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Global Bifurcation Theory and Hilbert's Sixteenth Problem

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ISBN-10: 1402075715

ISBN-13: 9781402075711

Edition: 2003

Authors: Valery A. Gaiko

List price: $109.00
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Description:

This volume is devoted to the qualitative investigation of two-dimensional polynomial dynamical systems and is aimed at solving Hilbert's Sixteenth Problem on the maximum number and relative position of limit cycles. The author presents a global bifurcation theory of such systems and suggests a new global approach to the study of limit cycle bifurcations. The obtained results can be applied to higher-dimensional dynamical systems and can be used for the global qualitative analysis of various mathematical models in mechanics, radioelectronics, in ecology and medicine. Audience: The book would be of interest to specialists in the field of qualitative theory of differential equations and…    
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Book details

List price: $109.00
Copyright year: 2003
Publisher: Springer
Publication date: 9/30/2003
Binding: Hardcover
Pages: 182
Size: 6.25" wide x 9.25" long x 0.75" tall
Weight: 1.034
Language: English

List of Figures
Preface
Acknowledgments
Foreword
Geometric Methods
On global bifurcation theory
Local bifurcations of limit cycles
On methods of the study of limit cycles
Geometric methods of bifurcations theory
Erugin's two-isocline method
On the method of isoclines
Isocline portraits
Classification of singular points
Other applications of the two-isocline method
The problem of center and focus
On the distinguishing problem
Lyapunov's focus quantities
Classification of symmetric cases
Geometric interpretation of quadratic centers
Poincare's topographical systems
Basic definitions
Construction of topographical systems
Topographical systems and limit cycles
On the control of semi-stable limit cycles
Canonical systems with limit cycles
Systems with field-rotation parameters
Reduction of quadratic systems
Andronov-Hopf Bifurcation
On the Andronov-Hopf bifurcation
On the history of the bifurcation
Bautin's result
The example by Shi Sonling
The example by E. A. Andronova
Construction of systems
Canonical systems with two singular points
Properties of the canonical systems
Bifurcations of limit cycles
Bifurcations of algebraic limit cycles
The case of a saddle and an antisaddle
A quadratic system with four limit cycles
Asymptotic behavior of limit cycles
Numerical results
Classification of Separatrix Cycles
On separatrix cycles
Dulac's theorem
Existential problem
On application of canonical systems
One saddle and three antisaddles (the first case)
One saddle and three antisaddles (the second case)
Classification of separatrix cycles
Other cases of singular points
The complete classification
Multiple Limit Cycles
On multiple limit cycles
Local analysis of one-parameter families
Basic notions
Multiple limit cycles and Puiseux series
Arches and paths of limit cycles
Global analysis of one-parameter families
Planar termination principle
The proof of the principle
Bifurcation surfaces of multiple limit cycles
Fold and cusp bifurcation surfaces
A swallow-tail bifurcation surface
General bifurcation surfaces
Wintner-Perko termination principle
General discussion
Monotonic families and rotated vector fields
The limit cycle problem for quadratic systems
Applications, Open Problems, Alternatives
On polynomial models of dynamical systems
A generalized Lotka-Volterra system
Some open problems of qualitative theory
Abelian integrals and limit cycles
On a work by N. P. Erugin
References
Index