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El. knyga: Method of Averaging for Differential Equations on an Infinite Interval: Theory and Applications

(Yaroslavl State University, Yaroslavl, Russia)
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In recent years, mathematicians have detailed simpler proofs of known theorems, have identified new applications of the method of averaging, and have obtained many new results of these applications. Encompassing these novel aspects, Method of Averaging of the Infinite Interval: Theory and Applications rigorously explains the modern theory of the method of averaging and provides a solid understanding of the results obtained when applying this theory.

The book starts with the less complicated theory of averaging linear differential equations (LDEs), focusing on almost periodic functions. It describes stability theory and Shtokalo's method, and examines various applications, including parametric resonance and the construction of asymptotics. After establishing this foundation, the author goes on to explore nonlinear equations. He studies standard form systems in which the right-hand side of a system is proportional to a small parameter and proves theorems similar to Banfi's theorem. The final chapters are devoted to systems with a rapidly rotating phase.

Covering an important asymptotic method of differential equations, this book provides a thorough understanding of the method of averaging theory and its resulting applications.

Recenzijos

". . . a very readable book . . . clearly written book can be highly recommended to students with interests in ordinary differential equations. Non-experts and researchers in natural sciences will also find interesting methods that are useful in applications."

In EMS Newsletter, March 2008

"Covering an important asymptotic method of differential equations, this book provides a thorough understanding of the method of averaging theory and its resulting applications."

in LEnseignment Math, 2007

Preface xii
I. Averaging of Linear Differential Equations
1(92)
Periodic and Almost Periodic Functions. Brief Introduction
3(10)
Periodic Functions
3(2)
Almost Periodic Functions
5(4)
Vector-Matrix Notation
9(4)
Bounded Solutions
13(10)
Homogeneous System of Equations with Constant Coefficients
13(1)
Bounded Solutions of Inhomogeneous Systems
14(6)
The Bogoliubov Lemma
20(3)
Lemmas on Regularity and Stability
23(14)
Regular Operators
23(1)
Lemma on Regularity
24(4)
Lemma on Regularity for Periodic Operators
28(2)
Lemma on Stability
30(7)
Parametric Resonance in Linear Systems
37(10)
Systems with One Degree of Freedom. The Case of Smooth Parametric Perturbations
37(3)
Parametric Resonance in Linear Systems with One Degree of Freedom. Systems with Impacts
40(3)
Parametric Resonance in Linear Systems with Two Degrees of Freedom. Simple and Combination Resonance
43(4)
Higher Approximations. The Shtokalo Method
47(18)
Problem Statement
47(1)
Transformation of the Basic System
48(2)
Remark on the Periodic Case
50(3)
Stability of Solutions of Linear Differential Equations with Near Constant Almost Periodic Coefficients
53(2)
Example. Generalized Hill's Equation
55(3)
Exponential Dichotomy
58(3)
Stability of Solutions of Systems with a Small Parameter and an Exponential Dichotomy
61(2)
Estimate of Inverse Operator
63(2)
Linear Differential Equations with Fast and Slow Time
65(10)
Generalized Lemmas on Regularity and Stability
65(4)
Example. Parametric Resonance in the Mathieu Equation with a Slowly Varying Coefficient
69(1)
Higher Approximations and the Problem of the Stability
70(5)
Asymptotic Integration
75(12)
Statement of the Problem
75(1)
Transformation of the Basic System
76(4)
Asymptotic Integration of an Adiabatic Oscillator
80(7)
Singularly Perturbed Equations
87(6)
II. Averaging of Nonlinear Systems
93(202)
Systems in Standard Form. First Approximation
95(30)
Problem Statement
95(1)
Theorem of Existence. Almost Periodic Case
96(3)
Theorem of Existence. Periodic Case
99(3)
Investigation of the Stability of an Almost Periodic Solution
102(5)
More General Dependence on a Parameter
107(1)
Almost Periodic Solutions of Quasi-Linear Systems
108(6)
Systems with Fast and Slow Time
114(6)
One Class of Singularly Perturbed Systems
120(5)
Systems in the Standard Form. First Examples
125(44)
Dynamics of Selection of Genetic Population in a Varying Environment
125(1)
Periodic Oscillations of Quasi-Linear Autonomous Systems with One Degree of Freedom and the Van der Pol Oscillator
126(6)
Van der Pol Quasi-Linear Oscillator
132(1)
Resonant Periodic Oscillations of Quasi-Linear Systems with One Degree of Freedom
133(4)
Subharmonic Solutions
137(2)
Duffing's Weakly Nonlinear Equation. Forced Oscillations
139(7)
Duffing's Equation. Forced Subharmonic Oscillations
146(4)
Almost Periodic Solutions of the Forced Undamped Duffing's Equation
150(1)
The Forced Van der Pol Equation. Almost Periodic Solutions in Non-Resonant Case
151(4)
The Forced Van de Pol Equation. A Slowly Varying Force
155(2)
The Forced Van der Pol Equation. Resonant Oscillations
157(1)
Two Weakly Coupled Van der Pol Oscillators
158(3)
Excitation of Parametric Oscillations by Impacts
161(8)
Pendulum Systems with an Oscillating Pivot
169(26)
History and Applications in Physics
169(3)
Equation of Motion of a Simple Pendulum with a Vertically Oscillating Pivot
172(1)
Introduction of a Small Parameter and Transformation into Standard Form
173(2)
Investigation of the Stability of Equilibria
175(3)
Stability of the Upper Equilibrium of a Rod with Distributed Mass
178(1)
Planar Vibrations of a Pivot
179(2)
Pendulum with a Pivot Whose Oscillations Vanish in Time
181(4)
Multifrequent Oscillations of a Pivot of a Pendulum
185(4)
System Pendulum-Washer with a Vibrating Base (Chelomei's Pendulum)
189(6)
Higher Approximations of the Method of Averaging
195(30)
Formalism of the Method of Averaging for Systems in Standard Form
195(3)
Theorem of Higher Approximations in the Periodic Case
198(3)
Theorem of Higher Approximations in the Almost Periodic Case
201(4)
General Theorem of Higher Approximations in the Almost Periodic Case
205(3)
Higher Approximations for Systems with Fast and Slow Time
208(1)
Rotary Regimes of a Pendulum with an Oscillating Pivot
209(6)
Critical Case Stability of a Pair of Purely Imaginary Roots for a Two-Dimensional Autonomous System
215(4)
Bifurcation of Cycle (the Andronov-Hopf Bifurcation)
219(6)
Averaging and Stability
225(20)
Basic Notation and Auxiliary Assertions
225(2)
Stability under Constantly Acting Perturbations
227(5)
Integral Convergence and Closeness of Solutions on an Infinite Interval
232(2)
Theorems of Averaging
234(4)
Systems with Fast and Slow Time
238(2)
Closeness of Slow Variables on an Infinite Interval in Systems with a Rapidly Rotating Phase
240(5)
Systems with a Rapidly Rotating Phase
245(20)
Near Conservative Systems with One Degree of Freedom
245(3)
Action-Angle Variables for a Hamiltonian System with One Degree of Freedom
248(2)
Autonomous Perturbations of a Hamiltonian System with One Degree of Freedom
250(3)
Action-Angle Variables for a Simple Pendulum
253(3)
Quasi-Conservative Vibro-Impact Oscillator
256(3)
Formal Scheme of Averaging for the Systems with a Rapidly Rotating Phase
259(6)
Systems with a Fast Phase. Resonant Periodic Oscillations
265(14)
Transformation of the Main System
266(2)
Behavior of Solutions in the Neighborhood of a Non-Degenerate Resonance Level
268(1)
Forced Oscillations and Rotations of a Simple Pendulum
269(6)
Resonance Oscillations in Systems with Impacts
275(4)
Systems with Slowly Varying Parameters
279(16)
Problem Statement. Transformation of the Main System
279(2)
Existence and Stability of Almost Periodic Solutions
281(9)
Forced Oscillations and Rotations of a Simple Pendulum. The Action of a Double-Frequency Perturbation
290(5)
III. Appendices
295(34)
Almost Periodic Functions
297(10)
Stability of the Solutions of Differential Equations
307(12)
Basic Definitions
307(3)
Theorems of the Stability in the First Approximation
310(4)
The Lyapunov Functions
314(5)
Some Elementary Facts from the Functional Analysis
319(10)
Banach Spaces
319(2)
Linear Operators
321(2)
Inverse Operators
323(3)
Principle of Contraction Mappings
326(3)
References 329(13)
Index 342


Yaroslavl State University, Yaroslavl, Russia
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