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Canard Cycles and Center Manifolds
 
Freddy Dumortier University Centrum Limburg
Robert Roussarie University of Bourgogne
Canard Cycles and Center Manifolds
eBook ISBN:  978-1-4704-0162-7
Product Code:  MEMO/121/577.E
List Price: $44.00
MAA Member Price: $39.60
AMS Member Price: $26.40
Canard Cycles and Center Manifolds
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Canard Cycles and Center Manifolds
Freddy Dumortier University Centrum Limburg
Robert Roussarie University of Bourgogne
eBook ISBN:  978-1-4704-0162-7
Product Code:  MEMO/121/577.E
List Price: $44.00
MAA Member Price: $39.60
AMS Member Price: $26.40
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 1211996; 100 pp
    MSC: Primary 34; 58;

    In this book, the “canard phenomenon” occurring in Van der Pol's equation \(\epsilon \ddot x+(x^2+x)\dot x+x-a=0\) is studied. For sufficiently small \(\epsilon >0\) and for decreasing \(a\), the limit cycle created in a Hopf bifurcation at \(a = 0\) stays of “small size” for a while before it very rapidly changes to “big size”, representing the typical relaxation oscillation. The authors give a geometric explanation and proof of this phenomenon using foliations by center manifolds and blow-up of unfoldings as essential techniques. The method is general enough to be useful in the study of other singular perturbation problems.

    Readership

    Graduate students, mathematicians, physicists, and engineers interested in ordinary differential equations, specifically singular perturbation problems.

  • Table of Contents
     
     
    • Chapters
    • 1. Statement of the result: The “canard phenomenon” for the singular Van der Pol equation
    • 2. Global desingularization
    • 3. Foliations by center manifolds
    • 4. The canard phenomenon
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 1211996; 100 pp
MSC: Primary 34; 58;

In this book, the “canard phenomenon” occurring in Van der Pol's equation \(\epsilon \ddot x+(x^2+x)\dot x+x-a=0\) is studied. For sufficiently small \(\epsilon >0\) and for decreasing \(a\), the limit cycle created in a Hopf bifurcation at \(a = 0\) stays of “small size” for a while before it very rapidly changes to “big size”, representing the typical relaxation oscillation. The authors give a geometric explanation and proof of this phenomenon using foliations by center manifolds and blow-up of unfoldings as essential techniques. The method is general enough to be useful in the study of other singular perturbation problems.

Readership

Graduate students, mathematicians, physicists, and engineers interested in ordinary differential equations, specifically singular perturbation problems.

  • Chapters
  • 1. Statement of the result: The “canard phenomenon” for the singular Van der Pol equation
  • 2. Global desingularization
  • 3. Foliations by center manifolds
  • 4. The canard phenomenon
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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