# Theory of Oscillators

### Adiwes International Series in Physics

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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.

## Table of Contents

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

1. The Phase Plane

2. Equation Not Involving Time

3. Singular Points. Center

4. Isoclines

5. State of Equilibrium and Periodic Motion

§ 3. Stability of a State of Equilibrium

§ 4. Linear Oscillator in the Presence of Friction

1. Damped Oscillatory Process

2. Representation of a Damped Oscillatory Process on the Phase Plane

3. Direct Investigation of the Differential Equation

4. Damped Aperiodic Process

5. Representation of an Aperiodic Process on the Phase Plane

§ 5. Oscillator with Small Mass

1. Linear Systems with Half Degree of Freedom

2. Initial Conditions and Their Relations to the Idealization

3. Conditions for a Jump

4. Other Examples

§ 6. Linear Systems with "Negative Friction"

1. Mechanical Example

2. Electrical Example

3. Portrait on the Phase Plane

4. Behavior of the System for a Variation of the Feedback

§ 7. Linear System with Repulsive Force

1. Portrait on the Phase Plane

2. An Electrical System

3. 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

1. Motion of a Point Mass along a Circle which Rotates about a Vertical Axis

2. Motion of a Material Point along a Parabola Rotating about Its Vertical Axis

3. Motion of a Conductor Carrying a Current

§ 6. The Equations of Motion

1. Oscillating Circuit with Iron Core

2. Oscillating Circuit having a Rochelle Salt Capacitor

§ 7. General Properties of Conservative Systems

1. Periodic Motions and Their Stability

2. Single-Valued Analytic Integral and Conservativeness

3. Conservative Systems and Variational Principle

4. Integral Invariant

5. Basic Properties of Conservative Systems

6. 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

1. The Clock with Linear Friction

2. Valve Generator with a Discontinuous ⌡ Characteristic

3. Model of the Clock with Coulomb Friction

§ 5. Theory of the Clock. Model of a "Recoil Escapement" without Impulses

1. Model of Clock with a Balance-Wheel without Natural Period

2. 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 ƒ(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

1. Voltaic Arc in a Circuit with Resistance and Self-Induction

2. Dynatron Circuit with Resistance and Capacitance

3. Valve Relay (Bi-Stable Trigger Circuit)

4. Motion of a Hydroplane

5. Single-Phase Induction Motor

6. Frictional Speed Regulator

§ 6. Periodic Motions

1. Two-Position Temperature Regulator

2. 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

1. Small Oscillations of a Dynatron Generator

2. The "Universal" Circuit

§ 4. States of Equilibrium and Their Stability

1. The Case of Real Roots of the Characteristic Equation

2. 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

1. Sequence Function and Point Transformation

2. Stability of the Fixed Point. Koenigs's Theorem

3. A Condition of Stability of the Limit Cycle

§ 8. Poincare's Indices

§ 9. Systems without Closed Paths

1. Symmetrical Valve Relay (Trigger)

2. Dynamos Working in Parallel on a Common Load

3. Oscillator with Quadratic Terms

4. 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

1. Limit Points of Half-Paths and Paths

2. The First Basic Theorem on the Set of Limit Points of a Half-Path

3. Auxiliary Propositions

4. Second Basic Theorem on the Set of the Limit Points of a Half-Path

5. Possible Types of Half-Paths and Their Limit Sets

§ 3. Qualitative Features of the Phase Portrait on the Phase Plane. Singular Paths

1. Topologically Invariant Properties and Topological Structure of the Phase Portrait

2. Orbitally Stable and Orbitally Unstable (Singular) Paths

3. The Possible Types of Singular and Non-Singular Paths

4. Elementary Cell Regions Filled with Non-Singular Paths having the Same Behavior

5. Simply Connected and Doubly Connected Cells

§ 4. Coarse Systems

1. Coarse Dynamic Systems

2. Coarse Equilibrium States

3. Simple and Multiple Limit Cycles. Coarse Limit Cycles

4. Behavior of a Separatrix of Saddle Points in Coarse Systems

5. Necessary and Sufficient Conditions of Coarseness

6. Classification of the Paths Possible in Coarse Systems

7. Types of Cells Possible in Coarse Systems

§ 5. Effect of a Parameter Variation on the Phase Portrait

1. Branch Value of a Parameter

2. The Simplest Branchings at Equilibrium States

3. Limit Cycles Emerging from Multiple Limit Cycles

4. Limit Cycles Emerging from a Multiple Focus

5. Physical Example

6. Limit Cycles Emerging from a Separatrix Joining Two Saddle-Points, and from a Separatrix of a Saddle-Node Type when this Disappears

VII. Systems with a Cylindrical Phase Surface

§ 1. Cylindrical Phase Surface

§ 2. Pendulum with Constant Torque

§ 3. Pendulum with Constant Torque. The Non-Conservative Case

§ 4. Zhukovskii's Problem of Gliding Flight

VIII. the Method of the Point Transformations in Piece-Wise Linear Systems

§ 1. Introduction

§ 2. A Valve Generator

1. Equation of the Oscillations

2. Point Transformation

3. The Fixed Point and Its Stability

4. Limit Cycle

§ 3. Valve Generator (the Symmetrical Case)

1. The Equations of the Oscillations and Phase Plane

2. Point Transformation

3. Fixed Point and Limit Cycle

§ 4. Valve Generator with a Biassed ⌡ Characteristic

1. The Equation of the Oscillations

2. Point Transformation

3. Fixed Points and Limit Cycles

4. The Case of Small Values of a Andy

§ 5. Valve Generator with a Two-Mesh Rc Circuit

1. The Phase Plane

2. The Correspondence Functions

3. Lamerey's Diagram

4. Discontinuous Oscillations

5. Period of Self-Oscillations for Small Values of µ

§ 6. Two-Position Automatic Pilot for Ships Controller

1. Formulation of the Problem

2. The Phase Plane

3. The Point Transformation

4. Automatic Pilot with Parallel Feedback

5. Other Automatic Controlling Systems

§ 7. Two-Position Automatic Pilot with Delay

1. Ship's Automatic Pilot with "Spatial" Delay

2. Automatic Ship's Pilot with Pure Time Delay

§ 8. Relay Operated Control Systems (with Dead Zone Backlash and Delay)

1. The Equations of Motion of Certain Relay Systems

2. The Phase Surface

3. The Point Transformation for ß<1

4. Lamerey's Diagram

5. Structure of the Phase Portrait

6. The Dynamics of the System with Large Velocity Correction

§ 9. Oscillator with Square-Law Friction

§ 10. Steam-Engine

1. Engine Working with a "Constant" Load and without a Regulator

2. Steam-Engine Working on a "Constant" Load but with a Speed Regulator

3. Engine with a Speed Dependent Load Torque

IX. Non-Linear Systems with Approximately Sinusoidal Oscillation

§ 1. Introduction

§ 2. Van Der Pol's Method

§ 3. Justification of Van Der Pol's Method

1. The Justification of Van Der Pol's Method for Transient Processes

2. Justification of Van Der Pol's Method for Steady-State Oscillations

§ 4. Application of Van Der Pol's Method

1. The Valve Generator with Soft Operating Conditions

2. The Valve Generator Whose Characteristic is Represented by a Polynomial of the Fifth Degree

3. Self-Oscillations in a Valve Generator with a Two Mesh RC Circuit

§ 5. Poincare's Method of Perturbations

1. the Procedure in Poincare's Method

2. Poincare's Method for Almost Linear Systems

§ 6. Application of Poincare's Method

1. A Valve Generator with Soft Self-Excitation

2. The Significance of the Small Parameter µ

§ 7. A Valve Generator with a Segmented Characteristic

1. A Valve Generator with a Discontinuous ⌡ Characteristic

2. A Valve Oscillator with a Segmented Characteristic without Saturation

§ 8. The Effect of Grid Currents on the Performance of a Valve Oscillator

§ 9. The Bifurcation Or Branch Theory for a Self-Oscillating System Close to a Linear Conservative System

§ 10. Application of Branch Theory in the Investigation of the Modes of Operation of a Valve Oscillator

1. Soft Excitation of Oscillations

2. Hard Excitation of Oscillations

X. Discontinuous Oscillations

§ 1. Introduction

§ 2. Small Parameters and Stability of States of Equilibrium

1. Circuit with a Voltaic Arc

2. Self-Excitation of a Multivibrator

§ 3. Small Parasitic Parameters and Discontinuous Oscillations

1. The Mapping of the "Complete" Phase Space by the Paths

2. Condition for Small (Parasitic) Parameters to be Unimportant

3. Discontinuous Oscillations

§ 4. Discontinuous Oscillations in Systems of the Second Order

§ 5. Multivibrator with One RC Circuit

1. Equations of the Oscillations

2. The x,y Phase Plane for µ→+0

§ 6. Mechanical Discontinuous Oscillations

§ 7. Two Electrical Generators of Discontinuous Oscillations

1. Circuit with a Neon Tube

2. Dynatron Generator of Discontinuous Oscillations

§ 8. Fruhhauf's Circuit

1. "Degenerate" Model

2. The Jump Postulate

3. Discontinuous Oscillations in the Circuit

4. Including the Stray Capacitances

§ 9. A Multivibrator with an Inductance in the Anode Circuit

1. The Equations of "Slow" Motions

2. Equations of the Multivibrator with Stray Capacitance Ca

3. Discontinuous Oscillations of the Circuit

§ 10. The "Universal" Circuit

§ 11. The Blocking Oscillator

1. The Equations of the Oscillations

2. Jumps of Voltages and Currents

3. Discontinuous Oscillations

4. Discontinuous Self-Oscillations of the Blocking Oscillator

§12. Symmetrical Multivibrator

1. The Equations of the Oscillations

2. Jumps of the Voltages µ1 and µ2

3. Discontinuous Oscillations of the Multivibrator

§ 13. Symmetrical Multivibrator (with Grid Currents)

1. Equations of the Oscillations

2. Discontinuous Oscillations

3. The Point Transformation Π

4. Lamerey's Diagram

5. Selfoscillations of the Multivibrator for Eg≥0.

X . Comments on More Recent Works

Appendix: Basic Theorems of the Theory of Differential Equations

References

Index

## Product details

- No. of pages: 848
- Language: English
- Copyright: © Pergamon 1966
- Published: January 1, 1966
- Imprint: Pergamon
- eBook ISBN: 9781483194721