Relative equilibria and bifurcations in the generalized van der Waals 4D oscillator

Bifurcations of the Hamiltonian Fourfold 1:1 Resonance with Toroidal Symmetry

“…In that respect we have been working also with the generalized van der Waals problem in 4 dimensions, another uniparametric family which, versus the Stark-Zeeman, presents several integrable cases. This fact introduces another ingredient in the analysis which we are exploring [8].…”

“…π 3 π 13 = π 10 π 12 + π 6 π 16 π 3 π 9 = π 8 π 10 − π 14 π 16 π 3 π 15 = π 10 π 14 + π 8 π 16 π 4 π 5 = π 7 π 9 + π 13 π 15 π 4 π 11 = π 9 π 13 − π 7 π 15 π 4 π 6 = π 7 π 10 + π 13 π 16 π 4 π 12 = π 10 π 13 − π 7 π 16 π 4 π 8 = π 9 π 10 + π 15 π 16 π 4 π 14 = π 10 π 15 − π 9 π 16 π 5 π 10 − π 7 π 8 = π 13 π 14 + π 11 π 16 π 10 π 11 + π 5 π 16 = π 8 π 13 − π 7 π 14 π 7 π 8 − π 6 π 9 = π 13 π 14 − π 12 π 15 π 9 π 12 + π 6 π 15 = π 7 π 14 + π 8 π 13 π 5 π 10 + π 11 π 16 = π 6 π 9 + π 12 π 15 π 10 π 11 − π 5 π 16 = π 9 π 12 − π 6 π 15 (8) Note that from the third element of the first column of (8) to the last in the third column we have the complex version of the two columns which, given in terms of π i , are appearing in (1). If we replace now in (8) each σ by its corresponding complex expressions z ∈ C we obtain the definition of CP 3 given in for instance [6].…”

“…Then, we built the resultant R 1 of P λ h,n,ξ (K) and d dK P λ h,n,ξ (K) with respect to K, and the resultant R 2 of P λ h,n,ξ (K) and d 2 dK 2 P λ h,n,ξ (K) with resepct to K, where R 1 and R 2 are polynomials in (n, ξ, λ, h). It remains to obtain the resultant R 3 of R 1 and R 2 with respect to h, and finally we obtain R 3 which is a polynomial depending on (n, ξ, λ, h), one of whose factors R 3a , given by R 3a = 64ξ 12 + 192(3λ − 50)ξ 10 + 144(λ(15λ − 388) + 268)ξ 8 + 32(9λ(3λ(5λ − 138) + 244) − 4168)ξ 6 (37) + 36 λ 3λ 45λ 2 − 984λ − 296 + 13408 + 5360 ξ 4 + 12 3λ 3λ 3λ 9λ 2 − 78λ − 152 − 656 − 6320 − 20000 ξ 2 + (3λ + 2) 3 (3λ + 10) 3 , allows to obtain the bifurcation curve, by means of an implicit representation, which determines the double roots represented in Fig. (8).…”

Generalized Hopf Fibration and Geometric SO(3) Reduction of the 4DOF Harmonic Oscillator

“…Note that these same polynomials also play a role in the reduction of the Kepler system [4] and the reduction of the 4DOF isotropic harmonic oscillator [5,6,7].…”

“…where q, Q is the hermitian inner product defined by (5). Therefore, the components of A R are defined as follows…”