Qualitative analysis of the periodically forced relaxation oscillations

American Mathematical Society

1981

163 p.

Memoirs of the American Mathematical Society ; 244

""Table of Contents""; ""Â1. Introduction""; ""1.1. A short history of the problem""; ""1.2. Classical results""; ""1.3. Later developments""; ""1.4. Some questions posed by the classical results""; ""1.5. Outline of the paper""; ""Â2. Results""; ""2.1. Geometry of Poincare map (the main result)""; ""2.2. Qualitative analysis of the equation-results""; ""2.3. An informal explanation of the shape of Poincare map; geometry of Levinson's results""; ""A. Global behaviour of the phase flow""; ""B. Geometry of Levinson's results""; ""C. An application to the large amplitude case""""2.4. A family of annulus maps""""A. An example of a family of annulus maps""; ""B. Projected circle maps; notations""; ""C. Hypothesis""; ""D. Analysis of the map""; ""Â3. Proof of the Hypothesis for an equation of van der Pol type""; ""3.0. Statement of the main result""; ""3.1. Construction of the window map""; ""A. Factorization""; ""B. Construction of the window r""; ""3.2. Global behaviour of the solutions; Linearized and Hessian systems""; ""A. Behaviour of the solutions""; ""B. Linearized and Hessian systems""; ""3.3. Deformation of the window r with the flow"" ""3.4. Proof of H1 """"3.5. Proof of H2""; ""3.6. Nondegeneracy of the homoclinic tangency""; ""Â4. Qualitative analysis of the equation â€? proof of Theorem 2.2""; ""Appendix to Â2.4. â€? Proof of Theorem 2.3""; ""Appendix to Â3.1""; ""A. Two auxiliary lemmas""; ""B. Proof of Proposition 3.1""; ""C. Proof of Proposition 3.2""; ""D. Proof of the auxiliary lemmas""; ""Appendix to Â3.2""; ""A. Proof of theorem 3.2""; ""B. Proof of Lemmas 3.2 â€? 3.6""; ""Appendix to Â3.4. â€? Proof of Proposition 3.4""; ""A. Connection between f[sub(Ï?)] and g[sub(Ï?)]""; ""B. Proof of Proposition 3.4""""C. Proofs of Lemmas A3.4 â€? 1,2,3""""References""

Английский