Fixed points and chaotic dynamics for expansive-contractive maps in Euclidean spaces, with some applications

In this work we introduce a topological method for the search of fixed points and periodic points for continuous maps defined on generalized rectangles in finite dimensional Euclidean spaces. We name our technique "Stretching Along the Paths" method, since we deal with maps that expand the arcs along one direction. Such theory was developed in the planar case by Papini and Zanolin in [11,12] and it has been extended to the N-dimensional framework by the author and Zanolin in [16]. In the bidimensional setting, elementary theorems from plane topology suffice, while in the higher dimension some results from degree theory are needed, leading to the study of the so-called "Cutting Surfaces" [16]. Our method is also significant from a dynamical point of view, as it allows to detect complex dynamics. As it is well-known, a prototypical example of chaotic system is represented by the Smale horseshoe. However, in order to prove conjugacy with the shift map, it requires the verification of hyperbolicity conditions, which are difficult or impossible to prove in practical cases. For such reason more general and less stringent definitions of horseshoe have been suggested so as to reproduce some geometrical features typical of the Smale horseshoe while discarding the hyperbolicity hypotheses. This led to the study of the so-called "topological (or geometrical) horseshoes" [2,5]. In particular, different characterizations have been proposed by various authors in order to establish the presence of complex dynamics for continuous maps defined on subsets of the N-dimensional Euclidean space (see, for instance, [10,21,23] and the references therein). The tools employed in these and related works range from the Conley index [10] to the Lefschetz fixed point theory [20]. On the other hand, our approach, although mathematically rigorous, avoids the use of more advanced topological theories and it is relatively easy to apply to specific models arising in applications. For example we have employed such method to study discrete and continuous-time models arising from economics and biology [9,18]. In more details, the topics considered along the thesis can be summarized as follows. The description of the Stretching Along the Paths method and suitable variants of it can be found in Chapter 1. In Chapter 2 we discuss which are the chaotic features that can be obtained for a given map when our technique applies. In particular, we are able to prove semi-conjugacy to the Bernoulli shift and thus positivity of the topological entropy, the presence of topological transitivity and sensitivity with respect to initial conditions, density of periodic points. Moreover we show the mutual relationships among various classical notions of chaos (such as those by Devaney, Li-Yorke, etc.). We also introduce an alternative geometrical framework related to the so-called "Linked Twist Maps" [3,4,22], where it is possible to employ our method in order to detect complex dynamics. The theoretical results obtained so far find an application to discrete and continuous-time systems in Chapters 3 and 4. As regards the former, in Chapter 3 we deal with some one-dimensional and planar discrete economic models, both of the Overlapping Generation and of the Duopoly Game classes. The bidimensional models are taken from [8,19] and [1], respectively. On the other hand, in Chapter 4, with respect to continuous-time models, we study some nonlinear ODEs with periodic coefficients through a combination of a careful but elementary phase-plane analysis with the results on chaotic dynamics for Linked Twist Maps from Chapter 2. In more details, we consider a modified version of the Volterra predator-prey model, in which a periodic harvesting is included, as well as a simplification of the Lazer-McKenna suspension bridges model [6,7] from [13,14]. When dealing with ODEs with periodic coefficients, our method is applied to the associated Poincaré map. The contents of the present thesis are based on the papers [9,13,16,17,18] and partially on [14], where maps expansive along several directions were considered. [1] H.N. Agiza and A.A. Elsadany, Chaotic dynamics in nonlinear duopoly game with heterogeneous players, Appl. Math. Comput. 149 (2004), 843-860. [2] K. Burns and H. Weiss, A geometric criterion for positive topological entropy, Comm. Math. Phys. 172 (1995), 95-118. [3] R. Burton and R.W. Easton, Ergodicity of linked twist maps, In: Global theory of dynamical systems (Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1979), pp. 35-49, Lecture Notes in Math., 819, Springer, Berlin, 1980. [4] R.L. 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