The 0:1 resonance bifurcation associated with the supercritical Hamiltonian pitchfork bifurcation

We consider the non-semisimple 0:1 resonance (i.e. the unperturbed equilibrium has two purely imaginary eigenvalues +/- i alpha (alpha is an element of R and alpha > 0) and a non-semisimple double-zero one) Hamiltonian bifurcation with one distinguished parameter, which corresponds to the supercritical Hamiltonian pitchfork bifurcation. Based on BCKV singularity theory established by [H.W. Broer, S. -N. Chow, Y. Kim, and G. Vegter, A normally elliptic Hamiltonian bifurcation, Z. Angew. Math. Phys. 44 (3) (1993), pp. 389-432], this bifurcation essentially triggered by the reversible universal unfolding M = 1/2 p(2) + 1/4 q(4) + (lambda + I-1)q(2) with respect to BCKV-restricted morphisms of the planar non-semisimple singularity 1/2 p(2) + 1/4 q(4) (the I-1 is regarded as distinguished parameter with respect to the external parameter lambda). We first give the plane bifurcation diagram of the integrable Hamiltonian on each level of integral in detail, which is related to the usual supercritical Hamiltonian pitchfork bifurcation. Then, we use the S-1-symmetry generated by the additional pair of imaginary eigenvalues +/- i alpha to reconstruct the above plane bifurcation phenomenon caused by the zero eigenvalue pair into the case with two degrees of freedom. Finally, we prove the persistence of typical bifurcation scenarios (e.g. 2-dimensional invariant tori and the symmetric homoclinic orbit) under the small Hamiltonian perturbations, as proposed by [H.W. Broer, S. -N. Chow, Y. Kim, and G. Vegter, A normally elliptic Hamiltonian bifurcation, Z. Angew. Math. Phys. 44 (3) (1993), pp. 389-432]. An example system (the coupled Duffing oscillator) with strong linear coupling and weak local nonlinearity is given for this bifurcation.