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Theory of bifurcations of dynamics system on a plane
Mechanism and Machine Theory Vol. 14, pp. 75-78 © Pergamon Press Ltd., 1979, Printed in Great Britain
0094-114X/79/0101-0075/$02.00/0
Book Reviews A. A. ANDRONOV,E. H. LEONTOVICH,I. E. GORDONand A. G. /~dER, Theory of Bifurcations of Dynamic Systems on a Plane (in Russian). Izdat. Nauka, Moskow, 1967. (English translation, 482 pages, 208 diagrams, 6~ x ~ in. Price $42.00. Israel Program for Sciencific Translations Ltd. New York (1973)). This monograph may be considered as a continuation of the authors book Qualitative Theory of Dynamic Systems of Second-Order (Russian, Izdat. Nauka, Moskow, 1966). However, it can be studied independently provided the reader has an understanding of the basic concepts of the qualitative theory of second-order differential equations. The main feature of the book is the coherent presentation of some results on structural stability obtained in the last 30 yr mainly in U.S.S.R. by A. A. Andronov, L. S. Pontryagin and others. In the references 38 titles are listed. The First Part of the book deals with the theory of structurally stable systems: Chapter I, Multiplicity of roots of functions and multiplicity of intersection points of two curves; Chapter II, Dynamic systems close to a given system and properties of their phase portraits; Chapter III, The space of dynamic systems and structurally stable systems; Chapter IV, Equilibrium states of structurally stable systems. Saddle-to-saddle separatrix; Chapter V, Closed paths in structurally stable systems; Chapter VI, Necessary and sufficient conditions of structural stability of systems; Chapter VII, Cells of structurally stable systems. An addition to the theory of structurally stable systems. Roughly speaking a dynamic system on a plane
dx P(x, y), d--[=
dy d--t= Q(x, y)
(1)
defined in a region G is said to be structurally stable if its topological structures does not change under small modifications of the right hand side of the system. Structurally stable systems are important in physical problems governed by equations of type (1) when some parameters exhibit small changes within the experimental margin of error. However, specific applications are not presented since that is not the author's aim. The Second .Part of the book is concerned with the theory of bifurcations: Chapter VIII, Bifurcations of dynamic systems. Decomposition of a multiple equilibrium state into structurally stable equilibrium states; Chapter IX, Creation of limit cycles from a multiple focus; Chapter X, Creation of closed paths from a multiple limit cycle; Chapter XI, Creation of limit cycles from the loop of a saddle-point separatrix; Chapter XII, Creation of a limit cycle from the loop of a saddle-node separatrix. Systems of first degree of structural instability and their bifurcations; Chapter XIII, Limit cycles of some dynamic systems depending on a parameter; Chapter XIV, The application of the theory of bifurcations to the investigation of particular dynamic systems. A good deal of consideration is given to structurally unstable systems, which are also of interest in applications especially the conservative systems. The book is written precisely and carefully and gives an exhaustive treatment on the subject. It can be used in a twofold way: for self-study and for a seminar for graduate work. To my best knowledge this is the first monograph on structural stability.
Department of Mathematics Simon Fraser University Burnaby 2, British Columbia Canada
GEORGEBOJADZIEV
75
0094-114X/79/0101-0075/$02.00/0
Book Reviews A. A. ANDRONOV,E. H. LEONTOVICH,I. E. GORDONand A. G. /~dER, Theory of Bifurcations of Dynamic Systems on a Plane (in Russian). Izdat. Nauka, Moskow, 1967. (English translation, 482 pages, 208 diagrams, 6~ x ~ in. Price $42.00. Israel Program for Sciencific Translations Ltd. New York (1973)). This monograph may be considered as a continuation of the authors book Qualitative Theory of Dynamic Systems of Second-Order (Russian, Izdat. Nauka, Moskow, 1966). However, it can be studied independently provided the reader has an understanding of the basic concepts of the qualitative theory of second-order differential equations. The main feature of the book is the coherent presentation of some results on structural stability obtained in the last 30 yr mainly in U.S.S.R. by A. A. Andronov, L. S. Pontryagin and others. In the references 38 titles are listed. The First Part of the book deals with the theory of structurally stable systems: Chapter I, Multiplicity of roots of functions and multiplicity of intersection points of two curves; Chapter II, Dynamic systems close to a given system and properties of their phase portraits; Chapter III, The space of dynamic systems and structurally stable systems; Chapter IV, Equilibrium states of structurally stable systems. Saddle-to-saddle separatrix; Chapter V, Closed paths in structurally stable systems; Chapter VI, Necessary and sufficient conditions of structural stability of systems; Chapter VII, Cells of structurally stable systems. An addition to the theory of structurally stable systems. Roughly speaking a dynamic system on a plane
dx P(x, y), d--[=
dy d--t= Q(x, y)
(1)
defined in a region G is said to be structurally stable if its topological structures does not change under small modifications of the right hand side of the system. Structurally stable systems are important in physical problems governed by equations of type (1) when some parameters exhibit small changes within the experimental margin of error. However, specific applications are not presented since that is not the author's aim. The Second .Part of the book is concerned with the theory of bifurcations: Chapter VIII, Bifurcations of dynamic systems. Decomposition of a multiple equilibrium state into structurally stable equilibrium states; Chapter IX, Creation of limit cycles from a multiple focus; Chapter X, Creation of closed paths from a multiple limit cycle; Chapter XI, Creation of limit cycles from the loop of a saddle-point separatrix; Chapter XII, Creation of a limit cycle from the loop of a saddle-node separatrix. Systems of first degree of structural instability and their bifurcations; Chapter XIII, Limit cycles of some dynamic systems depending on a parameter; Chapter XIV, The application of the theory of bifurcations to the investigation of particular dynamic systems. A good deal of consideration is given to structurally unstable systems, which are also of interest in applications especially the conservative systems. The book is written precisely and carefully and gives an exhaustive treatment on the subject. It can be used in a twofold way: for self-study and for a seminar for graduate work. To my best knowledge this is the first monograph on structural stability.
Department of Mathematics Simon Fraser University Burnaby 2, British Columbia Canada
GEORGEBOJADZIEV
75