Abstract
An axially symmetric perturbed isotropic harmonic oscillator undergoes several bifurcations as the parameter λ adjusting the relative strength of the two terms in the cubic potential is varied. We show that three of these bifurcations are Hamiltonian Hopf bifurcations. To this end we analyse an appropriately chosen normal form. It turns out that the linear behaviour is not that of a typical Hamiltonian Hopf bifurcation as the eigen-values completely vanish at the bifurcation. However, the nonlinear structure is that of a Hamiltonian Hopf bifurcation. The result is obtained by formulating geometric criteria involving the normalized Hamiltonian and the reduced phase space.
Similar content being viewed by others
REFERENCES
Aizawa, Y., and Saitô, N. (1972). On the stability of isolating integrals. I. Effect of the perturbation in the potential function. J. Phys. Soc. Japan 32, 1636–1640.
Braun, M. (1973). On the applicability of the third integral of motion. J. Differential Equations 13, 300–314.
Cotter, C. (1986). The 1:1 Semisimple Resonance. Ph.D. thesis, University of California at Santa Cruz.
Cushman, R. (1982). Geometry of the bifurcations of the normalized reduced Hénon-Heiles family. Proc. Roy. Soc. London Ser. A 382, 361–371.
Cushman, R., and Van der Meer, J. C. (1990). The Hamiltonian hopf bifurcation in the lagrange top. In Albert, C. (ed.), Géométric Symplectique et Mécanique, La Grande Motte 1988, LNM, Vol. 1416, Springer-Verlag, Berlin, pp. 26–38.
Cushman, R., Ferrer, S., and Hanßmann, H. (1999). Singular reduction of axially symmetric perturbations of the isotropic harmonic oscillator. Nonlinearity 12, 389–410.
Farrelly, D., Humpherys, J., and Uzer, T. (1994). Normalization of resonant Hamiltonians. In Seimenis, J. (ed.), Hamiltonian Mechanics, Integrability and Chaotic Behavior, Torun 1993, Plenum, New York, pp. 237–244.
Ferrer, S., Lara, M., Palacián, J., San Juan, J. F., Viartola, A., and Yanguas, P. (1998). The Hénon and Heiles problem in three dimensions. I. Periodic orbits near the origin. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 8, 1199–1213.
Ferrer, S., Lara, M., Palacián, J., San Juan, J. F., Viartola, A., and Yanguas, P. (1998). The Hénon and Heiles problem in three dimensions. II. Relative equilibria and bifurcations in the reduced system. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 8, 1215–1229.
Ferrer, S., Palacián, J., and Yanguas, P. (2000). Hamiltonian oscillators in 1:1:1 resonance: Normalization and integrability. J. Nonlinear Sci. 10, 145–174.
Ferrer, S., Hanßmann, H., Palacián, J., and Yanguas, P. (2002). On perturbed oscillators in 1:1:1 resonance: The case of axially symmetric cubic potentials. J. Geom. Phys. 40, 320–369.
Fradkin, D. M. (1965). Three-dimensional isotropic harmonic oscillator and SU 3. Amer. J. Phys. A 33, 207–211.
Hanßmann, H., and Sommer, B. (2001). A degenerate bifurcation in the Hénon-Heiles family. Celestial Mech. Dynam. Astronom. 81, 249–261.
Hénon, M., and Heiles, C. (1964). The applicability of the third integral of motion: Some numerical experiments. Astron. J. 69, 73–79.
Kummer, M. (1976). On resonant nonlinearly coupled oscillators with two equal frequencies. Comm. Math. Phys. 48, 53–79.
Van der Meer, J. C. (1985). The Hamiltonian Hopf Bifurcation, LNM, Vol. 1160, Springer-Verlag, Berlin.
Van der Meer, J. C. (1990). Hamiltonian Hopf bifurcation with symmetry. Nonlinearity 3, 1041–1056.
Van der Meer, J. C. (1996). Degenerate Hamiltonian Hopf bifurcations. Fields Inst. Commun. 8, 159–176.
Miller, B. R. (1991). The lissajous transformation-III. parametric bifurcations. Celestial Mech. Dynam. Astronom. 51, 251–270.
Moser, J. (1970). Regularization of Kepler's problem and the averaging method on a manifold. Comm. Pure Appl. Math. 23, 609–636.
Verhulst, F. (1979). Discrete symmetric dynamical systems at the main resonances with applications to axi-symmetric galaxies. Philos. Trans. Roy. Soc. London Ser. A 290, 435–465.
Yanguas, P. (1998). Integrability, Normalization and Symmetries of Hamiltonian Systems in 1:1:1 Resonance. Ph.D. thesis, Universidad Pu´ blica de Navarra.
De Zeeuw, T. (1985). Motion in the core of a triaxial potential. Mon. Not. R. Astr. Soc. 215, 731–760.
De Zeeuw, T., and Franx, M. (1991). Structure and dynamics of elliptical galaxies. Annu. Rev. Astron. Astrophys. 29, 239–274.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hanßmann, H., van der Meer, JC. On the Hamiltonian Hopf Bifurcations in the 3D Hénon–Heiles Family. Journal of Dynamics and Differential Equations 14, 675–695 (2002). https://doi.org/10.1023/A:1016343317119
Issue Date:
DOI: https://doi.org/10.1023/A:1016343317119