Sansone / Conti / Sneddon Non-Linear Differential Equations

1. Auflage 2014
ISBN: 978-1-4831-8505-7
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band Volume 67, 550 Seiten, Web PDF

Reihe: International Series in Pure and Applied Mathematics

Non-Linear Differential Equations covers the general theorems, principles, solutions, and applications of non-linear differential equations. This book is divided into nine chapters. The first chapters contain detailed analysis of the phase portrait of two-dimensional autonomous systems. The succeeding chapters deal with the qualitative methods for the discovery of periodic solutions in periodic systems. The remaining chapters describe a synthetical approach to the study of asymptotic properties, especially stability properties, of the solutions of general n-dimensional systems. This book will be of great value to mathematicians, researchers, and students.

Autoren/Hrsg.

Sansone, G.

Conti, R.

Sneddon, I. N.

Stark, M.

Ulam, S.

Inhaltsverzeichnis

1;Front Cover;1
2;Non-Linear Differential Equations;4
3;Copyright Page;5
4;Table of Contents;6
5;PREFACE;14
6;CHAPTER I. GENERAL THEOREMS ABOUT SOLUTIONS OF DIFFERENTIAL SYSTEMS;16
6.1;§ 1. Integral Curves;16
6.2;§ 2. Lipschitzian and Carathéodory Systems;26
6.3;§ 3. The Solution f(t, t0, x0) of the System (1.1.1);31
6.4;§ 4. Periodic Solutions;40
6.5;§ 5. Autonomous Systems;43
6.6;COMPLEMENTS;47
6.7;BIBLIOGRAPHY;49
7;CHAPTER II. PARTICULAR PLANE AUTONOMOUS SYSTEMS;52
7.1;§ 1. The Linear Case;52
7.2;§ 2. Homogeneous Systems;64
7.3;§ 3. The Analytic Case;81
7.4;§ 4. The Problem of the Center;99
7.5;§ 5. Singular Points at Infinity;113
7.6;COMPLEMENTS;122
7.7;BIBLIOGRAPHY;125
8;CHAPTER III. THE SINGULARITIES OF BRIOT–BOUQUET;126
8.1;§ 1. Theorem of Briot—Bouquet for the Analytic Case;126
8.2;§ 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;133
8.3;§ 3. Equation of Briot—Bouquet in the Nodal Case in the Real Domain. Theorems of A. Wintner;142
8.4;COMPLEMENTS;149
8.5;BIBLIOGRAPHY;154
9;CHAPTER IV. PLANE AUTONOMOUS SYSTEMS;156
9.1;§ 1. Limiting Sets;156
9.2;§ 2. Plane Cycles;174
9.3;§ 3. Isolated Singular Points;193
9.4;§ 4. The Index;203
9.5;§ 5. The Cylinder as Phase Space;212
9.6;§ 6. The Torus as Phase Space;215
9.7;§ 7. A Short Account on Dynamical;219
9.8;BIBLIOGRAPHY;221
10;CHAPTER V. AUTONOMOUS PLANE SYSTEMS WITH PERTURBATIONS;225
10.1;§ 1. Homogeneous Perturbed Systems;225
10.2;§ 2. Isolated Singular Points of System of Class C1. Elementary Points;241
10.3;§ 3. An Asymptotic Study of a Node with two Tangents and a Saddle Point, of H. Weyl;250
10.4;§ 4. Isolated Singular Points of Systems of Class C1. Non-Elementary Points;271
10.5;§ 5. Structurally Stable Systems. Systems With a Parameter;284
10.6;BIBLIOGRAPHY;289
11;CHAPTER VI. ON SOME AUTONOMOUS SYSTEMS WITH ONE DEGREE OF FREEDOM;292
11.1;§ 1. Trajectories of the Equation of Linear Motion of a Point Under Viscous Resistance;292
11.2;§ 2. The Equation;293
11.3;§ 3. Equations of van der Pol and Liénard of the Oscillations of Relaxation;318
11.4;§ 4. Periodic Solutions of the Generalized Equation of Liénard;336
11.5;§ 5. Periodic Solutions of the Equation x + f(x) x + g(x) = 0 without the Hypothesis x g(x) > 0 for | x | > 0;349
11.6;§ 6. The Equation of Damped Vibrations: Ax+f(x)x+Cx = 0;356
11.7;§ 7. On an Equation of Dynamics and Aerodynamics of Wires;359
11.8;COMPLEMENTS;365
11.9;BIBLIOGRAPHY;368
12;CHAPTER VII. NON-AUTONOMOUS SYSTEMS WITH ONE DEGREE OF FREEDOM;374
12.1;§ 1. The Problem of Forced Oscillations. Linear Case;374
12.2;§ 2. The Fixed Point Theorem of L. E. J. Brouwer and the Theorems of M. L. Cartwright, J· E. Littlewood and J. L. Massera;378
12.3;§ 3. Theorems of T. Yoshizawa;386
12.4;§ 4. Harmonic Solutions out of Phase of the Equation x = F(x, cos . t). Theorem of F. John;404
12.5;§ 5. The Equation x + f(x) x + g(x) = p(t);414
12.6;§ 6. The Equation x + F(x) + x = p(t);423
12.7;§ 7. Theorems of G. E. H. Reuter on the Equations;428
12.8;§ 8. The Equation x + f(x, x) x + g(x) = p(t);433
12.9;§ 9. Non-Linear Systems with Subharmonic Solutions;438
12.10;§ 10. General Remarks Concerning Periodic Solutions;443
12.11;COMPLEMENTS;444
12.12;BIBLIOGRAPHY;450
13;CHAPTER VIII. LINEAR SYSTEMS;456
13.1;§ 1. The Adjoint System. The Inequalities of T. Wazewski;456
13.2;§ 2. Linear Autonomous Systems with Constant Coefficients;459
13.3;§ 3. Linear Periodic Systems;466
13.4;§ 4. Reducible Systems;470
13.5;§ 5. Type Numbers of a Function. Relation of t- Similitude;472
13.6;§ 6. Regular Systems;477
13.7;§ 7. Periodic Solutions;481
13.8;COMPLEMENTS;487
13.9;BIBLIOGRAPHY;487
14;CHAPTER IX. STABILITY;491
14.1;§ 1. The method of V functions;491
14.2;§ 2. Stability of Linear Systems;503
14.3;§ 3. Stability in the First Approximation;517
14.4;§ 4. Asymptotic Equivalence;529
14.5;COMPLEMENTS AND PROBLEMS;535
14.6;BIBLIOGRAPHY;538
15;INDEX;546
16;OTHER TITLES IN THE SERIES IN PURE AND APPLIED MATHEMATICS;550