Theory of Oscillators presents the applications and exposition of the qualitative theory of differential equations. This book discusses the idea of a discontinuous transition in a dynamic process.
Organized into 11 chapters, this book begins with an overview of the simplest type of oscillatory system in which the motion is described by a linear differential equation. This text then examines the character of the motion of the representative point along the hyperbola. Other chapters consider examples of two basic types of non-linear non-conservative systems, namely, dissipative systems and self-oscillating systems. This book discusses as well the discontinuous self-oscillations of a symmetrical multi-vibrator neglecting anode reaction. The final chapter deals with the immense practical importance of the stability of physical systems containing energy sources particularly control systems.
This book is a valuable resource for electrical engineers, scientists, physicists, and mathematicians.

Contenu

Preface to the Second Russian Edition

Note from the English Editor

Introduction

I. Linear Systems

§ 1. A Linear System without Friction (Harmonic Oscillator)

§ 2. The Concept of the Phase Plane. Representation on the Phase Plane of the Totality of Motions of a Harmonic Oscillator

The Phase Plane
Equation Not Involving Time
Singular Points. Center
Isoclines
State of Equilibrium and Periodic Motion
§ 3. Stability of a State of Equilibrium
§ 4. Linear Oscillator in the Presence of Friction
Damped Oscillatory Process
Representation of a Damped Oscillatory Process on the Phase Plane
Direct Investigation of the Differential Equation
Damped Aperiodic Process
Representation of an Aperiodic Process on the Phase Plane
§ 5. Oscillator with Small Mass
Linear Systems with Half Degree of Freedom
Initial Conditions and Their Relations to the Idealization
Conditions for a Jump
Other Examples
§ 6. Linear Systems with "Negative Friction"
Mechanical Example
Electrical Example
Portrait on the Phase Plane
Behavior of the System for a Variation of the Feedback
§ 7. Linear System with Repulsive Force
Portrait on the Phase Plane
An Electrical System
Singular Point of the Saddle Type

II. Non-Linear Conservative Systems

§ 1. Introduction

§ 2. The Simplest Conservative System

§ 3. Investigation of the Phase Plane Near States of Equilibrium

§ 4. Investigation of the Character of the Motions on the Whole Phase Plane

§ 5. Dependence of the Behavior of the Simplest Conservative System upon a Parameter

Motion of a Point Mass along a Circle which Rotates about a Vertical Axis
Motion of a Material Point along a Parabola Rotating about Its Vertical Axis
Motion of a Conductor Carrying a Current
§ 6. The Equations of Motion
Oscillating Circuit with Iron Core
Oscillating Circuit having a Rochelle Salt Capacitor
§ 7. General Properties of Conservative Systems
Periodic Motions and Their Stability
Single-Valued Analytic Integral and Conservativeness
Conservative Systems and Variational Principle
Integral Invariant
Basic Properties of Conservative Systems
Example. Simultaneous Existence of Two Species

III. Non-Conservative Systems

§ 1. Dissipative Systems

§ 2. Oscillator with Coulomb Friction

§ 3. Valve Oscillator with a ) Characteristic

§ 4. Theory of the Clock. Model with Impulses

The Clock with Linear Friction
Valve Generator with a Discontinuous ) Characteristic
Model of the Clock with Coulomb Friction
§ 5. Theory of the Clock. Model of a "Recoil Escapement" without Impulses
Model of Clock with a Balance-Wheel without Natural Period
Model of Clock with a Balance-Wheel having a Natural Period
§ 6. Properties of the Simplest Self-Oscillating Systems
§ 7. Preliminary Discussion of Nearly Sinusoidal Self-Oscillations

IV. Dynamic Systems with a First Order Differential Equation

§ 1. Theorems of Existence and Uniqueness

§ 2. Qualitative Character of the Curves on the t, x Plane Depending on the Form of the Function f(x)

§ 3. Motion on the Phase Line

§ 4. Stability of the States of Equilibrium

§ 5. Dependence of the Character of the Motions on a Parameter

Voltaic Arc in a Circuit with Resistance and Self-Induction
Dynatron Circuit with Resistance and Capacitance
Valve Relay (Bi-Stable Trigger Circuit)
Motion of a Hydroplane
Single-Phase Induction Motor
Frictional Speed Regulator
§ 6. Periodic Motions
Two-Position Temperature Regulator
Oscillations in a Circuit with a Neon Tube
§ 7. Multivibrator with One RC Circuit

V. Dynamic Systems of the Second Order

§ 1. Phase Paths and Integral Curves on the Phase Plane

§ 2. Linear Systems of the General Type

§ 3. Examples of Linear Systems

Small Oscillations of a Dynatron Generator
The "Universal" Circuit
§ 4. States of Equilibrium and Their Stability
The Case of Real Roots of the Characteristic Equation
The Characteristic Equation with Complex Roots
§ 5. Example: States of Equilibrium in the Circuit of a Voltaic Arc
§ 6. Limit Cycles and Self-Oscillations
§ 7. Point Transformations and Limit Cycles
Sequence Function and Point Transformation
Stability of the Fixed Point. Koenigs's Theorem
A Condition of Stability of the Limit Cycle
§ 8. Poincare's Indices
§ 9. Systems without Closed Paths
Symmetrical Valve Relay (Trigger)
Dynamos Working in Parallel on a Common Load
Oscillator with Quadratic Terms
One More Example of Non-Self-Oscillating System
§10. The Behavior of the Phase Paths Near Infinity
§11. Estimating the Position of Limit Cycles
§12. Approximate Methods of Integration

VI. Fundamentals of the Qualitative Theory of Differential Equations of the Second Order

§ 1. Introduction

§ 2. General Theory of the Behavior of Paths on the Phase Plane. Limit Paths and Their Classification

Limit Points of Half-Paths and Paths
The First Basic Theorem on the Set of Limit Points of a Half-Path
Auxiliary Propositions
Second Basic Theorem on the Set of the Limit Points of a Half-Path
Possible Types of Half-Paths and Their Limit Sets
§ 3. Qualitative Features of the Phase Portrait on the Phase Plane. Singular Paths
Topologically Invariant Properties and Topological Structure of the Phase Portrait
Orbitally Stable and Orbitally Unstable (Singular) Paths
The Possible Types of Singular and Non-Singular Paths
Elementary Cell Regions Filled with Non-Singular Paths having the Same Behavior
Simply Connected and Doubly Connected Cells
§ 4. Coarse Systems
Coarse Dynamic Systems
Coarse Equilibrium States
Simple and Multiple Limit Cycles. Coarse Limit Cycles
Behavior of a Separatrix of Saddle Points in Coarse Systems
Necessary and Sufficient Conditions of Coarseness
Classification of the Paths Possible in Coarse Systems
Types of Cells Possible in Coarse Systems
§ 5. Effect of a Parameter Variation on the Phase Portrait
Branch Value of a Parameter
The Simplest Branchings at Equilibrium States
Limit Cycles Emerging from Multiple Limit Cycles
Limit Cycles Emerging from a Multiple Focus
Physical Example
Limit Cycles Emerging from a Separatrix Joining Two…