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This monograph deals with some of the latest results in nonlinear mechanics, obtained recently by the use of a modernized version of Bogoljubov's method of successive changes of variables which ensures rapid convergence. This method visualised as early as 1934 by Krylov and Bogoljubov provides an effective tool for solving many interesting problems of nonlinear mechanics. It led, in particular, to the solution of the problem of the existence of a quasi periodic regime, with the restriction that approximate solutions obtained in the general case involved divergent series. Recently, making use of the research of Kolmogorov and Arno'ld, Bogoljubov has modernised the method of successive substitutions in such a way that the convergence of the corresponding expansions is ensured. This book consists of a short Introduction and seven chapters. The first chapter presents the results obtained by BogoIjubov in 1963 on the extension of the method of successive substitutions and the study of quasi periodic solutions applied to non-conservative systems (inter alia making explicit the dependence of these solutions on the parameter, indicating methods of obtaining asymptotic and convergent series for them, etc.).
List of contents
0. Introduction.- 1. Quasi-Periodic Solutions in Problems of Nonlinear Mechanics.-
1. Statement of the Problem. The Existence of an Invariant Manifold.-
2. Auxiliary Theorems.-
3. Lemma on Iterations.-
4. Theorem on Quasi-Periodic Solutions.-
5. Parametric Dependence of Quasi-Periodic Solutions. Asymptotic and Convergent Expansions.-
6. Quasi-Periodic Solutions in Second Order Systems.- 2. General Solutions of Nonlinear Differential Equations in the Neighbourhood of Quasi-Periodic Solutions.-
7. Statement of the Problem.-
8. Some Auxiliary Statements.-
9. Inductive Theorem.-
10. Iteration Process and its Convergence.-
11. Theorem on Reducibility of Nonlinear Equations.- 3. A Smoothing Technique.-
12. Loss of Derivatives.-
13. Examples of Smoothing Operators.-
14. The Basic Properties of a Smoothing Operator.-
15. Iteration Process with Smoothing.- 4. Trajectories on a Torus.-
16. Behaviour of Trajectories on a Two-Dimensional Torus.-
17. Behaviour of Trajectories on an m-Dimensional Torus.-
18. Inductive Theorem.-
19. Proof of the Theorem on the Reducibility of Equations on a Torus.- 5. Linear Systems with Quasi-Periodic Coefficients.-
20. Reducibility Theorem.-
21. Solution of the Auxiliary Equation.-
22. Proof of Reducibility Theorem.-
23. Construction of a Fundamental Matrix of Solutions.-
24. The Measure of Reducible Systems. Statement of the Problem.-
25. A Generalized Reducibility Theorem.-
26. Metric Propositions.-
27. Proof of the Measure Theorem.-
28. Linear Systems with Smooth Right-Hand Sides.- 6. Neighbourhood of an Invariant Smooth Toroidal Manifold.-
29. Behaviour of Integral Curves in the Neighbourhood of Toroidal Manifolds.-
30. Auxiliary Propositions.-
31.Iteration Theorem.-
32. Reducibility Theorem in the Neighbourhood of a Toroidal Manifold.-
33. Behaviour under Perturbation of Integral Curves in the Neighbourhood of an Invariant Manifold.- 7. Neighbourhood of a Compact Invariant Manifold of a Non-Autonomous System.-
34. Statement of the Problem and Basic Postulates.-
35. Lemma on the Solutions of an Auxiliary System.-
36. Inductive Theorem.-
37. Neighbourhood of an Invariant Manifold.-
38. Behaviour of Solutions of a System of Two Equations in the Neighbourhood of Equilibrium Positions.- Appendices I to XV.- References.
