# Non-Linear Differential Equations

### International Series of Monographs in Pure and Applied Mathematics

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International Series of Monographs in Pure and Applied Mathematics, Volume 67: Non-Linear Differential Equations, Revised Edition focuses on the analysis of the phase portrait of two-dimensional autonomous systems; qualitative methods used in finding periodic solutions in periodic systems; and study of asymptotic properties. The book first discusses general theorems about solutions of differential systems. Periodic solutions, autonomous systems, and integral curves are explained. The text explains the singularities of Briot-Bouquet theory. The selection takes a look at plane autonomous systems. Topics include limiting sets, plane cycles, isolated singular points, index, and the torus as phase space. The text also examines autonomous plane systems with perturbations and autonomous and non-autonomous systems with one degree of freedom. The book also tackles linear systems. Reducible systems, periodic solutions, and linear periodic systems are considered. The book is a vital source of information for readers interested in applied mathematics.

## Table of Contents

Contents

Preface

Chapter I. General Theorems About Solutions of Differential Systems

1. Integral Curves 1

1. Integral Curves. Extreme Time in the Future t+

2. Conditions for t+—B. The Case N = 1

3. Conditions for t+= 6. General Case

4. Boundedness of the Integral Curves

5. Integral Curves in the Sense of Caratheodory

2. Lipschitzian and Caratheodory Systems

1. Gronwall's Lemma (Generalized)

2. Lipschitzian Systems. Evaluation of x(t) — y(t)\ for Two Arcs of Integral

Curves

3. Uniqueness Theorem. Continuous Dependence on the Initial Point P0 and on f

4. Caratheodory Systems

3. The Solution Φ(t,t0,x°) of the System (1.1.1)

1. The Function ϕ(t,t0,x°). Cases of Uniqueness

2. Continuity of (ϕ(t,t0,x°)

3. Stability

4. The Function Φ(t, t09 x°) for Linear Systems

5. Differentiability of Φ (t, t0, x°)

6. Systems with Parameters

4. Periodic Solutions

1. Periodic Integral Curves. Periodic Orbits

2. Exceptional Periodic Solutions

5. Autonomous Systems

1. Autonomous Systems. Properties of their Integral Curves

2. Trajectories. Phase Space

3. Singular Points. Cycles. Open Trajectories Complements

Bibliography

Chapter II. Particular Plane Autonomous Systems

1. The Linear Case

1. Singular Points

2. Canonical Forms of Isolated Singular Points of Linear Systems

3. Affine Transformations of the Phase Plane

4. Classification of the Types of Singular Points

2. Homogeneous Systems

1. Homogeneous Systems

2. Invariant Rays. Stellar Node

3. The Center and the Focus

4. Isolated Invariant Rays. Normal Angles

5. Behavior of Trajectories In A Normal Angle

6. Examples

3. The Analytic Case

1. Introductory Remarks

2. Examples

3. The Functions Z(x, y), N(x, y)

4. A Lemma

5. Trajectories Tending To 0. Focus

6. The Equation Ν(Θ) = 0. Dicritical Points

7. Study of Z(x, y). Case of the Fixed Sign for Z(x, y)

8. Classification of Z-Sectors

4. The Problem of the Center

1. The Problem of the Center

2. The Problem of the Center for Ν(Θ)≠θ

3. The Case m = 1. Method of Poincare

4. The Case m = 1. Theorem of Poincare for the Center. The Proof of E. Picard-J. Chazy

5. The Case m = 1. Evaluation of the Period

6. Sufficient Condition of Poincare for the Center. Applications To Delaunay's Equations of Lunar Motion

7. Bibliographic Notes on the Problem of the Center

5. Singular Points at Infinity

1. Poincare's Sphere. Singular Points at Infinity

2. Examples

3. Singular Points at Infinity for Homogeneous Systems

Complements

Bibliography

Chapter III. The Singularities of Briot-Bouquet

1. Theorem of Briot-Bouquet for the Analytic Case

1. Introductory Remarks

2. The Equation of Briot—Bouquet in the Case Where P Is Not A Positive Integer. Study of Holomorphic Solutions

3. The Case of A Positive Integer P. Existence of Holomorphic Solutions

4. Solutions of the Equation for the Case p = 0

2. Reduction of Differential Equations with An Isolated Singular Point To A Typical Form in the Analytic Case. The Theorem of I. Bendixson on the Behavior of the Trajectories of the Reduced Equations of the Second Type

1. Reduced Forms of the First and Second Type

2. Results of I. Bendixson on the Behaviour of the Trajectories of the Reduced Equations of the Second Type

3. Equation of Briot-Bouquet in the Nodal Case in the Real Domain. Theorems of A. Wintner

1. Lemma of A. Wintner

2. First Theorem of A. Wintner

3. Second Theorem of A. Wintner

Complements

Bibliogr4phy

Chapter IV. Plane Autonomous Systems

1. Limiting Sets

1. Limiting Sets Α(λ), Ω(λ) of A Trajectory λ. General Properties

2. Classification of Trajectories

3. Regular Points and Trajectories

4. Closed (Plane) Trajectories. Stability Properties of Plane Cycles

5. Regular (Plane) Limiting Trajectories

6. Structure of Bounded Limiting Sets Ω(λ)

7. Limiting Sets Consisting of A Single (Singular) Point

8. Structure of Unbounded Sets Ω(λ)

2. Plane Cycles

1. Limit Cycles

2. Classification of Limit Cycles. Orbital Stability

3. Examples

4. Bendixson's Theorem

5. Systems of Class C1. Characteristic Exponent of A Cycle

6. Cycles of Analytic Systems

7. Limit Cycles of A System with Polynomial Right Sides

8. Regions with No Plane Cycles

9. Periodic Solutions of A Plane Autonomous System. Existence of Limit Cycles

10. Uniqueness of (Limit) Cycles

3. Isolated Singular Points

1. Classification of Isolated Singular Points. Points of the First Kind (Centerfocus) Center

2. The Neighborhood of A Point of the Second Kind

3. The Focus

4. Exceptional Directions

5. Normal Sectors

4. The Index

1. Kronecker's Index

2. Index of A Point

3. Evaluation of the Index for Particular Singular Points

4. Index on the Sphere and on A Surface of Genus p

5. The Cylinder As Phase Space

1. The Cylinder As Phase Space

2. An Example

6. The Torus As Phase Space

1. The Torus As Phase Space

2. Examples

3. Systems with No Singular Points on the Torus

4. Other Results

7. A Short Account on Dynamical Systems

Bibliography

Chapter V. Autonomous Plane Systems with Perturbations

1. Homogeneous Perturbed Systems

1. The General Problem

2. The Case Ν(Θ)≠0

3. Trajectories Tending to 0. Exceptional Directions

4. In Variance of Normal Sectors. Normal Sectors of the First Type

5. Normal Sectors of the Second Type. First Decision Problem

6. Normal Sectors of the Third Type. Second Decision Problem

7. The Case Ν(Θ) Identically Zero

8. Some Remarks

2. Isolated Singular Points of Systems of Class C1. Elementary Points

1. Introductory Remarks

2. Foci and Weak Foci

3. Attractors. Stellar Node

4. Node with One Tangent

5. Node with Two Tangents

6. Saddle Points

7. Remarks

3. An Asymptotic Study of A Node with Two Tangents and A Saddle Point of H. Weyl

1. Statement of the Problem. Notations 235

2. The Nodal Case (0 < I < k). The First Theorem of H. Weyl

3. Bounds for \elty(t) — 6|, \x(t) — x0e-kt

4. The Case %(r) = Crδ Bounds for \Y(T) — be-lt\, \x(t)\

5. The Case k ≥ l, k > 0. Second Theorem of H. Weyl

6. Parametrized Systems

7. The Case of the Node with Two Tangents. Third Theorem of H. Weyl

8. The Case of A Saddle Point (l < 0 < k). Fourth Theorem of H. Weyl

4. Isolated Singular Points of Systems of Class C1. Non-Elementary Points

1. Introductory Remarks

2. First Theorem of K. A. Keil for the Systems x = x + f(x, y), y = g(x, y)

3. Lemmas on the Isoclines

4. K. A. Keil's Second and Third Theorems for the System x = x + f(x, y), y = g(x, y)

5. Further Results of K. A. Keil for the System x = y + f(x, y), y = g(x, y)

6. Bibliographical Notes To Sec. 1.2.4

5. Structurally Stable Systems. Systems with A Parameter

1. Structurally Stable Systems

2. Structurally Unstable Systems. Generation of Limit Cycles

3. Variation of the Cycles of Systems with A Parameter

Bibliography

Chapter VI. on Some Autonomous Systems with One Degree of Freedom

1. Trajectories of the Equation of Linear Motion of A Point Under Viscous Resistance

1. Trajectories of the Equation of Linear Motion of A Point Under Viscous Resistance

2. The Equation Ӫ + αθ + Sin θ — β = 0, α ≥ θ, β ≥ 0

1. Introductory Remarks

2. The Case β > 1. Existence of A Periodic Solution Z= z(θ ) of (6.2.3)

3. The Case 0 < β < 1. Classification of Singular Points 281

4. The Trajectories for the Limiting Case α — 0

5. The Case 0 < α , 0 < β < 1. Periodic Solutions of (6.2.3) and Critical Values α(θ0)

6. The Case θo = Π/2, (β = 1)

7. The Trajectories for 0 < θo < Π/2, (0 < α; 0 < β < 1)

8. Inequalities for the Critical Value α(θ0)

9. Procedure of M. Urabe for Calculating α(θ0)

3. Equations of Van Der Pol and Lienard of the Oscillations of Relaxation

1. Preliminary Results 303

2. Existence of Periodic Solutions of Lienard's Equation

3. Sufficient Conditions for the Uniqueness of Periodic Solutions of Lienard's Equation

4. Case of Non-Uniqueness of Periodic Solutions of A Lienard Equation

5. Theorem of Existence of Periodic Solutions of Lienard's Equation in the Case Where f(x) Has Ordinary Discontinuities

6. Theorem of Comparison for Lienard's Equation

7. Calculation of the Period

8. Van Der Pol's Equation. Behavior of Trajectories at Infinity

9. Behavior of the Limit Cycle of Van Der Pol's Equation When The Parameter Tends To Infinity. Theorem of D. A. Flanders and J. J. Stoker

10. Asymptotic Evaluation of the Period and Amplitude of Periodic Solutions Of Van Der Pol's Equation for Large Values of the Parameter

11. Inequalities for Limit Cycles Due To R. Gomory and D. E. Richmond

4. Periodic Solutions of the Generalized Equation of Lienard

1. First Theorem of A. F. Filippov

2. Second Theorem of A. F. Filippov

3. Theorem of Uniqueness

4. Study of the Equation x + f(x, x) x + g(x) = 0

5. Periodic Solutions of the Equation, x -f- t(x) x + g(x) = 0 Without the Hypothesis x g(x) > 0 for \x\> 0

1. Introductory Remarks

2. Singular Points

3. Cycles. Their Properties

4. A Case of Non-Existence of Periodic Solutions

5. Existence of Cycles

6. A Criterion for the Uniqueness of A Cycle

6. The Equation of Damped Vibrations: Äx + f(x)x + Cx = 0

1. Introductory Remarks

2. Conditions for the Origin To Be A Stable Point

3. A Theorem of G. Malgarini

7. on An Equation of Dynamics and Aerodynamics of Wires

1. Singular Points

2. The Field Relative To System (6.7.2)

3. Existence of Periodic Solutions for Sufficiently Small Values of the Parameter p

Complements

Bibliography

Chapter VII. Non-Autonomous Systems with One Degree of Freedom

1. The Problem of Forced Oscillations. Linear Case

1. Forced Oscillations in the Harmonic Case

2. Forced Oscillations in the Non-Harmonic Case

3. The Problem of Forced Oscillations

2. The Fixed Point Theorem of L. E. J. Brouwer and the Theorems of M. L. Cartwright, J. E. Littlewood and J. L. Massera

1. The Fixed Point Theorem of L. E. J. Brouwer 3

2. Brouwer's Theorem in the Proofs of Existence of Periodic Solutions

3. Theorem of M. L. Cartwright-J. E. Littlewood

4. Theorem of J. L. Massera

3. Theorems of T. Yoshizawa

1. Criterion of Ultimate Boundedness

2. Theorem of Existence of Periodic Solutions

3. Stability of Solutions

4. A Theorem on Uniqueness and Stability of Periodic Solutions

5. A Criterion for Boundedness of Individual Solutions

6. Derivation of A Criterion for Existence of Periodic Solutions From The Theorem of J. L. Massera. A Theorem of S. Mizohata and M. Yamaguti

4. Harmonic Solutions Out of Phase of the Equation x = F(x, Cos ωt). Theorem of F. John

1. The Interval (—∞, -F- ∞) As Domain of Existence of Solutions

2. Theorem of F. John on the Existence of Harmonic Solutions Out of Phase

5. The Equation x + f(x)x + g(x) = p(t)

1. Results of S. Lefschetz, N. Levinson, M. L. Cartwright, and J. E. Littlewood

2. An Existence Theorem and A Theorem on Asymptotic Stability of N. Levinson

3. The Equation x + g(x) = p(t) for p(t) Even. Theorem of G. R. Morris

4. Odd Periodic Harmonic Solutions. Theorem of W. S. Loud on the Duffing Equation with Forcing Term

5. Inequalities of D. Graffi for the Periodic Solutions of the Equation x + f(x) x + λ2x = F Sin ωt, λ > 0, ω > 0, F > 0

6. The Equation x + F(x) + x = p(t)

1. Criteria of R. Caccioppoli, A. Ghizzetti and A. Ascari for Existence, Uniqueness and Stability of A Periodic Solution

2. on A Differential Equation in the Mechanics of Wires. Results of J . Cecconi and F. Stoppelli

7. Theorems of G. E. H. Reuter on the Equations x + k f(x) x + g(x) = k p(t)> x + k F(x) + g(x) = k p(t)

1. The Equation x + kf(x) x + g(x) = k p(t)

2. The Equation x + kf(x) x + g(x) = k p(t)

8. The Equation x + f(x, x) x -f- g(x) = p(t)

1. Criterion on Boundedness in the Future of H. A. Antosiewicz

2. Criteria of N. Levinson and C. E. Langenhop for Existence of A Periodic Solution

9. Non-Linear Systems with Subharmonic Solutions

1. Subharmonic Solutions

2. Class D Systems

3. Classifications of Fixed Points Relative To The Transformations of Systems of Class D

4. Theorems of N. Levinson and J . L. Massera on the Number of Subharmonic Solutions

10. General Remarks Concerning Periodic Solutions

1. Autonomous Systems

2. Periodic N On-Autonomous Systems

Complements

Bibliography

Chapter VIII. Linear Systems

1. The Adjoint System. The Inequalities of T. Wazewski

1. The Adjoint System

2. Wazewski's Inequality

2. Linear Autonomous Systems with Constant Coefficients

1. The Principal Fundamental Matrix

2. Form of Solutions of the Homogeneous System. Characteristic Exponents. Type Numbers

3. Singular Points in the Real Case

4. The Case N = 3 (Real Case)

3. Linear Periodic Systems

1. The Principal Fundamental Matrix. Theorems of Floquet and Lyapunov

2. Characteristic Exponents. Type Numbers

4. Reducible Systems

1. Reducible Systems. Characteristic Exponents and Type Numbers

5. Type Numbers of A Function. Relation of T-Similitude

1. Type Number of A Function

2. The Relation of I-Similitude (Or Kinematic Similarity)

3. Type Number of A Non-Vanishing Solution

4. Normal Systems of Solutions. The Number Smin

5. Inequality for Smin. Constant of Irregularity

6. Regular Systems

1. Regular Systems

2. Theorems of Perron

3. Triangular Matrices

7. Periodic Solutions

1. Linear Homogeneous Systems

2. Linear Non-Homogeneous Systems

3. The Existence of Harmonic Solutions of Quasi-Linear Periodic Systems

Complements

Bibliography

Chapter IX. Stability

1. The Method of V Functions

1. Introductory Remarks

2. The V Functions

3. A Lemma of T. Wazewski

4. Sufficient Conditions for Stability

5. Necessary Conditions for Stability. The Inverse Problem

6. Asymptotic Stability

7. Asymptotic Stability in the Large

8. Other Kinds of Stability

9. Instability

10. V Functions for Boundedness

2. Stability of Linear Systems

1. Stable and Unstable Linear Systems

2. Uniformly Stable Linear Systems

3. Uniform Stability and T ∞-Similarity

4. Criteria of Uniform Stability

5. Linear Systems Reducible To Zero and Restrictive Stability

6. Asymptotic Stability of Linear Systems

7. V Functions for Linear Systems with Constant Coefficients

3. Stability in the First Approximation

1. Introductory Remarks

2. Stability By A Linear First Approximation

3. Some Generalizations and Remarks. The L(v, n) Property

4. Asymptotic Stability. Cases of A Non-Linear First Approximation

5. Analytical Systems. The Critical Cases

6. Orbital (Asymptotic) Stability. The Behavior of Solutions Near Integral Manifolds

4. Asymptotic Equivalence

1. Asymptotic Equivalence

2. The Theorem of H. Weyl

3. Other Results on Asymptotic Equivalence

Complements and Problems

Bibliography

Index

Other Volumes In This Series (Pure and Applied Mathematics)

## Product details

- No. of pages: 550
- Language: English
- Copyright: © Pergamon 1964
- Published: January 1, 1964
- Imprint: Pergamon
- eBook ISBN: 9781483135960

## About the Authors

### G. Sansone

### R. Conti

## About the Editors

### I. N. Sneddon

### M. Stark

### S. Ulam

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