Published in:Mathematical Concepts and Methods in Science and Engineering ; 13 SpringerLink ; Bücher Springer eBook Collection ; Mathematics and Statistics
Description:
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 the Trajectory 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.