Heterodimensional cycles derived from homoclinic tangencies via Hopf bifurcations

2025/05/19 by Tomizawa, Shuntaro
#37C20 #37C29 #37D30 #37G15 #37G25 #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2505.12596

Abstract

We analyze three-dimensional C^r diffeomorphisms (r≥ 5) exhibiting a quadratic focus-saddle homoclinic tangency whose multipliers satisfy |λγ| = 1. For a proper three-parameter unfolding that splits the tangency, varies the argument of the stable multipliers, and controls the modulus |λγ|, we show that a Hopf bifurcation occurs on this curve and that a homoclinic point to the bifurcating periodic orbit is present. As a consequence, the original map f can be C^r-approximated by a diffeomorphism exhibiting a coindex-one heterodimensional cycle in the saddle case.

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