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INTRODUCTION 1

1. AVERAGING METHOD IN OSCILLATION SYSTEMS

WITH VARIABLE FREQUENCIES 9

1. Uniform Estimates for One-Dimensional Oscillation Integrals . . 9

2. Justification of Averaging Method for Oscillation Systems

with ω = ω(τ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3. Investigation of Two-Frequency Systems . . . . . . . . . . . . . . 39

4. Justification of Averaging Method for Oscillation Systems

with ω = ω(x, τ ) . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5. Averaging over All Fast Variables in Multifrequency Systems

of Higher Approximation . . . . . . . . . . . . . . . . . . . . . . 63

2. AVERAGING METHOD IN MULTIPOINT PROBLEMS 71

6. Boundary-Value Problems for Oscillation Systems with

Frequencies Dependent on Time Variable . . . . . . . . . . . . . 71

7. Theorem on Justification of Averaging Method on Entire Axis . . 80

8. Multipoint Problem for Resonance Multifrequency Systems . . . 89

9. Estimates of the Error of Averaging Method for Multipoint

Problems in Critical Case . . . . . . . . . . . . . . . . . . . . . . 99

10. Theorems on Existence of Solutions of Boundary-Value

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

11. Boundary-Value Problems with Parameters . . . . . . . . . . . . 120

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3. INTEGRAL MANIFOLDS 133

12. Auxiliary Statements . . . . . . . . . . . . . . . . . . . . . . . . 133

13. Construction of Successive Approximations . . . . . . . . . . . . 145

14. Existence of Integral Manifold . . . . . . . . . . . . . . . . . . . 157

15. Conditional Asymptotic Stability of Integral Manifold . . . . . . . 165

16. Smoothness of Integral Manifold . . . . . . . . . . . . . . . . . . 174

17. Asymptotic Expansion of Integral Manifold . . . . . . . . . . . . 194

18. Decomposition of Equations in a Neighborhood of

Asymptotically Stable Integral Manifold . . . . . . . . . . . . . . 206

19. Proof of Theorem 18.1 . . . . . . . . . . . . . . . . . . . . . . . 218

20. Investigation of Second-Order Oscillation Systems . . . . . . . . 224

21. Weakening of Conditions in the Theorem on Integral Manifold . . 233

4. INVESTIGATION OF A DYNAMICAL SYSTEM IN

A NEIGHBORHOOD OF A QUASIPERIODIC TRAJECTORY 243

22. Statement and General Description of the Problem . . . . . . . . 243

23. Theorem on Reducibility . . . . . . . . . . . . . . . . . . . . . . 248

24. Variational Equation and Theorem on Attraction to

Quasiperiodic Solutions . . . . . . . . . . . . . . . . . . . . . . . 261

25. Behavior of Trajectories under Small Perturbations of

a Dynamical System . . . . . . . . . . . . . . . . . . . . . . . . 266

26. The Case of a Toroidal Manifold Filled with Trajectories of

General Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

27. Discrete Dynamical System in the Neighborhood of

a Quasiperiodic Trajectory . . . . . . . . . . . . . . . . . . . . . 282

REFERENCES 297

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