Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 7 (2011), 042, 20 pages      arXiv:1011.6056
Contribution to the Proceedings of the Workshop “Supersymmetric Quantum Mechanics and Spectral Design”

Potentials Unbounded Below

Thomas Curtright a, b
a) CERN, CH-1211 Geneva 23, Switzerland
b) Department of Physics, University of Miami, Coral Gables, FL 33124-8046, USA

Received December 21, 2010, in final form March 27, 2011; Published online April 26, 2011

Continuous interpolates are described for classical dynamical systems defined by discrete time-steps. Functional conjugation methods play a central role in obtaining the interpolations. The interpolates correspond to particle motion in an underlying potential, V. Typically, V has no lower bound and can exhibit switchbacks wherein V changes form when turning points are encountered by the particle. The Beverton-Holt and Skellam models of population dynamics, and particular cases of the logistic map are used to illustrate these features.

Key words: classical dynamical systems; functional conjugation methods; Beverton-Holt model; Skellam model.

pdf (626 kb)   tex (495 kb)


  1. Beverton R.J.H., Holt S.J., On the dynamics of exploited fish populations, Fishery Investigations Series II, Vol. XIX, Ministry of Agriculture, Fisheries and Food, 1957.
  2. Collet P., Eckmann J.-P., Iterated maps on the interval as dynamical systems, Progress in Physics, Vol. 1, Birkhäuser, Boston, Mass., 1980.
  3. Curtright T., Veitia A., Logistic map potentials, Phys. Lett. A 375 (2011), 276-282, arXiv:1005.5030.
  4. Curtright T., Zachos C., Evolution profiles and functional equations, J. Phys. A: Math. Theor. 42 (2009), 485208, 16 pages, arXiv:0909.2424.
  5. Curtright T., Zachos C., Chaotic maps, hamiltonian flows and holographic methods, J. Phys. A: Math. Theor. 43 (2010), 445101, 15 pages, arXiv:1002.0104.
  6. Curtright T., Zachos C., Renormalization group functional equations, Phys. Rev. D 83 (2011), 065019, 17 pages, arXiv:1010.5174.
  7. Devaney R.L., An introduction to chaotic dynamical systems, 2nd ed., Addison-Wesley Studies in Nonlinearity, Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1989.
  8. Erdös P., Jabotinsky E., On analytic iteration, J. Analyse Math. 8 (1960), 361-376.
  9. Feigenbaum M.J., Quantitative universality for a class of nonlinear transformations, J. Statist. Phys. 19 (1978), 25-52.
  10. Geritz S.A.H., Kisdi E., On the mechanistic underpinning of discrete-time population models with complex dynamics, J. Theor. Biology 228 (2004), 261-269.
  11. Goldstone J., Field theories with "superconductor" solutions, Nuovo Cimento 19 (1961), 154-164.
  12. Julia G., Mémoire sur l'itération des fonctions rationnelles, Journ. de Math. (8) 1 (1918), 47-245.
  13. Kuczma M., Choczewski B., Ger R., Iterative functional equations, Encyclopedia of Mathematics and its Applications, Vol. 32, Cambridge University Press, Cambridge, 1990.
  14. Nambu Y., Quasi-particles and gauge invariance in the theory of superconductivity, Phys. Rev. 117 (1960), 648-663.
  15. Patrascioiu A., Classical Euclidean solutions, Phys. Rev. D 15 (1977), 3051-3053.
  16. Poincaré H., Sur une classe étendue de transcendantes uniformes, C.R. Acad. Sci. Paris 103 (1886), 862-864.
  17. Schröder E., Über iterierte Funktionen, Math. Ann. 3 (1870), 296-322.
  18. Skellam J.G., Random dispersal in theoretical populations, Biometrika 38 (1951), 196-218.

Previous article   Next article   Contents of Volume 7 (2011)