This is a condensed and edited translation of the first comprehen­sive treatment of non-linear oscilla­tions ever to be published. The original work, by two leading mem­ bers of the Soviet Institute of Os­cillations, was published in Russian in 1937, and did not become gen­erally known among English-speaking engineers and mathematicians because of the language difficulty. Professor Lefschetz has now edited an English translation of this im­portant work.Most studies of oscillatory phe­nomena have made idealizing as­sumptions allowing them to be treated linearly, but of course truly linear oscillations rarely if ever oc­cur in nature; and thus the study of non-linear oscillations assumes great practical importance. Since the field is not well known, numer­ ous examples are given in this book, and a special Index of Physical Ex­amples is provided. Practical ap­plications are emphasized, and all physical examples are preserved from the original Russian.Theory of Oscillations begins with a treatment of linear systems, giving the newcomer to the field fa­miliar ground from which to work, but it rapidly progresses to various types of non-linear systems using both mechanical and electrical ex­amples. The book will be useful to electrical, aeronautical, and me­chanical engineers, to mathemati­cians and physicists, and to any scientists working with oscillatory phenomena.I. Linear Systems 5§1. Linear systems without friction.§2. The phase plane. Application to harmonic oscillators. §3. Stability of equilibrium.§4. Linear oscillator without friction.§5. Degenerate linear systems.§6. Linear systems with “negative friction.”§7. Linear system with repulsive force.II. Non-Linear Conservative Systems 61§1. Introduction.§2. The simplest conservative system.§3. The phase plane in the neighborhood of the singular points.§4. Discussion of the motion in the whole phase plane.§5. Dependence of the behavior of a conservative system upon certain parameters.§6. Equations of motion.§7. Periodic motions in conservative systems.III. Non-Conservative Systems .102§1. Introduction.§2. Dissipative systems.§3. Constant friction.§4. Vacuum tubes with discontinuous characteristic and their self­ oscillations.§5. Theory of the clock.§6. Self-oscillating systems.§7. Preliminary investigation of approximately sinusoidal self-oscillations.IV. Dynamical Systems Described by a Single Differen­tial Equation of the First Order  138§1. Introduction.§2. Existence and uniqueness theory.§3. Dependence of the solution upon the initial conditions.§4. Variation of the qualitative character of the curves in the (t,x) plane with the form of the function f(x). §5. Motion on the phase line.§6. Stability of states of equilibrium.§7. Dependence of a motion upon a parameter.§8. Relaxation oscillations. Mechanical example.§9. Relaxation oscillations. Circuit with a neon tube.§10. Relaxation oscillations. Circuit containing a vacuum tube.§11. Voltaic arc in parallel with capacitance.V. Dynamical Systems Described by Two Differential Equations of the First Order .§1. Introduction.§2. Linear systems.§3. An example; the universal circuit.§4. States of equilibrium and their stability. §5. Application to a circuit with voltaic arc. §6. Periodic motions and their stability.§7. The limiting configurations of the paths. §8. The index in the sense of Poincaré.VI. Dynamical Systems Described by Two Differential Equations of the First Order (continued). . .§1. Effect of the variation of a parameter upon the phase portrait. §2. Appearance of a limit-cycle around a focus.§3. Systems without limit-cycles.§4. Behavior of the paths at infinity.§5. Position of the limit-cycles.§6. Approximate methods of integration.VII. Discontinuous Oscillations. and Stability. 182§1. Systems with one degree of freedom described by two equations of the first order.§2. Systems with two degrees of freedom described by two equations of the first order.§3. Small parasitic parameters and stability.VIII. Systems with Cylindrical Phase Surface. 287§1. Cylindrical phase space.§2. A conservative system.§3. A non-conservative system.§4. Other systems which give rise to a cylindrical phase surface.IX. Quantitative Investigation of Non-Linear Systems. 302§1. The van der Pol method of approximation.§2. Applications of the van der Pol method.§3. The method of Poincaré (Method of Perturbations).§4. Applications of the Poincaré method.§5. The vacuum tube with a Broken line characteristic.§6. Influence of the grid current on the performance of vacuum tubes. §7. Branch points for a self-oscillatory system close to a linear con­servative system.§8. Application of the theory of branch points to the performance ofvacuum tubes.Appendix A. Structural Stability. 337 Appendix B. Justification of the van der Pol Approximations.Appendix C. T he van der Pol Values of the ParameterSupplementary Reading List 349List Index of Physical Examples 351 Index of Mathematical Terms. 353 Equation for Arbitrary. 353