Hidden toric symmetry and structural stability of singularities in integrable systems

Structurally stable non-degenerate singularities of integrable systems

“…B. Theorem 1.4 (a) was proved by J. Vey [44], and its equivariant generalization was proved by the first author [30, Lemma 6.2]. For a compact orbit O, Theorem 1.4 (b) was proved in [38,Theorem 2.1] for C ∞ and real-analytic cases (see also [30,Example 4.2 (A)] for real-analytic case), and its equivariant generalization was proved [38,Theorem 4.3] for C ∞ and real-analytic cases.…”

“…However they are not symplectically structurally stable, because of the presence of "moduli" in their symplectic classifications (see [6] for real-analytic case, [32] for the smooth and real-analytic cases). 8) Structural stability under integrable perturbations preserving a Hamiltonian (S 1 ) n−1 -action (for ndegree of freedom integrable systems) was proved for many degenerate local singularities, e.g., parabolic orbits with resonances [22] (which are smoothly structurally stable when the resonance order is different from 4 [18,30]), their parametric bifurcations [18], periodic integrable Hamiltonian Hopf bifurcation [42,18] and its hyperbolic analogue [36,Sec. 2], periodic integrable Hamiltonian Hopf bifurcations with resonances and their parametric bifurcations [12], normally-elliptic parabolic orbits [8], normallyhyperbolic parabolic orbits etc.…”

“…2], periodic integrable Hamiltonian Hopf bifurcations with resonances and their parametric bifurcations [12], normally-elliptic parabolic orbits [8], normallyhyperbolic parabolic orbits etc. The above F -preserving Hamiltonian (S 1 ) n−1 -action is generated by n−1 functions, some of which are real-analytic functions multiplied with √ −1 (in the real-analytic case) [30,Example 3.12]. It is conjectured in [30,Example 4.2 (B)] that, using "hidden" torus actions, one can prove structural stability in a strong sense of the singular orbits mentioned above in real-analytic case.…”

“…The above F -preserving Hamiltonian (S 1 ) n−1 -action is generated by n−1 functions, some of which are real-analytic functions multiplied with √ −1 (in the real-analytic case) [30,Example 3.12]. It is conjectured in [30,Example 4.2 (B)] that, using "hidden" torus actions, one can prove structural stability in a strong sense of the singular orbits mentioned above in real-analytic case.…”

“…Proof. Our proof follows [30, Example 4.2(A)] and is based on a strengthening (Lemma B.1) of the Vey Theorem 1.4 (see also [30,Theorem 3.10]).…”