon_ 1=-1\) and \(b\neq 2\sqrt{2}\). A regular set of 3-jets is defined by these conditions and \(5\varepsilon_ 1a-3bd\neq 0\). Under these conditions the authors prove that the 4-jet is equivalent to \[ y{\partial\over\partial x}+(\varepsilon_ 1x^ 3+bxy+\varepsilon_ 2x^ 2y+fx^ 3y){\partial\over\partial y}. \] Then three cases are treated separately, the saddle case (\(\varepsilon_ 1=1\), \(\varepsilon_ 2\), \(b\) arbitrary), the focus case (\(\varepsilon_ 1=-1\), \(0 < b < 2\sqrt{2}\)), and finally the elliptic case \(\varepsilon = -1\), \(b > 2\sqrt{2}\). In the article the bifurcations for generic families \(X_ \lambda\) of germs of vector fields with \(X_ 0\) in one of these sets is studied in detail. The authors define the term fiber equivalence for germs of families of vector fields. The present work is, according to the authors, devoted to establish facts which support the conjecture that any two families \(X_ \lambda\), \(Y_ \lambda\) with \(X_ 0\), \(Y_ 0\) belonging to the same set of regular germs are fiber equivalent. This study is divided into seven chapters. In chapter one a general introduction is given. Chapter 2 gives the notations and definitions. In chapter 3 certain transformations to normal forms are performed. A concise review of codimension 1 and 2 bifurcations is presented in chapter 4.

The main body of the work is in chapters four to seven. The points with codimension 1 and 2 bifurcations are computed. Two rescaling techniques are introduced (principal and central rescaling). In chapter six these are applied to the three cases. In chapter seven the authors discuss conclusions and remaining problems. The authors produced a clear and concise study of this problem. It is an invitation to work on this beautiful subject.

The second article in this volume by H. Żoładek is subdivided into three parts. In the first part the question of finding the number of zeros of Abelian integrals is related to two weakened versions of Hilbert’s 16th problem. After reviewing some finiteness properties for the number of limit cycles of planar vector fields the Petrov bounds and Chebyshev property of certain function spaces derived from Abelian integrals are discussed. In paragraph 4 of the first part, the author looks at bifurcation problems where Abelian integrals proved to be very useful. In the remaining two parts the saddle, elliptic and focus case are investigated again. Here, the emphasis is on the number of limit cycles when the system is nearly conservative. If the unperturbed problem has a first integral the number of limit cycles for the perturbed problem is given by the number of zeros of the corresponding Abelian integral. For the number of zeros precise bounds are given. The author uses methods from algebraic geometry and monodromy techniques. The theory of zeros of Abelian integrals is rather new. For different applications individual methods have been developed. The present paper could be a starting point towards a unified theory.

In part I the authors continue and complete a previous study. Here three unfoldings of codimension 3 singularities of planar vector fields are presented. Together with the unfolding in the earlier paper they constitute a complete list of all topological models for all possible \(k\) parameter families with \(k \leq 3\). Linear planar vector fields with nilpotent linear part have (up to conjugacy) the form \(y{\partial\over\partial x}\). Normal form theory allows to transform a smooth, parameter dependent vector field with this linear part into \[ y{\partial\over\partial x}+(F(x,\lambda)+yG(x,\lambda)){\partial\over\partial y}+Q_ 1{\partial\over\partial x}+Q_ 2{\partial\over\partial y}. \] Here \(F\), \(G\) can be assumed to be polynomials of order \(N\) and \(Q_ 1\), \(Q_ 2\) are of order \(o(\|(x,y)\|^ N)\) in the variables \((x,y)\). The 2- jet of such a vector field has the form \(y{\partial\over\partial x}+(\alpha x^ 2+\beta xy){\partial\over\partial y}\). The cusp case (\(\alpha\neq 0\), \(\beta=0\)) is dealt with in the earlier paper. The other codimension 3 case (\(\alpha=0\), \(\beta\neq 0\)) is the subject of the present work. The goal is to study 3-parameter local families which generically unfold such a germ. By normal form theory the 4-jet of a germ with the above 2-jet is conjugate to \[ y{\partial\over\partial x}+(\varepsilon_ 1x^ 3+dx^ 4+bxy+ax^ 2y+ex^ 3y){\partial\over\partial y} \] with \(b > 0\), \(\varepsilon_ 1 = 0, \pm 1\). By a previous result of Dumortier the 3-jet determines the topological type if \(\varepsilon_ 1\neq 0\) or \(\varepsilon_ 1=-1\) and \(b\neq 2\sqrt{2}\). A regular set of 3-jets is defined by these conditions and \(5\varepsilon_ 1a-3bd\neq 0\). Under these conditions the authors prove that the 4-jet is equivalent to \[ y{\partial\over\partial x}+(\varepsilon_ 1x^ 3+bxy+\varepsilon_ 2x^ 2y+fx^ 3y){\partial\over\partial y}. \] Then three cases are treated separately, the saddle case (\(\varepsilon_ 1=1\), \(\varepsilon_ 2\), \(b\) arbitrary), the focus case (\(\varepsilon_ 1=-1\), \(0 < b < 2\sqrt{2}\)), and finally the elliptic case \(\varepsilon = -1\), \(b > 2\sqrt{2}\). In the article the bifurcations for generic families \(X_ \lambda\) of germs of vector fields with \(X_ 0\) in one of these sets is studied in detail. The authors define the term fiber equivalence for germs of families of vector fields. The present work is, according to the authors, devoted to establish facts which support the conjecture that any two families \(X_ \lambda\), \(Y_ \lambda\) with \(X_ 0\), \(Y_ 0\) belonging to the same set of regular germs are fiber equivalent. This study is divided into seven chapters. In chapter one a general introduction is given. Chapter 2 gives the notations and definitions. In chapter 3 certain transformations to normal forms are performed. A concise review of codimension 1 and 2 bifurcations is presented in chapter 4.

The main body of the work is in chapters four to seven. The points with codimension 1 and 2 bifurcations are computed. Two rescaling techniques are introduced (principal and central rescaling). In chapter six these are applied to the three cases. In chapter seven the authors discuss conclusions and remaining problems. The authors produced a clear and concise study of this problem. It is an invitation to work on this beautiful subject.

The second article in this volume by H. Żoładek is subdivided into three parts. In the first part the question of finding the number of zeros of Abelian integrals is related to two weakened versions of Hilbert’s 16th problem. After reviewing some finiteness properties for the number of limit cycles of planar vector fields the Petrov bounds and Chebyshev property of certain function spaces derived from Abelian integrals are discussed. In paragraph 4 of the first part, the author looks at bifurcation problems where Abelian integrals proved to be very useful. In the remaining two parts the saddle, elliptic and focus case are investigated again. Here, the emphasis is on the number of limit cycles when the system is nearly conservative. If the unperturbed problem has a first integral the number of limit cycles for the perturbed problem is given by the number of zeros of the corresponding Abelian integral. For the number of zeros precise bounds are given. The author uses methods from algebraic geometry and monodromy techniques. The theory of zeros of Abelian integrals is rather new. For different applications individual methods have been developed. The present paper could be a starting point towards a unified theory.

Reviewer: R.Lauterbach (Berlin)

##### MSC:

58-02 | Research exposition (monographs, survey articles) pertaining to global analysis |

34-02 | Research exposition (monographs, survey articles) pertaining to ordinary differential equations |

34C23 | Bifurcation theory for ordinary differential equations |

37G15 | Bifurcations of limit cycles and periodic orbits in dynamical systems |

37G05 | Normal forms for dynamical systems |

34C20 | Transformation and reduction of ordinary differential equations and systems, normal forms |

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

34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |