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Advances in the Mathematical Sciences
Methods of Qualitative Theory of Differential Equations and Related Topics
Edited by: L. Lerman, Research Institute for Applied Mathematics and Cybernetics, Nizhnii Novgorod, Russia, G. Polotovskiĭ, Nizhnii Novgorod State University, Nizhni Novgorod, Russia, and L. Shilnikov, Research Institute for Applied Mathematics and Cybernetics, Nizhnii Novgorod, Russia

American Mathematical Society Translations--Series 2
Advances in the Mathematical Sciences
2000; 196 pp; hardcover
Volume: 200
ISBN-10: 0-8218-2663-8
ISBN-13: 978-0-8218-2663-8
List Price: US$109
Member Price: US$87.20
Order Code: TRANS2/200
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Dedicated to the memory of Professor E. A. Leontovich-Andronova, this book was composed by former students and colleagues who wished to mark her contributions to the theory of dynamical systems. A detailed introduction by Leontovich-Andronova's close colleague, L. Shilnikov, presents biographical data and describes her main contribution to the theory of bifurcations and dynamical systems.

The main part of the volume is composed of research papers presenting the interests of Leontovich-Andronova, her students and her colleagues. Included are articles on traveling waves in coupled circle maps, bifurcations near a homoclinic orbit, polynomial quadratic systems on the plane, foliations on surfaces, homoclinic bifurcations in concrete systems, topology of plane controllability regions, separatrix cycle with two saddle-foci, dynamics of 4-dimensional symplectic maps, torus maps from strong resonances, structure of 3 degree-of-freedom integrable Hamiltonian systems, splitting separatrices in complex differential equations, Shilnikov's bifurcation for \(C^1\)-smooth systems and "blue sky catastrophe" for periodic orbits.


Graduate students, research and applied mathematicians interested in qualitative theory of differential equations; physicists.

Table of Contents

  • L. P. Shilnikov -- Evgeniya Aleksandrovna Leontovich-Andronova (1905-1996)
  • V. Afraimovich and M. Courbage -- On the abundance of traveling waves in coupled expanding circle maps
  • S. A. Alekseeva and L. P. Shilnikov -- On cusp-bifurcations of periodic orbits in systems with a saddle-focus homoclinic curve
  • S. Aranson, V. Medvedev, and E. Zhuzhoma -- Collapse and continuity of geodesic frameworks of surface foliations
  • V. N. Belykh -- Homoclinic and heteroclinic linkages in concrete systems: Nonlocal analysis and model maps
  • A. A. Binstein and G. M. Polotovskiĭ -- On the mutual arrangement of a conic and a quintic in the real projective plane
  • N. N. Butenina -- The structure of the boundary curve for planar controllability domains
  • V. V. Bykov -- Orbit structure in a neighborhood of a separatrix cycle containing two saddle-foci
  • N. Gavrilov and A. Shilnikov -- Example of a blue sky catastrophe
  • S. V. Gonchenko -- Dynamics and moduli of \(\Omega\)-conjugacy of 4D-diffeomorphisms with a structurally unstable homoclinic orbit to a saddle-focus fixed point
  • V. Z. Grines and R. V. Plykin -- Topological classification of amply situated attractors of \(A\)-diffeomorphisms of surfaces
  • M. V. Shashkov and D. V. Turaev -- A proof of Shilnikov's theorem for \(C^1\)-smooth dynamical systems
  • L. P. Shilnikov and D. V. Turaev -- A new simple bifurcation of a periodic orbit of "blue sky catastrophe" type
  • V. P. Tareev -- On the splitting of the complex loop of a separatrix
  • A. A. E. -- Some topological structures of quadratic systems with at least four limit cycles
  • J. Guckenheimer and I. Khibnik -- Torus maps from weak coupling of strong resonances
  • L. M. Lerman -- Isoenergetical structure of integrable Hamiltonian systems in an extended neighborhood of a simple singular point: Three degrees of freedom
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