Persistence of Poincaré mappings in functional differential equations (with application to structural stability of complicated behavior)
The paper deals with complicated behavior associated with delay differential equations. The existence of chaos in F.D.E's was a fundamental issue in connection with some real-life problems, during the seventies. A still open conjecture is that chaos arises in a blood production model involving delays, first proposed by Mackey and Glass. As far as I know, the first analytical proof of complicated behavior for delay differential equations is due to \textit{H. Peters} [C. R. Acad. Sci., Paris, Sér. A 290, 1119-1122 (1980; Zbl 0435.34063)]. It was followed by works by Walther, and an der Heiden and Walther. Two main methods are known for proving chaotic behavior: one consists in applying the principle ``period three implies chaos'' due to Li and Yorke (1975). The second one is of a more general nature and is connected to the existence of Cantor-like sets near transversal homoclinic orbits. For F.D.E's, that method poses some natural difficulties: non-invertibility of the semiflow, and noninjectivity. These problems have been attacked successfully during the eighties in works by Walther, an der Heiden and Walther, Hale and Lin, and a few others. Severe restrictions have been made on the equations which, for most of them, are not too far from the example considered by Peters. In order to extend the validity of those works, one possibility is to show some kind of structural stability of the results. This is the main goal and achievement of the present work. One of the main tools for this is a result on dependence of Poincaré maps on the equation (theorem 1.7) which, according to the author's own words, ``is certainly not surprising but the author is not aware of a reference for it''. The statements in terms of structural stability for three classes of equations for which complicated behavior has been shown to hold are given in the last section of the paper. It should be noted however that the paper does not contain any proof of those results. For the main arguments of those proofs, the author refers to either his Ph.D. thesis or a (1992) preprint. Sections 2, 3 and 4 are devoted to a detailed presentation of the three classes of equations. These examples were introduced by Walther, an der Heiden and Walther, then revisited by Hale and Lin. In the course of his presentation, the author improves some results and completes earlier proofs by Hale and Lin (1986). This lengthy paper should be considered as a companion in the sense of supplying examples to the announced work by the author on ``Hyperbolic sets, Shadowing and Persistence for \({\mathcal C}^1\) mappings in Banach Spaces''.
scientific article; zbMATH DE number 740265
scientific article