Local Methods in Nonlinear Differential Equations

Bruno Alexander D.

ISBN 10: 364264788X / ISBN 13: 9783642647888

Editore: Springer, 2011

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Dati bibliografici

• Titolo: Local Methods in Nonlinear Differential Equations
• Editore: Springer
• Data di pubblicazione: 2011
• Lingua: Inglese
• ISBN 10: 364264788X
• ISBN 13: 9783642647888
• Rilegatura: Brossura
• Condizione: New

Descrizione articolo

Riassunto

The method of normal forms is usually attributed to Poincaré although some of the basic ideas of the method can be found in earlier works of Jacobi, Briot and Bouquet. In this book, A.D.Bruno gives an account of the work of these mathematicians and further developments as well as the results of his own extensive investigations on the subject. The book begins with a thorough presentation of the analytical techniques necessary for the implementation of the theory as well as an extensive description of the geometry of the Newton polygon. It then proceeds to discuss the normal form of systems of ordinary differential equations giving many specific applications of the theory. An underlying theme of the book is the unifying nature of the method of normal forms regarding techniques for the study of the local properties of ordinary differential equations. In the second part of the book it is shown, for a special class of equations, how the method of normal forms yields classical results of Lyapunov concerning families of periodic orbits in the neighborhood of equilibrium points of Hamiltonian systems as well as the more modern results concerning families of quasiperiodic orbits obtained by Kolmogorov, Arnold and Moser. The book is intended for mathematicians, theoretical mechanicians, and physicists. It is suitable for advanced undergraduate and graduate students.

Contenuti

I The Local Method of Nonlinear Analysis of Differential Equations.- I. Foundations of the Local Method.- § 1. Linear Inequalities in the Plane.- § 2. Zeros of an Analytic Function.- § 3. Level Curves of an Analytic Function.- II. A System of Two Differential Equations.- § 1. Simple Points and Elementary Singularities.- § 2. Generalized Normal Forms.- § 3. A Nonelementary Singular Point.- § 4. On Distinguishing Between a Center and a Focus.- III. The Normal Form of a System on n Differential Equations.- § 1. The Normalizing Transformation.- § 2. The Integration and Classification of Normal Forms.- § 3. Analytic Integral Sets.- § 4. The Normal Form and Methods of Averaging.- IV. On the Newton Polyhedron.- § 1. A System of Differential Equations.- § 2. Other Problems Using the Newton Polyhedron.- V. Applications of the Normal Form in Mechanics.- § 1. On the Motion of a Gyroscope in a Cardan Suspension.- § 2. On the Oscillations of a Satellite in the Plane of an Elliptical Orbit.- II The Sets of Analyticity of a Normalizing Transformation.- § 1. The Seminormal Form.- § 2. Questions of Convergence.- § 3. A Hamiltonian System.- § 4. Families of Periodic Solutions.- § 5. Integral Manifolds with Small Divisors.- Author’ Comments (1986).- References.