The 1:1 resonance in Hamiltonian systems

Bifurcations and monodromy of the axially symmetric 1:1:−2 resonance

“…x 1 x 2 + y 1 y 2 = q 1 q 2 + p 1 p 2 would yield the same detuning, but at the price of breaking the symmetry generated by N . A general detuning of the 1:1 subresonance has indeed co-dimension three and would lead to all the phenomena detailed in [17] concerning higher order terms in the normal form.…”

“…The semi-algebraic variety P µ given by (17) is smooth everywhere except possibly at its 'tip' point (R, X, Y ) = (R min , 0, 0). The latter point is a cusp (or cuspidal singularity of order 3) when µ = = 0; a cone (or conical singularity) when = |µ| > 0 or µ = 0, < 0; and smooth in all other cases.…”

Structure of the center manifold of the L1 and L2 collinear libration points in the restricted three-body problem

“…The normal form (18), when expressed in the coordinates (P 2 , P 3 , Q 2 , Q 3 ), is an example of this class of Hamiltonians. A general result concerning polynomials invariant under the action of a compact group is the Hilbert-Schwartz theorem (Hanßmann & Hoveijn 2018): it states that a finite number of polynomials, generating the Hilbert basis, exist such that any invariant polynomial can be expressed in terms of them. Cushman & Bates (1997) prove that the Hilbert basis of the polynomials in R 4 invariant under the S 1 -action ( 21) is given by…”

“…A most effective description is achieved by means of the invariants of the isotropic oscillator (Cushman & Bates 1997;Hanßmann & Hoveijn 2018) which plays the role of resonant unperturbed system. Then, a singular reduction process of the normal form (Efstathiou 2005) does the rest of the job.…”