Multifrequency Oscillations of Nonlinear Systems

ISBN-13: 9789048165742 / Angielski / Miękka / 2010 / 317 str.

Anatolii M. Samoilenko; R. Petryshyn

Opis

In contrast to other books devoted to the averaging method and the method of integral manifolds, in the present book we study oscillation systems with many varying frequencies. In the process of evolution, systems of this type can pass from one resonance state into another. This fact considerably complicates the investigation of nonlinear oscillations. In the present monograph, a new approach based on exact uniform estimates of oscillation integrals is proposed. On the basis of this approach, numerous completely new results on the justification of the averaging method and its applications are obtained and the integral manifolds of resonance oscillation systems are studied. This book is intended for a wide circle of research workers, experts, and engineers interested in oscillation processes, as well as for students and post-graduate students specialized in ordinary differential equations.

Recenzje

From the reviews:

"The present research monograph deals with the averaging method for multifrequency nonlinear systems of ordinary differential equations. It gives a careful and detailed exposition to recent achievements by the authors and their coworkers based on exact uniform estimates of oscillatory integrals. As a result, many new results on the justification of the averaging method and its applications are obtained ... ." (G. Teschl, Monashefte für Mathematik, Vol. 146 (2), 2005)

Spis treści

Introduction. 1: Averaging Method in Oscillation Systems with Variable Frequencies. 1. Uniform Estimates for One-Dimensional Oscillation Integrals. 2. Justification of Averaging Method for Oscillation Systems with omega = omega(tau). 3. Investigation of Two-Frequency Systems. 4. Justification of Averaging Method for Oscillation Systems with omega = omega(x, tau). 5. Averaging over All Fast Variables in Multifrequency Systems of Higher Approximation. 2: Averaging Method in Multipoint Problems. 6. Boundary-Value Problems for Oscillation Systems with Frequencies Dependent on Time Variable. 7. Theorem on Justification of Averaging Method on Entire Axis. 8. Multipoint Problem for Resonance Multifrequency Systems. 9. Estimates of the Error of Averaging Method for Multipoint Problems in Critical Case. 10. Theorems on Existence of Solutions of Boundary-Value Problems. 11. Boundary-Value Problems with Parameters. 3: Integral Manifolds. 12. Auxiliary Statements. 13. Construction of Successive Approximations. 14. Existence of Integral Manifold. 15. Conditional Asymptotic Stability of Integral Manifold. 16. Smoothness of Integral Manifold. 17. Asymptotic Expansion of Integral Manifold. 18. Decomposition of Equations in the Neighborhood of Asymptotically Stable Integral Manifold. 19. Proof of Theorem 18.1. 20. Investigation of Second-Order Oscillation Systems. 21. Weakening of Conditions in the Theorem on Integral Manifold. 4: Investigation of a Dynamical System in the Neighborhood of Quasiperiodic Trajectory. 22. Statement and General description of the Problem. 23. Theorem on Reducibility. 24. Variational Equation and Theorem on Attraction to Quasiperiodic Solutions. 25. Behavior of Trajectories under Small Perturbations of a Dynamical System. 26. The Case of a Toroidal Manifold Filled with Trajectories of General Form. 27. Discrete Dynamical System in the Neighborhood of a Quasiperiodic Trajectory. References.