Heteroclinic tangles and homoclinic snaking in the unfolding of a degenerate reversible Hamiltonian Hopf bifurcation

This paper considers an unfolding of a degenerate reversible $1-1$ resonance (or Hamiltonian Hopf) bifurcation for four-dimensional systems of time-reversible ordinary differential equations (ODEs). This bifurcation occurs when a complex quadruple of eigenvalues of an equilibrium coalesce on the imaginary axis to become two imaginary pairs. The degeneracy occurs via the vanishing of a normal form coefficient (q_2=0) that determines whether the bifurcation is super- or sub-critical. Of particular concern is the behaviour of homoclinic and heteroclinic connections between the trivial equilibrium and simple periodic orbits. A partial unfolding of such solutions already occurs in the work of Dias and Iooss (Eur.J.Mech.B-Fluids 15, (1996) 367-393), given a sign of the coefficient of a higher-order term (q_40 and -1