Global bifurcations close to symmetry

• Type: Article in International Scientific Journal
• Date: 2016

Authors

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

Journal

Title: Journal of Mathematical Analysis and Applications

Vol. 444

Pages: 648-671

ISSN: 0022-247X

Publisher: Elsevier

Indexing

ISI Web of Knowledge - 8 Citations

Scopus - 7 Citations

Other information

Authenticus ID: P-00K-TJD

DOI: 10.1016/j.jmaa.2016.06.032

Abstract (EN): Heteroclinic cycles involving two saddle-foci, where the saddle-foci share both invariant manifolds, occur persistently in some symmetric differential equations on the 3-dimensional sphere. We analyse the dynamics around this type of cycle in the case when trajectories near the two equilibria turn in the same direction around a 1-dimensional connection - the saddle-foci have the same chirality. When part of the symmetry is broken, the 2-dimensional invariant manifolds intersect transversely creating a heteroclinic network of Bykov cycles. We show that the proximity of symmetry creates heteroclinic tangencies that coexist with hyperbolic dynamics. There are n-pulse heteroclinic tangencies - trajectories that follow the original cycle n times around before they arrive at the other node. Each n-pulse heteroclinic tangency is accumulated by a sequence of (n + 1)-pulse ones. This coexists with the suspension of horseshoes defined on an infinite set of disjoint strips, where the first return map is hyperbolic. We also show how, as the system approaches full symmetry, the suspended horseshoes are destroyed, creating regions with infinitely many attracting periodic solutions.

Language: English

Type (Professor's evaluation): Scientific

No. of pages: 24