E-raamat: Differential Equations with Small Parameters and Relaxation Oscillations

I. Dependence of Solutions on Small Parameters. Applications of
Relaxation Oscillations.-
1. Smooth Dependence. Poincarés 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
Pols Equation. Dorodnitsyns 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.