MULTIFREQUENCY OSCILLATIONS OF NONLINEAR SYSTEMS

By JENNIFER GARCIA

CONTENTS INTRODUCTION 1 AVERAGING METHOD IN OSCILLATION SYSTEMS WITH VARIABLE FREQUENCIES 1 UNIFORM ESTIMATES FOR ONE-DIMENSIONAL OSCILLATION INTEGRALS 2 JUSTI?CATION OF AVERAGING METHOD FOR OSCILLATION SYSTEMS WITH ? = ?(?) 3 INVESTIGATION OF TWO-FREQUENCY SYSTEMS 4 JUSTI?CATION OF AVERAGING METHOD FOR OSCILLATION SYSTEMS WITH ? = ?(X, ?) 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 JUSTI?CATION 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 A 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 A NEIGHBORHOOD OF A 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