A large amount of work has been done on ordinary differ­ ential equations with small parameters multiplying deriv­ atives. This book investigates questions related to the asymptotic calculation of relaxation oscillations, which are periodic solutions formed of sections of both sl- and fast-motion parts of phase trajectories. A detailed discussion of solutions of differential equations involving small parameters is given for regions near singular points. The main results examined were obtained by L. S. Pontryagin and the authors. Other works have also been taken into account: A. A. Dorodnitsyn's investigations of Van der Pol's equation, results obtained by N. A. Zheleztsov and L. V. Rodygin concerning relaxation oscillations in electronic devices, and results due to A. N. Tikhonov and A. B. Vasil'eva concerning differential equations with small parameters multiplying certain derivatives. E. F. Mishchenko N. Kh. Rozov v CONTENTS Chapter I. Dependence of Solutions on Small Parameters. Applications of Relaxation Oscillations 1. Smooth Dependence. Poincare's Theorem . 1 2. Dependence of Solutions on a Parameter, on an Infinite Time Interval 3 3. Equations with Small Parameters 4 Multiplying Derivatives 4. Second-Order Systems. Fast and Slow Motion.

I. Dependence of Solutions on Small Parameters. Applications of Relaxation Oscillations.- 1. Smooth Dependence. Poincaré's Theorem.- 2. Dependence of Solutions on a Parameter, on an Infinite Time Interval.- 3. Equations with Small Parameters Multiplying Derivatives.- 4. Second-Order Systems. Fast and Slow Motion. Relaxation Oscillations.- 5. Systems of Arbitrary Order. Fast and Slow Motion. Relaxation Oscillations.- 6. Solutions of the Degenerate Equation System.- 7. Asymptotic Expansions of Solutions with Respect to a Parameter.- 8. A Sketch of the Principal Results.- II. Second-Order Systems. Asymptotic Calculation of Solutions.- 1. Assumptions and Definitions.- 2. The Zeroth Approximation.- 3. Asymptotic Approximations on Slow-Motion Parts of the Trajectory.- 4. Proof of the Asymptotic Representations of the Slow-Motion Part.- 5. Local Coordinates in the Neighborhood of a Junction Point.- 6. Asymptotic Approximations of the Trajectory on the Initial Part of a Junction.- 7. The Relation between Asymptotic Representations and Actual Trajectories in the Initial Junction Section.- 8. Special Variables for the Junction Section.- 9. A Riccati Equation.- 10. Asymptotic Approximations for the Trajectory in the Neighborhood of a Junction Point.- 11. The Relation between Asymptotic Approximations and Actual Trajectories in the Immediate Vicinity of a Junction Point.- 12. Asymptotic Series for the Coefficients of the Expansion Near a Junction Point.- 13. Regularization of Improper Integrals.- 14. Asymptotic Expansions for the End of a Junction Part of a Trajectory.- 15. The Relation between Asymptotic Approximations and Actual Trajectories at the End of a Junction Part.- 16. Proof of Asymptotic Representations for the Junction Part.- 17. Asymptotic Approximations of theTrajectory on the Fast-Motion Part.- 18. Derivation of Asymptotic Representations for the Fast-Motion Part.- 19. Special Variables for the Drop Part.- 20. Asymptotic Approximations of the Drop Part of the Trajectory.- 21. Proof of Asymptotic Representations for the Drop Part of the Trajectory.- 22. Asymptotic Approximations of the Trajectory for Initial Slow-Motion and Drop Parts.- III. Second-Order Systems. Almost-Discontinuous Periodic solutions.- 1. Existence and Uniqueness of an Almost-Discontinuous Periodic Solution.- 2. Asymptotic Approximations for the Trajectory of a Periodic Solution.- 3. Calculation of the Slow-Motion Time.- 4. Calculation of the Junction Time.- 5. Calculation of the Fast-Motion Time.- 6. Calculation of the Drop Time.- 7. An Asymptotic Formula for the Relaxation-Oscillation Period.- 8. Van der Pol's Equation. Dorodnitsyn's Formula.- IV. Systems of Arbitrary Order. Asymptotic Calculation of Solutions.- 1. Basic Assumptions.- 2. The Zeroth Approximation.- 3. Local Coordinates in the Neighborhood of a Junction Point.- 4. Asymptotic Approximations of a Trajectory at the Beginning of a Junction Section.- 5. Asymptotic Approximations for the Trajectory in the Neighborhood of a Junction Point.- 6. Asymptotic Approximation of a Trajectory at the End of a Junction Section.- 7. The Displacement Vector.- V. Systems of Arbitrary Order. Almost-Discontinuous Periodic Solutions.- 1. Auxiliary Results.- 2. The Existence of an Almost-Discontinuous Periodic Solution. Asymptotic Calculation of the Trajectory.- 3. An Asymptotic Formula for the Period of Relaxation Oscillations.- References.