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Newtonian dynamics in the plane corresponding to straight and cyclic motions on the hyperelliptic curve \(\mu^{2}= \nu^{n}-1, n \in \mathbb Z\): ergodicity, isochrony and fractals. (English) Zbl 1126.37040

Summary: We study the complexification of the one-dimensional Newtonian particle in a monomial potential. We discuss two classes of motions on the associated Riemann surface: the rectilinear and the cyclic motions, corresponding to two different classes of real and autonomous Newtonian dynamics in the plane. The rectilinear motion has been studied in a number of papers, while the cyclic motion is much less understood. For small data, the cyclic time trajectories lead to isochronous dynamics. For bigger data the situation is quite complicated; computer experiments show that, for sufficiently small degree of the monomial, the motion is generically isochronous with integer period, which depends in a quite sensitive way on the initial data. If the degree of the monomial is sufficiently high, computer experiments show essentially chaotic behavior. We suggest a possible theoretical explanation of these different behaviors. We also introduce a two-parameter family of two-dimensional mappings, describing the motion of the center of the circle, as a convenient representation of the cyclic dynamics; we call such a mapping the center map. Computer experiments for the center map show a typical multifractal behavior with periodicity islands. Therefore the above complexification procedure generates dynamics amenable to analytic treatment and possessing a high degree of complexity.

MSC:

37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
70F10 \(n\)-body problems
28A80 Fractals
37E99 Low-dimensional dynamical systems
70E55 Dynamics of multibody systems
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
37A25 Ergodicity, mixing, rates of mixing
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[1] Calogero, F., A class of integrable hamiltonian systems whose solutions are (perhaps) all periodic, J. Math. Phys., 38, 5711-5719 (1997) · Zbl 0892.58038
[2] Calogero, F., (Classical Many-Body Problems Amenable to Exact Treatment. Classical Many-Body Problems Amenable to Exact Treatment, Lecture Notes in Physics Monograph, vol. 66 (2001), Springer: Springer Berlin)
[3] Calogero, F., Partially superintegrable (indeed isochronous) systems are not rare, (Shabat, A. B.; Gonzalez-Lopez, A.; Manas, M.; Martinez Alonso, L.; Rodriguez, M. A., New trends in Integrability and Partial Solvability (Proceedings of the NATO Advanced Research Workshop held in Cadiz, Spain, 2-6 June 2002). New trends in Integrability and Partial Solvability (Proceedings of the NATO Advanced Research Workshop held in Cadiz, Spain, 2-6 June 2002), NATO Science Series, II. Mathematics, Physics and Chemistry, vol. 132 (2004), Kluwer), 49-77 · Zbl 1069.70009
[4] Calogero, F., Periodic solutions of a system of complex ODEs, Phys. Lett. A, 293, 146-150 (2002) · Zbl 0991.34040
[5] Calogero, F., Isochronous dynamical systems, Appl. Anal., 85, 5-22 (2006) · Zbl 1134.37321
[6] Calogero, F.; Francoise, J. P., Periodic motions galore: How to modify nonlinear evolution equations so that they feature a lot of periodic solutions, J. Nonlinear Math. Phys., 9, 99-125 (2002) · Zbl 0996.35003
[7] Calogero, F.; Francoise, J. P.; Sommacal, M., Periodic solutions of a many-rotator problem in the plane. II. Analysis of various motions, J. Nonlinear Math. Phys., 10, 157-214 (2003) · Zbl 1114.70329
[8] Calogero, F.; Gomez-Ullate, D.; Santini, P. M.; Sommacal, M., The transition from regular to irregular motion as travel on Riemann surfaces, J. Phys. A, 38, 8873-8896 (2005) · Zbl 1157.70309
[9] Calogero, F.; Sommacal, M., Periodic solutions of a system of complex ODEs. II. Higher periods, J. Nonlinear Math. Phys., 9, 483-516 (2002) · Zbl 1020.34037
[10] Dubrovin, B. A.; Fomenko, A. T.; Novikov, S. P., (Modern Geometry—Methods and Applications: Part 1. The Geometry of Surfaces, Transformation Groups, and Fields. Modern Geometry—Methods and Applications: Part 1. The Geometry of Surfaces, Transformation Groups, and Fields, Graduate Texts in Mathematics, vol. 93 (1992), Springer-Verlag) · Zbl 0751.53001
[11] Dynnikov, I. A.; Novikov, S. P., Topology of quasiperiodic functions on the plane, Russian Math. Surveys, 60, 1, 1-26 (2005) · Zbl 1148.37043
[12] Y. Fedorov, D. Gomez-Ullate, Dynamical systems on infinitely sheeted Riemann surfaces, Physica D: Nonlinear Phenom. 227 (2) 120-134; Y. Fedorov, D. Gomez-Ullate, Dynamical systems on infinitely sheeted Riemann surfaces, Physica D: Nonlinear Phenom. 227 (2) 120-134 · Zbl 1130.37384
[13] Flaschka, H., A remark on integrable Hamiltonian systems, Phys. Lett. A, 131, 9, 505-508 (1988)
[14] E. Induti, Sul moto nel piano complesso di particelle attratte verso l’origine con forze lineari ed interagenti a due corpi con forze proporzionali ad una potenza intera, negativa, dispari della loro distanza; Tesi di Laurea, Università di Roma “La Sapienza”, Dipartimento di Fisica, Anno Accademico 2003-04. Advisor: Francesco Calogero; E. Induti, Sul moto nel piano complesso di particelle attratte verso l’origine con forze lineari ed interagenti a due corpi con forze proporzionali ad una potenza intera, negativa, dispari della loro distanza; Tesi di Laurea, Università di Roma “La Sapienza”, Dipartimento di Fisica, Anno Accademico 2003-04. Advisor: Francesco Calogero
[15] Kontsevich, M., Lyapunov exponents and Hodge theory. The mathematical beauty of physics (Saclay, 1996), (in Honor of C. Itzykson), (Adv. Ser. Math. Phys., vol. 24 (1997), World Sci. Publishing: World Sci. Publishing River Edge, NJ), 318-332 · Zbl 1058.37508
[16] Gambier, B., Sur les equations differentielles du second ordre et du premier degré dont l’integrale generale est à points critiques fixes, Acta Math., 33, 1-55 (1910) · JFM 40.0377.02
[17] Maltsev, A.; Novikov, S., Dynamical systems, topology and conductivity in normal metals, J. Statist. Phys., 115, 1, 31-46 (2004) · Zbl 1157.82403
[18] Masur, H., Interval exchange transformations and measured foliations, Ann. of Math., 115, 169-200 (1982) · Zbl 0497.28012
[19] Novikov, S. P., The Hamiltonian formalism and many-valued analogs of the Morse theory, Russian Math. Surveys, 37, 5, 1-56 (1982) · Zbl 0571.58011
[20] Novikov, S. P., Topology of the generic Hamiltonian foliations on the Riemann surface · Zbl 1109.37036
[21] Strocchi, F., Complex coordinates and quantum mechanics, Rev. Modern Phys., 38, 36-40 (1966) · Zbl 0137.18303
[22] Abenda, S.; Marinakis, V.; Bountis, T., On the connection between hyperelliptic separability and Painlevé integrability, J. Phys. A, 34, 3521-3539 (2001) · Zbl 1156.34300
[23] Taimanov, I. A., An example of jump from chaos to integrability in magnetic geodesic flows, Math. Notes, 76, 587-589 (2004) · Zbl 1068.37019
[24] Veech, W. A., Gauss measures for transformations on the space of interval exchange maps, Ann. of Math., 115, 201-242 (1982) · Zbl 0486.28014
[25] Costin, R. D., Integrability properties of a generalized Lamé equation; applications to the Hénon-Heiles system, Methods Appl. Anal., 4, 113-123 (1997) · Zbl 0897.34004
[26] Zorich, A., Flat surfaces. Frontiers in number theory, physics and geometry, (Proceedings of Les Houches Winter School-2003 (2006), Springer-Verlag), 439-586
[27] Zorich, A., Deviation for interval exchange transformations, Ergodic Theory Dynam. Syst., 17, 1477-1499 (1997) · Zbl 0958.37002
[28] Zorich, A., (Asymptotic Flag of an Orientable Measured Foliation on a Surface. Asymptotic Flag of an Orientable Measured Foliation on a Surface, Geometric Study of Foliations (1994), World Scientific Pb. Co.), 479-498
[29] Veech, W. A., Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. Math., 97, 3, 553-583 (1989) · Zbl 0676.32006
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