Dense heteroclinic tangencies near a Bykov cycle

• Type: Article in International Scientific Journal
• Date: 2015

Authors

Isabel S Labouriau (Author) FCUP; Alexandre A P Rodrigues (Author) FCUP

Journal

Title: Journal of Differential Equations

Vol. 259

Pages: 5875-5902

ISSN: 0022-0396

Publisher: Elsevier

Indexing

ISI Web of Knowledge - 12 Citations

Scopus - 0 Citations

Other information

Authenticus ID: P-00G-NA2

DOI: 10.1016/j.jde.2015.07.017

Abstract (EN): This article presents a mechanism for the coexistence of hyperbolic and non-hyperbolic dynamics arising in a neighbourhood of a Bykov cycle where trajectories turn in opposite directions near the two nodes - we say that the nodes have different chirality. We show that in the set of vector fields defined on a three-dimensional manifold, there is a class where tangencies of the invariant manifolds of two hyperbolic saddle-foci occur densely. The class is defined by the presence of the Bykov cycle, and by a condition on the parameters that determine the linear part of the vector field at the equilibria. This has important consequences: the global dynamics is persistently dominated by heteroclinic tangencies and by Newhouse phenomena, coexisting with hyperbolic dynamics arising from transversality. The coexistence gives rise to linked suspensions of Cantor sets, with hyperbolic and non-hyperbolic dynamics, in contrast with the case where the nodes have the same chirality. We illustrate our theory with an explicit example where tangencies arise in the unfolding of a symmetric vector field on the three-dimensional sphere.

Language: English

Type (Professor's evaluation): Scientific

No. of pages: 28