Differential Equations with Small Parameters and Relaxation Oscillations: Mathematical Concepts and Methods in Science and Engineering, cartea 13

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.