In this post, we will see the book *Applied Methods in the Theory of Nonlinear Oscillations *by V. M. Starzhinskii.

About the book:

The book is aimed at engineers with a strong mathematical background, scientists working in mechanics and applied mathematics, and undergraduate and postgraduate students of Applied Physics and Physics and Mathematics departments. The book is based on a course of lectures presented by the author to engineering students at the Mechanics and Mathematics Department of Moscow University in 1956-1976.

The book has two parts

Part One of the book is devoted to the combination of the Lyapunov, Poincare, and averaging methods as applied to the analysis of oscillations in Lyapunov and nearly Lyapunov systems.

The second part of the book is also based on the results achieved in one of the classical methods developed in the years spanning the late 19th and early 20th centuries, the theory of normal forms (Poincare, Lyapunov, Dulac, Siegel, Moser, Arnold, Pliss, and others).

The book requires considerable mathematical background and is not an easy read for those who are not thorough with quite advanced and topical stuff regarding solving equations.

The book was translated from the Russian by V. I. Kisin and was first published by Mir Publishers in 1980.

Original upload was in djvu form (with OCR and Bookmarked), we converted to PDF, added bookmarks and cover.

The Internet Archive link.

Contents

PART ONE

OSCILLATIONS IN LYAPUNOV SYSTEMS

Chapter I. Introduction (13)

§ 1. Transformation of Lyapunov Systems (13)

1.1. General case (13).

1.2. Systems of second-order equations (16).

§ 2. On the Poincare Method of Finding Periodic Solutions of Non-autonomous

Quasilinear Systems (19)

2.1. Differential equations of the generating solution and first corrections (19).

2.2. Non-resonant case (20).

2.3. Resonant case (22).

2.4. Variational equations for periodic unperturbed motion (24).

2.5. Case of distinct multipliers of unperturbed system of variational equations (25).

2.6. Case of multiple multipliers (27).

2.7. Examples (28).

§ 3. Forced Vibrations of Centrifuges Used for Spinning (33)

3.1. Statement of the problem and equations of motion (33).

3.2. Determination of a periodic solution (35).

3.3. Stability analysis (37)

Chapter II. Oscillatory Chains (40)

§ 1. Completely Elastic Free Oscillatory Chains (40)

1.1. Definition of an oscillatory chain (40).

1.2. Determination of equilibrium positions (43).

1.3. Asymptotic stability in the large of the lower equilibrium position for distinct resistance forces (46).

1.4. Variational equations for Vertical oscillations of the system (47).

1.5. Conservative case (49).

1.6. Stability of vertical vibrations of a spring-loaded pendulum (50).

§ 2. Partly Elastic Free Oscillatory Chains (55)

2.1. Statement of the problem (55).

2.2. Kinetic and potential energies (57).

2.3. Example (59).

2.4. Pendulum subject to elastic free suspension (62).

2.5. Pendulum subject to elastic guided suspension (65).

Chapter III. Application of the Methods of Small Parameter to Oscillations in

Lyapunov Systems (67)

§ 1. Loss of Stability of Vertical Vibrations of a Spring-Loaded Pendulum (67)

1.1. Step 1 (68).

1.2. Step 2 (69).

1.3. Step 3 (72).

§ 2. On Coupling of Radial and Vertical Oscillations of Particles in Cyclic

Accelerators (75)

2.1. Step 1 (75).

2.2. Step 2 (77).

2.3. Step 3 (78).

§ 3. Loss of Stability of Vertical Oscillations of a Pendulum Subject to Elastic Guided suspension (79)

3.1. Determination of nontrivial periodic modes (Step 2) (79).

3.2. Transient process (Step 3) (80).

§ 4. Periodic Modes of a Pendulum Subject to Elastic Free Suspension (82)

4.1. Transformation of equations of motion (82).

4.2. Periodic solution (83).

Chapter IV. Oscillations in Modified Lyapunov Systems (84)

§ 1. Lyapunov Systems with Damping (84)

1.1. Transformation of Equations of motion (84).

1.2. Complete system of variational Equations in the Poincare parameter and its solution (86).

1.3. Vibration in mechanical systems with one degree of freedom and different types of nonlinearity (89).

1.4. The Duffing equation with linear damping (92).

1.5. Spring-loaded pendulum with linear damping (95).

§ 2. On Lyapunov Type Systems (!)8)

2.1. Statement of the problem (98).

2.2. Transformation of Lyapunov systems (100).

PART TWO

APPLICATION OF THE THEOHY OF NORMAL FORMS TO OSCILLATION PROBLEMS

Chapter V. Elements of the Theory of Normal Forms of Real Autonomous Systems of Ordinary Differential Equations (103)

§ 1. Introductory Information (103)

1.1. Statement of the problem (103).

1.2. The fundamental Brjuno theorem (144).

1.3. The Poincare theorem (106).

§ 2. Additional Information (107)

2.1. Some properties of normalizing transformations (107).

2.2. Classification of normal forms; integrable normal forms (107).

2.3. Concept of power transformations (109).

2.4. The Brjuno theorem on convergence and divergence of normalizing transformations ( 111).

§ 3. Practical Calculation of Coefficients of Normalizing Transformation and Normal Form. ( 112)

3.1. Fundamental identities (112).

3.2. Computational alternative (114).

3.3. Fundamental identities in general form and their transformation (116).

3.4. Computational alternative in general case (120).

3.5. Remark: on the transition from symmetrized coefficients to ordinary <Jill’S (122).

3.6. Formulas for coefficients of fourth-power Variables (123).

3.7. Case of composite elementary divisors of the matrix of the linear part (123).

Chapter VI. Normal Forms of Arbitrary-Order Systems in the Cast of Asymptotic Stability in Linear Approximation ( **128)**

§ 1. Damped Oscillatory Systems (128)

1.1. Reduction to diagonal form (128).

1.2. Calculation of coefficients of normalising transformation (129).

1.3. General solution of the initial system (general solution of the Cauchy problem) (130).

§ 2. Examples (132)

2.1. A system with one degree of freedom (132).

2.2. Oscillations of a spring suspended mass with linear damping (133).

Chapter VII. Normal Forms of Third-Order Systems (136)

§ 1. Case of Two Pure Imaginary Eigenvalues of the Matrix of the Linear Part (136)

1.1. Reduction to normal form (136).

1.2. Calculation of coefficients of normalizing transformation and normal form ( 138).

1.3. Application of power transformation (140).

1.4. Free oscillations of an electric servodrive (142).

§ 2. Case of Neutral Linear Approximation (146)

2.1. Normal form (146).

2.2. Calculation of coefficients of normalizing transformation and normal form (148).

2.3. Remark on convergence (150).

2.4. Conclusions on stability (15U).

2.5. Integration of normal form in Quadratic approximation (152).

2.6. Example (155).

§ 3. Case of a Zero Eigenvalue of the Matrix of the Linear Part (156)

3.1. Normal form and normalizing transformation (156).

3.2. Integration of normal form (158).

3.3. Remark on convergence (159).

3.4. Free oscillations in a tracking system with a TV sensor (159).

Chapter VIII. Normal Forms of Fourth- and Six-Order Systems in Neutral** **Linear Approximation ( 165)

§ 1. Fourth-Order Systems (165)

1.1. Remark on coefficients of systems of diagonal form (16:i).

1.2. Reduction to normal form (166).

1.3. Calculation of coefficients of normalizing transformation and normal forms (168).

1.4. The Molchanov criterion of oscillation stability (170).

1.5. The Bibikov-Pliss criterion (173).

§ 2. The Ishlinskii Problem (173)

2.1. Reduction of equations of mot ion to tho Lyapunov form (173).

2.2. Transformation of systems similar to Lyapunov (176).

2.3. Determination of periodic solutions (178).

2.4. Reduction of equations of motion to diagonal form and transformation to normal form (180).

2.5. General solution of the Cauchy problem (182).

2.6. Preliminary conclusions on stability (184).

2.7. Construction of tho Lyapunov function (185).

§ 3. The Trajectory Described by the Centre of a Shaft’s Cross Section in One Revolution (186)

3.1. Statement of tho problem and equations of motion (186).

3.2. Reduction to diagonal form (190).

3.3. Reduction to normal form (193).

3.4. General solution of the Cauchy problem (194).

§ 4. Sixth-Order Systems** **(196)

4.1. Solutions of the resonant equation (197).

4.2. Normal forms (200).

4.3. Calculation of coefficients of normalizing transformation and normal forms (201).

4.4. Stability in the third approximation. The Molchanov** **criterion (205).

Chapter IX. Oscillations of a Heavy Solid Body with a Fixed Point About the Lower Equilibrium Position (208)

§ 1. Case of Centroid Located in a Principal Plane of the Ellipsoid of Inertia with respect** **to a Fixed Point (208)

1.1. Reduction to diagonal form (208).

1.2. Reduction to the Lyapunov form (211).

1.3. Resonances (212).

1.4. Simplest motions (213).

1.5. Transformation of equations of diagonal form (214).

1.6. Possible generalizations (215).

1.7. Situation similar to the Kovalevskaya case (216).

1.8. Application of the method of successive approximations (218).

1.9. Remarks on the determination of tho position of a solid body with a fixed point (219).

§ 2. The General Case (219)

2.1. Base reference frame (220).

2.2. Special reference frame (222).

2.3. Equations of motion of a heavy solid body in the special reference frame (223).

2.4. Reduction to the Lyapunov form (226).

2.5. Resonances (228).

2.6. Application of the method of successive approximations (229).

Brief Bibliographical Notes (232)

References (236)

Subject Index (262)