Robust cycles unfolding from conservative bifocal homoclinic orbits. Issue 4 (1st October 2016)

• Record Type: Journal Article
• Authors: Barrientos, Pablo G.; Ibáñez, Santiago; Rodríguez, J. Ángel
• Is Part Of: Dynamical systems. Volume 31:Issue 4(2016)
• Page Start: 546
• Page End: 579
• Publication Date: 2016-10-01
• Subjects: Bifocal homoclinic orbits -- Hamiltonian vector fields -- robust heterodimensional cycles -- robust homoclinic tangencies -- strong homoclinic bifurcation -- nilpotent singularities; 37C05 -- 37C10 -- 37C29 -- 37G20 -- 37C20; Differentiable dynamical systems -- Periodicals; 515.35205
• DOI: 10.1080/14689367.2016.1170763
• Languages: English
• ISSNs: 1468-9367

Abstract

ABSTRACT: We prove that suspended robust heterodimensional cycles and suspended robust homoclinic tangencies can be found arbitrarily close to any non-degenerate bifocal homoclinic orbit of a Hamiltonian vector field. In order to achieve this result, we show that any diffeomorphism with a saddle-node periodic point, which has both quasi-transversal and tangential strong homoclinic intersections, can be approximated by diffeomorphisms with both, robust homoclinic tangencies and a robust heterodimensional cycles. As explained in the paper, Hamiltonian vector fields with non-degenerate bifocal homoclinic orbits are exhibited in the limit family of any generic unfolding of a four-dimensional nilpotent singularity of codimension four. Supported by the achieved results, we conjecture that suspended robust cycles can be generically unfolded from such singularities.