Methods of Accelerated Convergence in Nonlinear Mechanics

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