Method of Averaging for Differential Equations on an Infinite Interval

Transcription

1 Method of Averaging for Differential Equations on an Infinite Interval Theory and Applications SUB Gottingen X ;, ' Vladimir Burd Yaroslavl State University Yaroslavl, Russia 2 ' 08A14338

2 Contents Preface t xii I Averaging of Linear Differential Equations 1 1 Periodic and Almost Periodic Functions. Brief Introduction Periodic Functions Almost Periodic Functions Vector-Matrix Notation 9 2 Bounded Solutions Homogeneous System of Equations with Constant Coefficients Bounded Solutions of Inhomogeneous Systems The Bogoliubov Lemma 20 3 Lemmas on Regularity and Stability Regular Operators Lemma on Regularity Lemma on Regularity for Periodic Operators Lemma on Stability 30 4 Parametric Resonance in Linear Systems Systems with One Degree of Freedom. The Case of Smooth Parametric Perturbations Parametric Resonance in Linear Systems with One Degree of Freedom. Systems with Impacts Parametric Resonance in Linear Systems with Two Degrees of Freedom. Simple and Combination Resonance 43 5 Higher Approximations. The Shtokalo Method Problem Statement Transformation of the Basic System Remark on the Periodic Case Stability of Solutions of Linear Differential Equations with Near Constant Almost Periodic Coefficients Example. Generalized Hill's Equation Exponential Dichotomy 58 vn

3 Vlll 5.7 Stability of Solutions of Systems with a Small Parameter and an Exponential Dichotomy Estimate of Inverse Operator 63 6 Linear Differential Equations with Fast and Slow Time Generalized Lemmas on Regularity and Stability Example. Parametric Resonance in the Mathieu Equation with a Slowly Varying Coefficient Higher Approximations and the Problem of the Stability Asymptotic Integration Statement of the Problem Transformation of the Basic System 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 Problem Statement Theorem of Existence. Almost Periodic Case Theorem of Existence. Periodic Case Investigation of the Stability of an Almost Periodic Solution More General Dependence on a Parameter Almost Periodic Solutions of Quasi-Linear Systems Systems with Fast and Slow Time One Class of Singularly Perturbed Systems Systems in the Standard Form. First Examples Dynamics of Selection of Genetic Population in a Varying Environment Periodic Oscillations of Quasi-Linear Autonomous Systems with One Degree of Freedom and the Van der Pol Oscillator Van der Pol Quasi-Linear Oscillator Resonant Periodic Oscillations of Quasi-Linear Systems with One Degree of Freedom Subharmonic Solutions Duffing's Weakly Nonlinear Equation. Forced Oscillations Duffing's Equation. Forced Subharmonic Oscillations Almost Periodic Solutions of the Forced Undamped Duffing's Equation The Forced Van der Pol Equation. Almost Periodic Solutions in Non-Resonant Case The Forced Van de Pol Equation. A Slowly Varying Force. 155

4 10.11 The Forced Van der Pol Equation. Resonant Oscillations Two Weakly Coupled Van der Pol Oscillators Excitation of Parametric Oscillations by Impacts Pendulum Systems with an Oscillating Pivot History and Applications in Physics Equation of Motion of a Simple Pendulum with a Vertically Oscillating Pivot Introduction of a Small Parameter and Transformation into Standard Form Investigation of the Stability of Equilibria Stability of the Upper Equilibrium of a Rod with Distributed Mass Planar Vibrations of a Pivot Pendulum with a Pivot Whose Oscillations Vanish in Time Multifrequent Oscillations of a Pivot of a Pendulum System Pendulum-Washer with a Vibrating Base (Chelomei's Pendulum) Higher Approximations of the Method of Averaging Formalism of the Method of Averaging for Systems in Standard Form Theorem of Higher Approximations in the Periodic Case Theorem of Higher Approximations in the Almost Periodic Case General Theorem of Higher Approximations in the Almost Periodic Case Higher Approximations for Systems with Fast and Slow Time Rotary Regimes of a Pendulum with an Oscillating Pivot Critical Case Stability of a Pair of Purely Imaginary Roots for a Two-Dimensional Autonomous System Bifurcation of Cycle (the Andronov-Hopf Bifurcation) Averaging and Stability Basic Notation and Auxiliary Assertions Stability under Constantly Acting Perturbations Integral Convergence and Closeness of Solutions on an Infinite Interval Theorems of Averaging Systems with Fast and Slow Time Closeness of Slow Variables on an Infinite Interval in Systems with a Rapidly Rotating Phase 240

5 14 Systems with a Rapidly Rotating Phase Near Conservative Systems with One Degree of Freedom Action-Angle Variables for a Hamiltonian System with One Degree of Freedom Autonomous Perturbations of a Hamiltonian System with One Degree of Freedom Action-Angle Variables for a Simple Pendulum Quasi-Conservative Vibro-Impact Oscillator Formal Scheme of Averaging for the Systems with a Rapidly Rotating Phase Systems with a Fast Phase. Resonant Periodic Oscillations Transformation of the Main System Behavior of Solutions in the Neighborhood of a Non-Degenerate Resonance Level Forced Oscillations and Rotations of a Simple Pendulum Resonance Oscillations in Systems with Impacts Systems with Slowly Varying Parameters Problem Statement. Transformation of the Main System Existence and Stability of Almost Periodic Solutions 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 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