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arXiv · 2312.10720

On the Hausdorff dimension and Cantor set structure of sliding Shilnikov invariant sets

Matheus G. C. Cunha·Douglas D. Novaes·Gabriel Ponce
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Abstract

The concept of sliding Shilnikov connection has been recently introduced and represents an important notion in Filippov systems, because its existence implies chaotic behavior on an invariant subset of the system. The investigation of its properties has just begun, and understanding the topology and complexity of its invariant set is of interest. In this paper, we conduct a local analysis on the first return map associated to a Shilnikov sliding connection, which reveals a conformal iterated function system (CIFS) structure. By using the theory of CIFS, we estimate the Hausdorff dimension of the local invariant set of the first return map, showing, in particular, that it is strictly greater than $0$ and strictly less than $1$, and its one-dimensional Lebesgue measure is 0. Moreover, we prove that the closure of the local invariant set is a Cantor set and has the same Hausdorff dimension and Lebesgue measure of the original invariant set. Furthermore, it is given by the invariant set adjoined with the set of all pre-images of the regular-fold point.

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Matheus G. C. Cunha, Douglas D. Novaes, Gabriel Ponce. 2023-12-17. On the Hausdorff dimension and Cantor set structure of sliding Shilnikov invariant sets. https://doi.org/10.1088/1361-6544%2Fad8d9c

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