Generating hyperbolic singularities in semitoric systems via Hopf bifurcations (Q295321)

The paper considers two-degree of freedom integrable Hamiltonian systems \(F=(J,H)\) with only non-degenerate, non-hyperbolic singularities. Let \(p_1,\dots,p_n\) be the focus-focus singularities of such a system. It is shown that it is possible to smoothly modify the integrable system in a neighborhood of each \(p_j\) so that the resulting system \(\tilde{F} = (J,\tilde{H})\) is integrable, and its set of critical values contains, for each \(p_j\), an island, that is, a loop consisting of three piecewise smooth curves of critical values: a curve of non-degenerate transversally hyperbolic singularities and two curves of non-degenerate transversally elliptic singularities. The construction is based on inducing a subcritical Hamiltonian Hopf bifurcation at the desired focus-focus points \(p_j\) which become elliptic-elliptic through the bifurcation. The local character of the construction implies that one can control which focus-focus points go through the Hamiltonian Hopf bifurcation and ensure that the remaining focus-focus points do not change type. Moreover, one important aspect of the construction is that the integrability of the system, and the \(S^1\) action that always exists in a semi-local neighborhood of a focus-focus point, are preserved. In the particular case of semitoric systems, this implies that the method can be applied in such a way so that the global \(S^1\) action is preserved. As an example, the construction is applied to the Jaynes-Cummings model.