Molero, F. J.; Van Der Meer, J. C.; Ferrer, S.; Céspedes, F. J. The 2-D sextic Hamiltonian oscillator. (English) Zbl 1272.37027 Int. J. Bifurcation Chaos Appl. Sci. Eng. 23, No. 6, Article ID 1330019, 27 p. (2013). Summary: The 2-D sextic oscillator is studied as a family of axial symmetric parametric integrable Hamiltonian systems, presenting a bifurcation analysis of the different flows. It includes the “elliptic core” model in 1-D nonlinear oscillators, recently proposed in the literature. We make use of the energy-momentum mapping, which will give us the fundamental fibration of the four-dimensional phase space. Special attention is given to the singular values of the energy-momentum mapping connected with rectilinear and circular orbits. They are related to the saddle-center and pitchfork scenarios with the associated homoclinic and heteroclinic trajectories. We also study how the geometry of the phase space evolves during the transition from the one-dimensional to the two-dimensional model. Within an elliptic function approach, the solutions are given using Legendre elliptic integrals of the first and third kind and the corresponding Jacobi elliptic functions. MSC: 37J15 Symmetries, invariants, invariant manifolds, momentum maps, reduction (MSC2010) 37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests 34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations 37G10 Bifurcations of singular points in dynamical systems Keywords:integrable Hamiltonian systems; singular reduction; sextic Duffing oscillator; nonlinear oscillators; energy momentum mapping; bifurcations; Jacobian elliptic function PDFBibTeX XMLCite \textit{F. J. Molero} et al., Int. J. Bifurcation Chaos Appl. Sci. Eng. 23, No. 6, Article ID 1330019, 27 p. (2013; Zbl 1272.37027) Full Text: DOI References: [1] Akbarzade M., Adv. Theor. Appl. Mech. 3 pp 329– (2010) [2] Akbarzade M., Adv. Theor. Appl. Mech. 3 pp 321– (2010) [3] DOI: 10.1016/0001-8708(90)90058-U · Zbl 0721.53033 · doi:10.1016/0001-8708(90)90058-U [4] DOI: 10.1007/978-1-4613-9725-0_4 · doi:10.1007/978-1-4613-9725-0_4 [5] DOI: 10.1007/BF00944566 · Zbl 0755.58028 · doi:10.1007/BF00944566 [6] DOI: 10.1006/jsvi.1995.0448 · Zbl 1049.70644 · doi:10.1006/jsvi.1995.0448 [7] DOI: 10.1007/s11071-010-9797-0 · Zbl 1215.35108 · doi:10.1007/s11071-010-9797-0 [8] DOI: 10.1088/0305-4470/31/2/022 · Zbl 0956.81018 · doi:10.1088/0305-4470/31/2/022 [9] DOI: 10.1088/0305-4470/33/32/303 · Zbl 1009.81015 · doi:10.1088/0305-4470/33/32/303 [10] DOI: 10.1016/j.ijnonlinmec.2006.12.005 · Zbl 1160.35513 · doi:10.1016/j.ijnonlinmec.2006.12.005 [11] DOI: 10.1007/s11012-009-9205-3 · Zbl 1258.70040 · doi:10.1007/s11012-009-9205-3 [12] DOI: 10.1002/cpa.3160330602 · Zbl 0439.58014 · doi:10.1002/cpa.3160330602 [13] DOI: 10.1088/0951-7715/25/12/3409 · Zbl 1314.37041 · doi:10.1088/0951-7715/25/12/3409 [14] DOI: 10.1016/j.physleta.2007.04.088 · Zbl 1209.81148 · doi:10.1016/j.physleta.2007.04.088 [15] DOI: 10.2528/PIERM08061007 · doi:10.2528/PIERM08061007 [16] DOI: 10.1063/1.3272207 · Zbl 06437225 · doi:10.1063/1.3272207 [17] DOI: 10.1103/PhysRevE.80.046608 · doi:10.1103/PhysRevE.80.046608 [18] DOI: 10.1016/j.apm.2007.12.012 · Zbl 1168.34321 · doi:10.1016/j.apm.2007.12.012 [19] DOI: 10.1134/1.1555186 · doi:10.1134/1.1555186 [20] DOI: 10.1142/S0218127406015155 · Zbl 1149.34022 · doi:10.1142/S0218127406015155 [21] DOI: 10.1007/s12346-012-0081-1 · Zbl 1270.33011 · doi:10.1007/s12346-012-0081-1 [22] DOI: 10.1016/j.camwa.2008.10.082 · Zbl 1165.34310 · doi:10.1016/j.camwa.2008.10.082 [23] DOI: 10.1088/0951-7715/21/5/008 · Zbl 1153.34025 · doi:10.1088/0951-7715/21/5/008 [24] Van der Meer J. C., Fields Instit. Commun. 8 pp 159– (1996) [25] DOI: 10.1016/j.chaos.2005.04.048 · Zbl 1083.37062 · doi:10.1016/j.chaos.2005.04.048 [26] DOI: 10.1016/S0926-2245(96)00042-3 · Zbl 0887.58023 · doi:10.1016/S0926-2245(96)00042-3 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.