Some global bifurcations of two-dimensional endomorphisms by use of critical lines

Abstract

MANY STUDIES have been devoted to discrete dynamical systems represented by homeomorphisms and diffeomorphisms since the works by Smale [l] and Gavrilov and Shilnikov [2, 31, but comparatively few works have been devoted to endomorphisms, that is to maps with a nonunique inverse. However, the mathematical models derived from the description of evolutive phenomena in several applicative fields are often represented by discrete dynamical systems, or maps, with a nonunique inverse. This is particularly true in the socio-economic field [4-61, but examples can also be found in ecology [7, 81, biology [9-l l] and control systems [ 121. This work concerns the study of the global dynamics of recurrences represented by twodimensional endomorphisms of applicative interest in the economic context. Useful notions for the local and global analysis of nonlinear endomorphisms have been introduced by Gumowski and Mira [ll, 13, 141. For two-dimensional endomorphisms, in particular, this is the notion of critical lines (or critical curves), representing the twodimensional extension of the critical points in one-dimensional maps, and the notions of absorbing area and chaotic area (bounded by elements of critical lines), which determine phase plane domains where chaotic solutions are located. These notions, which will be recalled in the next section, together with some bifurcations involving the chaotic areas, are the basic concepts and properties that we shall use in the analysis of the subsequent sections. The aim of this paper is to illustrate by an example how powerful the critical lines may be in studying the global dynamics and bifurcations of plane nonlinear endomorphisms, particularly when they are used in combination with more traditional techniques. It is known that in diffeomorphisms, global bifurcations which qualitatively change the structure of a chaotic attractor, called interior crises or catastrophes by some authors [ 15-181, are characterized by the stable manifold of a saddle, approaching and touching the chaotic attractor. We shall find that this phenomenology also occurs in the endomorphisms of our model, with a particular role being played by the critical lines. In Section 3 we shall describe our mathematical model, a two-dimensional piecewise-linear map which we call map F, together with some of its elementary properties and the definition of three classes of endomorphisms into which the analysis can be conveniently subdivided. Endomorphisms of the first class have been the object of a previous work [ 191. Here we shall show, in Section 4, how several bifurcations involving chaotic areas can be explained by use of critical lines. The main result is the analytical determination of the bifurcation which discriminates between the global attractivity of an absorbing area and the nonglobal attractivity