Subsidiary bifurcations near bifocal homoclinic orbits

Abstract

Suppose that an ordinary differential equation in ℝ4 has an orbit Γ bi-asymptotic to a stationary point O of the flow. If the characteristic equation of the linear flow near O has roots (λ0 ± ιω0, λ1±ιω1) with —λ0 > λ1 > 0 and ωi > 0 for i = 0,1, we show that there are sequences of more complicated orbits bi-asymptotic to O in generic one-parameter perturbations of the equations. When ω0 ≠ 2nω1 for all n ∈ ℕ, there are sequences of such orbits on both sides of the bifurcation value of the parameter, in contrast to a similar case for flows in ℝ3.