Nonlinear Equations with Small Parameter: Volume 1 Oscillations and Resonances

• Publisher: De Gruyter
• Year: 2017
• Tongue: English
• Leaves: 358
• Series: De Gruyter Series in Nonlinear Analysis and Applications; 23/1
• Category: Library

✦ Synopsis

This two-volume monograph presents new methods of construction of global asymptotics of solutions to nonlinear equations with small parameter. These allow one to match the asymptotics of various properties with each other in transition regions and to get unified formulas for the connection of characteristic parameters of approximate solutions.

This approach underlies modern asymptotic methods and gives a deep insight into crucial nonlinear phenomena in the natural sciences. These include the outset of chaos in dynamical systems, incipient solitary and shock waves, oscillatory processes in crystals, engineering applications, and quantum systems. Apart from being of independent interest, such approximate solutions serve as a foolproof basis for testing numerical algorithms.

This first volume presents asymptotic methods in oscillation and resonance problems described by ordinary differential equations, whereby the second volume will be devoted to applications of asymptotic methods in waves and boundary value problems.

Contents

• Asymptotic expansions and series
• Asymptotic methods for solving nonlinear equations
• Nonlinear oscillator in potential well
• Autoresonances in nonlinear systems
• Asymptotics for loss of stability
• Systems of coupled oscillators

✦ Table of Contents

Contents
Introduction
1. Asymptotic expansions and series
1.1 Definitions of Asymptotic Series and Examples
1.1.1 An Example of Divergent Series
1.1.2 Order Operators
1.1.3 Calibration Sequence. Asymptotic Series
1.1.4 Problems
1.2 Summation of Asymptotic Series
1.2.1 Asymptotic Representation of Functions
1.2.2 Theorem on the Uniqueness of Asymptotic Expansion
1.2.3 Theorem on Existence of a Function with the Given Asymptotic Expansion
1.2.4 Problems
1.3 Laplace Method and Gamma Function
1.3.1 Asymptotic Expansion of Integral when Subintegral Function Exponent Does Not Contain Extrema
1.3.2 Asymptotic Expansion of Integral, When Integrand Exponent Contains Extrema
1.3.3 Derivation of Integral Formula for Gamma Function
1.3.4 Moivre–Stirling Formula
1.3.5 Problems
1.4 Fresnel Integral and Stationary Phase Method
1.4.1 Riemann Lemma
1.4.2 Fresnel Integral Formulae
1.4.3 Large Values of Argument
1.4.4 Method of Stationary Phase
1.4.5 Problems
1.5 Airy Function and Its Asymptotic Expansion
1.5.1 Airy’s Equation
1.5.2 An Integral Representation of General Solution for Airy Equation
1.5.3 Asymptotic Expansion for the Airy Function as z ? –8
1.5.4 Saddle-Point Method and the Airy Function Asymptotic Expansion as z?8
1.5.5 Problem
1.6 Functions of Parabolic Cylinder
1.6.1 Parabolic Cylinder Equation
1.6.2 Integral Representation
1.6.3 Connection Formulae at Different Values of Parameter
1.6.4 Values at Origin of Coordinates
1.6.5 Problems
1.7 WKB Method
1.7.1 Application of WKB Method for Ordinary Differential Equation of the Second Order
1.7.2 Justification of Constructed Asymptotic Expansion
1.7.3 Quasi-Classical Asymptotic Expansion and Transition Points
1.7.4 Essentially Singular Points of Differential Equation and the Stokes Phenomenon
1.7.5 Problems
2. Asymptotic methods for solving nonlinear equations
2.1 Fast Oscillating Asymptotic Expansions for Weak Nonlinear Case
2.1.1 Asymptotic Substitution
2.1.2 Equations for Leading-Order Term and First-Order Correction Term
2.1.3 Equation for the nth Term of Formal Series and Domain of Validity
2.1.4 Justification of Asymptotic Series
2.1.5 Problems
2.2 Boundary Layer Method
2.2.1 Asymptotic Solution
2.2.2 Boundary Layer
2.2.3 Justification of Asymptotic Solution
2.2.4 Problems
2.3 Catastrophes and Regular Expansions
2.3.1 A Formal Series with Respect to Powers of Small Parameter
2.3.2 Justification of Asymptotic Series
2.3.3 Non-analytic Perturbation
2.3.4 Matching of the Asymptotic Series for the Root of Cubic Equation
2.3.5 Compound Asymptotic Expansion for Root of Cubic Equation
2.3.6 Problems
2.4 Weierstrass Function
2.4.1 Differential Equation
2.4.2 Doubly Periodicity
2.4.3 A Representation by Series
2.4.4 Evenness
2.4.5 Liouville’s Theorem
2.4.6 Problem
2.5 Jacobi Elliptic Functions
2.5.1 Sin-Amplitude Function
2.5.2 Periodicity
2.5.3 Jacobi Elliptic Functions
2.5.4 Regular Expansion in the Neighbourhood of Zero Value of Argument
2.5.5 Regular Expansion in the Neighbourhood of Zero Value of Parameter
2.5.6 Regular Expansion in the Neighbourhood of k = 1
2.5.7 Problems
2.6 Uniform Asymptotic Behaviour of Jacobi-sn Near a Singular Point. The Lost Formula from Handbooks for Elliptic Functions
2.6.1 The Asymptotic Behaviour of the Period
2.6.2 Asymptotic Behaviour on a Regular Part of Trajectory
2.6.3 Asymptotic Behaviour Near Turning Point
2.6.4 Uniform Asymptotic Expansion
2.7 Mathieu’s and Lame’s Functions
2.7.1 Hill’s Equation and Floquet’s Theory
2.7.2 Examples
2.7.3 Mathieu Functions
2.7.4 Construction of Mathieu’s Functions
2.7.5 Lindeman form of Mathieu’s Equation
2.7.6 Special Case of Lame’s Equation
2.7.7 Degenerate Case
2.7.8 Problems
3. Perturbation of nonlinear oscillations
3.1 Regular Perturbation Theory for Nonlinear Oscillations
3.1.1 Properties of Solutions for Unperturbed Equation
3.1.2 Formal Asymptotic Expansion for Solution
3.1.3 Solution of Nonlinear Equation for Primary Term
3.1.4 Homogeneous Linearized Equation
3.1.5 Non-homogeneous Linearized Equation
3.1.6 First Correction for Perturbation Theory
3.1.7 Second Correction in Formula (3.8)
3.1.8 Causes Leading to the Growing of Corrections
3.1.9 Problems
3.2 Fast and Slow Variables
3.2.1 Two-Scaling Method
3.2.2 Isochronous Oscillations
3.2.3 An Averaging of Isochronous Oscillations
3.2.4 Transcendent Equation for Parameter
3.2.5 Equations for Parameters of Averaging
3.2.6 Problems
3.3 Krylov–Bogolyubov Method
3.3.1 Asymptotical Substitution
3.3.2 Formula for Leading-Order Term of Asymptotic Expansion
3.3.3 Solution of Linearized Equation
3.3.4 Periodic Solution for the First Correction
3.3.5 Problems
3.4 Higher-Order Terms in Krylov–Bogolyubov Method
3.4.1 Second Correction of Perturbation Theory
3.4.2 Periodic Solution of Equation for nth Correction
3.5 Interval of Validity for Krylov–Bogolyubov’s Ansatz
3.5.1 Small Neighbourhoods of a Centre
3.5.2 Neighbourhood of Separatrix and Saddle
3.5.3 Asymptotic Solution of a Cauchy Problem
4. Nonlinear oscillator in potential well
4.1 Nonlinear Oscillator Near Separatrix
4.1.1 Change to Simple Form
4.1.2 Qualitative Behaviour and Numerical Analysis
4.2 Asymptotic Solution Close to Separatrix
4.2.1 Construction of Germ Asymptotic Expansion
4.2.2 Behaviour of Correction Terms in the Neighbourhood of the Separatrix
4.2.3 Asymptotic Expansion Near Saddle Point
4.2.4 Asymptotic Solution in the Neighbourhood of Lower Separatrix
4.2.5 Parameters of Equation and Cantor Set
4.3 Oscillations with External Force into Potential Well
4.3.1 An Asymptotic Problem of a Capture
4.4 Non-resonant Regions of Parameter
4.4.1 An Equation for Averaged Action
4.4.2 The Substitution of Krylov–Bogolyubov
4.4.3 Linearized Equation
4.4.4 Construction of the First Correction Term
4.4.5 Construction of the Second Correction Term
4.4.6 Resonances in Higher-Order Correction Terms
4.5 Asymptotics in Resonant Regions
4.5.1 Formal Derivation of Nonlinear Resonance Equation
4.5.2 Inner Asymptotic Expansion
4.5.3 Capture into Resonance
4.5.4 Asymptotic Solutions of the Equation of Nonlinear Resonance
4.5.5 Matching of Asymptotic Expansions
4.5.6 Asymptotic Solution of the Capture Problem
5. Autoresonances in nonlinear systems
5.1 Problems of Autoresonance
5.1.1 The Arising of Autoresonance
5.1.2 Autoresonant Asymptotic Expansions and Scattering Problem
5.1.3 Cut-Off of a Resonant Growth
5.2 Threshold of Amplitude for Autoresonant Pumping
5.2.1 Autoresonant Solution
5.2.2 Asymptotic Substitution
5.2.3 Stability of Autoresonant Solution
5.3 Capture Into Autoresonance
5.3.1 Setting of the Problem for Trajectories of Large Amplitude
5.3.2 Numeric Results and Instability
5.3.3 Oscillations Far from the Capture
5.4 A Searching of Suitable Asymptotic Expansion
5.4.1 Asymptotic Expansion Towards Bifurcation
5.4.2 Asymptotic Expansion in Bottleneck
5.4.3 Connection Formulas for Perturbed System
5.5 A Thin Manifold of a Captured Trajectories
5.5.1 Slowly Varying Equilibrium Points
5.5.2 A Rough Conservation Law
5.5.3 Breaking Up of Separatrix
5.6 Asymptotic Solution of the Capture Problem
5.6.1 Matching to Bottleneck Asymptotic Expansion
5.6.2 Numerical investigations
5.6.3 Asymptotics and Numeric Points View
5.7 Capture into Parametric Resonance
5.7.1 Numeric Analysis
5.7.2 Qualitative Analysis
5.8 WKB Solution Before the Capture
5.8.1 The WKB Solution Closed to Zero
5.8.2 Constructing of the WKB Solution for Nonlinear Equation (5.40)
5.9 The Painlevé Layer
5.9.1 The Asymptotic Expansion in the Painlevé Layer
5.9.2 Matching with the WKB Asymptotic Expansion
5.10 The Captured WKB Asymptotic Solution
5.10.1 Slowly Varying Solutions
5.10.2 WKB Asymptotic Expansion Close to the Slowly Varying Centres
6. Asymptotics for loss of stability
6.1 Hard Loss of Stability in Painlevé-2 Equation
6.1.1 Naive Statement of the Problem
6.1.2 Matched Asymptotics for the Solution
6.1.3 The Outer Algebraic Asymptotics
6.1.4 The Domain of Validity of the Algebraic Asymptotic Solution
6.2 The Inner Asymptotics
6.2.1 First Inner Expansion
6.2.2 Second Inner Expansion
6.2.3 Dynamics in the Internal Layer
6.2.4 The Asymptotics of the Inner Expansions as 4 ? 8
6.3 Fast Oscillating Asymptotics
6.3.1 The Krylov–Bogolyubov Approximation
6.3.2 Degeneration of the Fast Oscillating Asymptotics
6.3.3 The Domain of Validity of the Fast Oscillating Asymptotics
6.3.4 The Matching of the Fast Oscillating Asymptotic Solution and the Inner Asymptotics
6.4 An Asymptotic Solution Slowly Crossing the Separatrix Near a Saddle-Centre Bifurcation Point
6.4.1 Typical Problems for the Autoresonance
6.4.2 Three Types of Algebraic Solutions
6.5 Expansions in Bifurcation Layer
6.5.1 Initial Interval
6.5.2 The Bifurcation Layer in the Case of Bounded k
6.5.3 The Intermediate Expansion for Large k
6.6 Fast Oscillating Asymptotic Expansion
6.6.1 Family of the Fast Oscillating Solutions
6.6.2 The Confluent Asymptotic Solution
6.6.3 The Domain of Validity of Confluent Asymptotic Solution as t ? t* – 0
6.6.4 Matching of the Asymptotics
6.6.5 Asymptotic Behaviours
6.7 Dissipation Is Cause for Halt of Resonant Growth
6.7.1 Setting of the Problem
6.7.2 Asymptotics of Autoresonant Growth Under Dissipation
6.7.3 Stability of Autoresonant Growth
6.7.4 Vicinity of the Break of Autoresonant Growth
6.8 Break of Autoresonant Growth
6.8.1 Fast Motion
6.8.2 Formal Approach to Answer
6.9 Open Problems
6.9.1 Hierarchy of Equations in Transition and Painlevé Equations
7. Systems of coupled oscillators
7.1 The Autoresonance Threshold into System of Weakly Coupled Oscillators
7.1.1 Statement of the Problem and Result
7.1.2 Asymptotic Reduction to the System of Primary Resonance Equations
7.1.3 Algebraic Asymptotic Solutions
7.1.4 Neighbourhoods of Equilibrium Positions
7.2 Forced Nonlinear Resonance in a System of Coupled Oscillators
7.2.1 Results
7.2.2 Formal Constructions for : « 1
7.2.3 Analysis of Asymptotic Solution for + » 1
Bibliography
Index