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Potentials Unbounded Below
Symmetry, integrability and geometry: methods and applications, Tome 7 (2011) Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: classical dynamical systems; functional conjugation methods; Beverton–Holt model; Skellam model. 
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Thomas Curtright. Potentials Unbounded Below. Symmetry, integrability and geometry: methods and applications, Tome 7 (2011). http://geodesic.mathdoc.fr/item/SIGMA_2011_7_a41/

[1] Beverton R. J. H., Holt S. J., On the dynamics of exploited fish populations, Fishery Investigations Series II, 19, Ministry of Agriculture, Fisheries and Food, 1957

[2] Collet P., Eckmann J.-P., Iterated maps on the interval as dynamical systems, v. 1, Progress in Physics, Birkhäuser, Boston, Mass., 1980 | MR | Zbl

[3] Curtright T., Veitia A., “Logistic map potentials”, Phys. Lett. A, 375 (2011), 276–282, arXiv: 1005.5030 | DOI | MR

[4] Curtright T., Zachos C., “Evolution profiles and functional equations”, J. Phys. A: Math. Theor., 42 (2009), 485208, 16 pp., arXiv: 0909.2424 | DOI | MR | Zbl

[5] Curtright T., Zachos C., “Chaotic maps, hamiltonian flows and holographic methods”, J. Phys. A: Math. Theor., 43 (2010), 445101, 15 pp., arXiv: 1002.0104 | DOI | MR | Zbl

[6] Curtright T., Zachos C., “Renormalization group functional equations”, Phys. Rev. D, 83 (2011), 065019, 17 pp., arXiv: 1010.5174 | DOI

[7] Devaney R. L., An introduction to chaotic dynamical systems, Addison-Wesley Studies in Nonlinearity, 2nd ed., Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1989 | MR | Zbl

[8] Erdös P., Jabotinsky E., “On analytic iteration”, J. Analyse Math., 8 (1960), 361–376 | DOI | MR

[9] Feigenbaum M. J., “Quantitative universality for a class of nonlinear transformations”, J. Statist. Phys., 19 (1978), 25–52 | DOI | MR | Zbl

[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 | DOI | MR

[11] Goldstone J., “Field theories with “superconductor” solutions”, Nuovo Cimento, 19 (1961), 154–164 | DOI | MR | Zbl

[12] Julia G., “Mémoire sur l'itération des fonctions rationnelles”, Journ. de Math. (8), 1 (1918), 47–245 | Zbl

[13] Kuczma M., Choczewski B., Ger R., Iterative functional equations, Encyclopedia of Mathematics and its Applications, 32, Cambridge University Press, Cambridge, 1990 | MR | Zbl

[14] Nambu Y., “Quasi-particles and gauge invariance in the theory of superconductivity”, Phys. Rev., 117 (1960), 648–663 | DOI | MR

[15] Patrascioiu A., “Classical Euclidean solutions”, Phys. Rev. D, 15 (1977), 3051–3053 | DOI

[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 | DOI | MR

[18] Skellam J. G., “Random dispersal in theoretical populations”, Biometrika, 38 (1951), 196–218 | DOI | MR | Zbl