The concept of chaos is widely used in the field of Dynamical Systems, and several approaches which aim to establish the presence of chaotic dynamics have been developed in the literature. At this juncture, a prototypical example comes from the geometric structure associated with the Smale’s horseshoe, cf. [4]. In recent years, several different approaches have been proposed to extend this classical geometry in a topological direction. This way, the so-called concept of “topological horseshoes” was introduced in [2].

The topological horseshoes along with symbolic dynamics provide a powerful tool to describe the time evolution of chaotic dynamics. In this framework, by exploiting techniques developed in [3], we investigate two families of chaotic dynamical systems. Firstly, we deal with a discrete application and prove the existence of topological horseshoes for the twisted horseshoe map [1] and its generalisation, cf. [5]. At last, we consider a continuous application and show analytically the presence of complex behaviours for a class of indefinite weight periodic boundary value problems, cf. [6].

References

1. [1]  J. Guckenheimer, G. Oster, A. Ipaktchi, The dynamics of density dependent pop- ulation models, J. Math. Biol. 4 (1977), 101–147.
2. [2]  J. Kennedy and J. A. Yorke, Topological horseshoes, Trans. Amer. Math. Soc., 353 (2001), 2513–2530.
3. [3]  A. Medio, M. Pireddu, F. Zanolin., Chaotic dynamics for maps in one and two dimensions: a geometrical method and applications to economics, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 19(10) (2009), 3283–3309.
4. [4]  S. Smale, Finding a horseshoe on the beaches of Rio, Math. Intelligencer, 20(1) (1998), 39–44.
5. [5]  E. Sovrano, About chaotic dynamics in the twisted horseshoe map, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 26(6) (2016), 1650092, 10.
6. [6]  E. Sovrano, How to get complex dynamics? A note on a topological approach, preprint.