Differential Equations with Small Parameters and Relaxation Oscillations

Inhaltsverzeichnis

• 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.
• 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.