Ebook: Bifurcations and Periodic Orbits of Vector Fields

The last thirty years were a period of continuous and intense growth in the subject of dynamical systems. New concepts and techniques and at the same time new areas of applications of the theory were found. The 31st session of the Seminaire de Mathematiques Superieures (SMS) held at the Universite de Montreal in July 1992 was on dynamical systems having as its center theme "Bifurcations and periodic orbits of vector fields". This session of the SMS was a NATO Advanced Study Institute (ASI). This ASI had the purpose of acquainting the participants with some of the most recent developments and of stimulating new research around the chosen center theme. These developments include the major tools of the new resummation techniques with applications, in particular to the proof of the non-accumulation of limit-cycles for real-analytic plane vector fields. One of the aims of the ASI was to bring together methods from real and complex dy­ namical systems. There is a growing awareness that an interplay between real and complex methods is both useful and necessary for the solution of some of the problems. Complex techniques become powerful tools which yield valuable information when applied to the study of the dynamics of real vector fields. The recent developments show that no rigid frontiers between disciplines exist and that interesting new developments occur when ideas and techniques from diverse disciplines are married. One of the aims of the ASI was to show these multiple interactions at work.

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The main topic of this book is the theory of bifurcations of vector fields, i.e. the study of families of vector fields depending on one or several parameters and the changes (bifurcations) in the topological character of the objects studied as parameters vary. In particular, one of the phenomena studied is the bifurcation of periodic orbits from a singular point or a polycycle. The following topics are discussed in the book:

• Divergent series and resummation techniques with applications, in particular to the proofs of the finiteness conjecture of Dulac saying that polynomial vector fields on R² cannot possess an infinity of limit cycles. The proofs work in the more general context of real analytic vector fields on the plane.
• Techniques in the study of unfoldings of singularities of vector fields (blowing up, normal forms, desingularization of vector fields). Local dynamics and nonlocal bifurcations.
• Knots and orbit genealogies in three-dimensional flows.
• Bifurcations and applications: computational studies of vector fields.
• Holomorphic differential equations in dimension two.
• Studies of real and complex polynomial systems and of the complex foliations arising from polynomial differential equations.
• Applications of computer algebra to dynamical systems.

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The main topic of this book is the theory of bifurcations of vector fields, i.e. the study of families of vector fields depending on one or several parameters and the changes (bifurcations) in the topological character of the objects studied as parameters vary. In particular, one of the phenomena studied is the bifurcation of periodic orbits from a singular point or a polycycle. The following topics are discussed in the book:

• Divergent series and resummation techniques with applications, in particular to the proofs of the finiteness conjecture of Dulac saying that polynomial vector fields on R² cannot possess an infinity of limit cycles. The proofs work in the more general context of real analytic vector fields on the plane.
• Techniques in the study of unfoldings of singularities of vector fields (blowing up, normal forms, desingularization of vector fields). Local dynamics and nonlocal bifurcations.
• Knots and orbit genealogies in three-dimensional flows.
• Bifurcations and applications: computational studies of vector fields.
• Holomorphic differential equations in dimension two.
• Studies of real and complex polynomial systems and of the complex foliations arising from polynomial differential equations.
• Applications of computer algebra to dynamical systems.

Content:
Front Matter....Pages i-xvii
Complex Foliations Arising from Polynomial Differential Equations....Pages 1-18
Techniques in the Theory of Local Bifurcations: Blow-Up, Normal Forms, Nilpotent Bifurcations, Singular Perturbations....Pages 19-73
Six Lectures on Transseries, Analysable Functions and the Constructive Proof of Dulac’s Conjecture....Pages 75-184
Knots and Orbit Genealogies in Three Dimensional Flows....Pages 185-239
Dynamical Systems: Some Computational Problems....Pages 241-277
Local Dynamics and Nonlocal Bifurcations....Pages 279-319
Singularit?s d’?quations diff?rentielles holomorphes en dimension deux....Pages 321-345
Techniques in the Theory of Local Bifurcations: Cyclicity and Desingularization....Pages 347-382
Bifurcation Methods in Polynomial Systems....Pages 383-428
Algebraic and Geometric Aspects of the Theory of Polynomial Vector Fields....Pages 429-467
Back Matter....Pages 469-472

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The main topic of this book is the theory of bifurcations of vector fields, i.e. the study of families of vector fields depending on one or several parameters and the changes (bifurcations) in the topological character of the objects studied as parameters vary. In particular, one of the phenomena studied is the bifurcation of periodic orbits from a singular point or a polycycle. The following topics are discussed in the book:

• Divergent series and resummation techniques with applications, in particular to the proofs of the finiteness conjecture of Dulac saying that polynomial vector fields on R² cannot possess an infinity of limit cycles. The proofs work in the more general context of real analytic vector fields on the plane.
• Techniques in the study of unfoldings of singularities of vector fields (blowing up, normal forms, desingularization of vector fields). Local dynamics and nonlocal bifurcations.
• Knots and orbit genealogies in three-dimensional flows.
• Bifurcations and applications: computational studies of vector fields.
• Holomorphic differential equations in dimension two.
• Studies of real and complex polynomial systems and of the complex foliations arising from polynomial differential equations.
• Applications of computer algebra to dynamical systems.

Content:
Front Matter....Pages i-xvii
Complex Foliations Arising from Polynomial Differential Equations....Pages 1-18
Techniques in the Theory of Local Bifurcations: Blow-Up, Normal Forms, Nilpotent Bifurcations, Singular Perturbations....Pages 19-73
Six Lectures on Transseries, Analysable Functions and the Constructive Proof of Dulac’s Conjecture....Pages 75-184
Knots and Orbit Genealogies in Three Dimensional Flows....Pages 185-239
Dynamical Systems: Some Computational Problems....Pages 241-277
Local Dynamics and Nonlocal Bifurcations....Pages 279-319
Singularit?s d’?quations diff?rentielles holomorphes en dimension deux....Pages 321-345
Techniques in the Theory of Local Bifurcations: Cyclicity and Desingularization....Pages 347-382
Bifurcation Methods in Polynomial Systems....Pages 383-428
Algebraic and Geometric Aspects of the Theory of Polynomial Vector Fields....Pages 429-467
Back Matter....Pages 469-472
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