Method of Averaging for Differential Equations on an Infinite Interval Theory and Applications SUB Gottingen 7 222 045 71X ;, ' Vladimir Burd Yaroslavl State University Yaroslavl, Russia 2 ' 08A14338
Contents Preface t xii I Averaging of Linear Differential Equations 1 1 Periodic and Almost Periodic Functions. Brief Introduction 3 1.1 Periodic Functions 3 1.2 Almost Periodic Functions 5 1.3 Vector-Matrix Notation 9 2 Bounded Solutions 13 2.1 Homogeneous System of Equations with Constant Coefficients 13 2.2 Bounded Solutions of Inhomogeneous Systems 14 2.3 The Bogoliubov Lemma 20 3 Lemmas on Regularity and Stability 23 3.1 Regular Operators 23 3.2 Lemma on Regularity 24 3.3 Lemma on Regularity for Periodic Operators 28 3.4 Lemma on Stability 30 4 Parametric Resonance in Linear Systems 37 4.1 Systems with One Degree of Freedom. The Case of Smooth Parametric Perturbations 37 4.2 Parametric Resonance in Linear Systems with One Degree of Freedom. Systems with Impacts 40 4.3 Parametric Resonance in Linear Systems with Two Degrees of Freedom. Simple and Combination Resonance 43 5 Higher Approximations. The Shtokalo Method 47 5.1 Problem Statement 47 5.2 Transformation of the Basic System 48 5.3 Remark on the Periodic Case 50 5.4 Stability of Solutions of Linear Differential Equations with Near Constant Almost Periodic Coefficients 53 5.5 Example. Generalized Hill's Equation 55 5.6 Exponential Dichotomy 58 vn
Vlll 5.7 Stability of Solutions of Systems with a Small Parameter and an Exponential Dichotomy 61 5.8 Estimate of Inverse Operator 63 6 Linear Differential Equations with Fast and Slow Time 65 6.1 Generalized Lemmas on Regularity and Stability 65 6.2 Example. Parametric Resonance in the Mathieu Equation with a Slowly Varying Coefficient 69 6.3 Higher Approximations and the Problem of the Stability.. 70 7 Asymptotic Integration 75 7.1 Statement of the Problem 75 7.2 Transformation of the Basic System 76 7.3 Asymptotic Integration of an Adiabatic Oscillator 80 8 Singularly Perturbed Equations 87 II Averaging of Nonlinear Systems 93 9 Systems in Standard Form. First Approximation 95 9.1 Problem Statement 95 9.2 Theorem of Existence. Almost Periodic Case 96 9.3 Theorem of Existence. Periodic Case 99 9.4 Investigation of the Stability of an Almost Periodic Solution 102 9.5 More General Dependence on a Parameter 107 9.6 Almost Periodic Solutions of Quasi-Linear Systems 108 9.7 Systems with Fast and Slow Time 114 9.8 One Class of Singularly Perturbed Systems 120 10 Systems in the Standard Form. First Examples 125 10.1 Dynamics of Selection of Genetic Population in a Varying Environment 125 10.2 Periodic Oscillations of Quasi-Linear Autonomous Systems with One Degree of Freedom and the Van der Pol Oscillator... 126 10.3 Van der Pol Quasi-Linear Oscillator 132 10.4 Resonant Periodic Oscillations of Quasi-Linear Systems with One Degree of Freedom 133 10.5 Subharmonic Solutions 137 10.6 Duffing's Weakly Nonlinear Equation. Forced Oscillations.. 139 10.7 Duffing's Equation. Forced Subharmonic Oscillations... 146 10.8 Almost Periodic Solutions of the Forced Undamped Duffing's Equation 150 10.9 The Forced Van der Pol Equation. Almost Periodic Solutions in Non-Resonant Case... 151 10.10 The Forced Van de Pol Equation. A Slowly Varying Force. 155
10.11 The Forced Van der Pol Equation. Resonant Oscillations. 157 10.12 Two Weakly Coupled Van der Pol Oscillators 158 10.13 Excitation of Parametric Oscillations by Impacts 161 11 Pendulum Systems with an Oscillating Pivot 169 11.1 History and Applications in Physics 169 11.2 Equation of Motion of a Simple Pendulum with a Vertically Oscillating Pivot 172 11.3 Introduction of a Small Parameter and Transformation into Standard Form 173 11.4 Investigation of the Stability of Equilibria 175 11.5 Stability of the Upper Equilibrium of a Rod with Distributed Mass 178 11.6 Planar Vibrations of a Pivot 179 11.7 Pendulum with a Pivot Whose Oscillations Vanish in Time. 181 11.8 Multifrequent Oscillations of a Pivot of a Pendulum 185 11.9 System Pendulum-Washer with a Vibrating Base (Chelomei's Pendulum) 189 12 Higher Approximations of the Method of Averaging 195 12.1 Formalism of the Method of Averaging for Systems in Standard Form 195 12.2 Theorem of Higher Approximations in the Periodic Case.. 198 12.3 Theorem of Higher Approximations in the Almost Periodic Case 201 12.4 General Theorem of Higher Approximations in the Almost Periodic Case 205 12.5 Higher Approximations for Systems with Fast and Slow Time 208 12.6 Rotary Regimes of a Pendulum with an Oscillating Pivot.. 209 12.7 Critical Case Stability of a Pair of Purely Imaginary Roots for a Two-Dimensional Autonomous System 215 12.8 Bifurcation of Cycle (the Andronov-Hopf Bifurcation)... 219 13 Averaging and Stability 225 13.1 Basic Notation and Auxiliary Assertions 225 13.2 Stability under Constantly Acting Perturbations 227 13.3 Integral Convergence and Closeness of Solutions on an Infinite Interval 232 13.4 Theorems of Averaging 234 13.5 Systems with Fast and Slow Time 238 13.6 Closeness of Slow Variables on an Infinite Interval in Systems with a Rapidly Rotating Phase 240
14 Systems with a Rapidly Rotating Phase 245 14.1 Near Conservative Systems with One Degree of Freedom.. 245 14.2 Action-Angle Variables for a Hamiltonian System with One Degree of Freedom 248 14.3 Autonomous Perturbations of a Hamiltonian System with One Degree of Freedom 250 14.4 Action-Angle Variables for a Simple Pendulum 253 14.5 Quasi-Conservative Vibro-Impact Oscillator 256 14.6 Formal Scheme of Averaging for the Systems with a Rapidly Rotating Phase 259 15 Systems with a Fast Phase. Resonant Periodic Oscillations 265 15.1 Transformation of the Main System 266 15.2 Behavior of Solutions in the Neighborhood of a Non-Degenerate Resonance Level 268 15.3 Forced Oscillations and Rotations of a Simple Pendulum.. 269 15.4 Resonance Oscillations in Systems with Impacts 275 16 Systems with Slowly Varying Parameters 279 16.1 Problem Statement. Transformation of the Main System.. 279 16.2 Existence and Stability of Almost Periodic Solutions 281 16.3 Forced Oscillations and Rotations of a Simple Pendulum. The Action of a Double-Frequency Perturbation 290 III Appendices 295 A Almost Periodic Functions 297 B Stability of the Solutions of Differential Equations 307 B.I Basic Definitions 307 B.2 Theorems of the Stability in the First Approximation... 310 B.3 The Lyapunov Functions 314 C Some Elementary Facts from the Functional Analysis 319 C.I Banach Spaces 319 C.2 Linear Operators 321 C.3 Inverse Operators 323 C.4 Principle of Contraction Mappings 326 References 329 Index 342