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Volumn 232, Issue 1, 2007, Pages 22-32

Newtonian dynamics in the plane corresponding to straight and cyclic motions on the hyperelliptic curve μ2 = νn - 1, n ∈ Z: Ergodicity, isochrony and fractals

Author keywords

Completely integrable system; Complex time dynamics; Fractal; Transition to chaos

Indexed keywords

COMPLEXATION; FRACTALS; NEWTONIAN LIQUIDS; ONE DIMENSIONAL;

EID: 34347406715     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.physd.2007.05.002     Document Type: Article
Times cited : (21)

References (41)
  • 1
    • 0031518164 scopus 로고    scopus 로고
    • A class of integrable hamiltonian systems whose solutions are (perhaps) all periodic
    • Calogero F. A class of integrable hamiltonian systems whose solutions are (perhaps) all periodic. J. Math. Phys. 38 (1997) 5711-5719
    • (1997) J. Math. Phys. , vol.38 , pp. 5711-5719
    • Calogero, F.1
  • 3
    • 25844527602 scopus 로고    scopus 로고
    • Partially superintegrable (indeed isochronous) systems are not rare
    • New trends in Integrability and Partial Solvability (Proceedings of the NATO Advanced Research Workshop held in Cadiz, Spain, 2-6 June 2002). Shabat A.B., Gonzalez-Lopez A., Manas M., Martinez Alonso L., and Rodriguez M.A. (Eds), Kluwer
    • Calogero F. Partially superintegrable (indeed isochronous) systems are not rare. In: Shabat A.B., Gonzalez-Lopez A., Manas M., Martinez Alonso L., and Rodriguez M.A. (Eds). 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
    • (2004) NATO Science Series, II. Mathematics, Physics and Chemistry , vol.132 , pp. 49-77
    • Calogero, F.1
  • 4
    • 0037185453 scopus 로고    scopus 로고
    • Periodic solutions of a system of complex ODEs
    • Calogero F. Periodic solutions of a system of complex ODEs. Phys. Lett. A 293 (2002) 146-150
    • (2002) Phys. Lett. A , vol.293 , pp. 146-150
    • Calogero, F.1
  • 5
    • 33646418460 scopus 로고    scopus 로고
    • Isochronous dynamical systems
    • Calogero F. Isochronous dynamical systems. Appl. Anal. 85 (2006) 5-22
    • (2006) Appl. Anal. , vol.85 , pp. 5-22
    • Calogero, F.1
  • 6
    • 0035994828 scopus 로고    scopus 로고
    • Periodic motions galore: How to modify nonlinear evolution equations so that they feature a lot of periodic solutions
    • Calogero F., and 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 (2002) 99-125
    • (2002) J. Nonlinear Math. Phys. , vol.9 , pp. 99-125
    • Calogero, F.1    Francoise, J.P.2
  • 7
    • 0038391419 scopus 로고    scopus 로고
    • Periodic solutions of a many-rotator problem in the plane. II. Analysis of various motions
    • Calogero F., Francoise J.P., and Sommacal M. Periodic solutions of a many-rotator problem in the plane. II. Analysis of various motions. J. Nonlinear Math. Phys. 10 (2003) 157-214
    • (2003) J. Nonlinear Math. Phys. , vol.10 , pp. 157-214
    • Calogero, F.1    Francoise, J.P.2    Sommacal, M.3
  • 8
    • 25844467896 scopus 로고    scopus 로고
    • The transition from regular to irregular motion as travel on Riemann surfaces
    • Calogero F., Gomez-Ullate D., Santini P.M., and Sommacal M. The transition from regular to irregular motion as travel on Riemann surfaces. J. Phys. A 38 (2005) 8873-8896
    • (2005) J. Phys. A , vol.38 , pp. 8873-8896
    • Calogero, F.1    Gomez-Ullate, D.2    Santini, P.M.3    Sommacal, M.4
  • 9
    • 0036436207 scopus 로고    scopus 로고
    • Periodic solutions of a system of complex ODEs. II. Higher periods
    • Calogero F., and Sommacal M. Periodic solutions of a system of complex ODEs. II. Higher periods. J. Nonlinear Math. Phys. 9 (2002) 483-516
    • (2002) J. Nonlinear Math. Phys. , vol.9 , pp. 483-516
    • Calogero, F.1    Sommacal, M.2
  • 11
    • 20444499701 scopus 로고    scopus 로고
    • Topology of quasiperiodic functions on the plane
    • Dynnikov I.A., and Novikov S.P. Topology of quasiperiodic functions on the plane. Russian Math. Surveys 60 1 (2005) 1-26
    • (2005) Russian Math. Surveys , vol.60 , Issue.1 , pp. 1-26
    • Dynnikov, I.A.1    Novikov, S.P.2
  • 12
    • 33947574729 scopus 로고    scopus 로고
    • Y. Fedorov, D. Gomez-Ullate, Dynamical systems on infinitely sheeted Riemann surfaces, Physica D: Nonlinear Phenom. 227 (2) 120-134
  • 13
    • 0001263696 scopus 로고
    • A remark on integrable Hamiltonian systems
    • Flaschka H. A remark on integrable Hamiltonian systems. Phys. Lett. A 131 9 (1988) 505-508
    • (1988) Phys. Lett. A , vol.131 , Issue.9 , pp. 505-508
    • Flaschka, H.1
  • 14
    • 34347397434 scopus 로고    scopus 로고
    • 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
    • 0009918930 scopus 로고    scopus 로고
    • Lyapunov exponents and Hodge theory. The mathematical beauty of physics (Saclay, 1996), (in Honor of C. Itzykson)
    • World Sci. Publishing, River Edge, NJ
    • 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, River Edge, NJ 318-332
    • (1997) Adv. Ser. Math. Phys. , vol.24 , pp. 318-332
    • Kontsevich, M.1
  • 16
    • 34250579023 scopus 로고
    • Sur le probleme de la rotation d'un corps solid autour d'un point fixé
    • Kowalevski S. Sur le probleme de la rotation d'un corps solid autour d'un point fixé. Acta Math. 12 (1889) 177-232
    • (1889) Acta Math. , vol.12 , pp. 177-232
    • Kowalevski, S.1
  • 17
    • 34250594266 scopus 로고
    • Sur les équations differentielles du second ordre et d'ordre superieur dont l'integrale generale est uniforme
    • Painlevé P. Sur les équations differentielles du second ordre et d'ordre superieur dont l'integrale generale est uniforme. Acta Math. 25 (1902) 1-85
    • (1902) Acta Math. , vol.25 , pp. 1-85
    • Painlevé, P.1
  • 18
    • 34250588266 scopus 로고
    • Sur les equations differentielles du second ordre et du premier degré dont l'integrale generale est à points critiques fixes
    • 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 (1910) 1-55
    • (1910) Acta Math. , vol.33 , pp. 1-55
    • Gambier, B.1
  • 19
    • 3543008379 scopus 로고    scopus 로고
    • Dynamical systems, topology and conductivity in normal metals
    • Maltsev A., and Novikov S. Dynamical systems, topology and conductivity in normal metals. J. Statist. Phys. 115 1 (2004) 31-46
    • (2004) J. Statist. Phys. , vol.115 , Issue.1 , pp. 31-46
    • Maltsev, A.1    Novikov, S.2
  • 20
    • 0000742775 scopus 로고
    • Interval exchange transformations and measured foliations
    • Masur H. Interval exchange transformations and measured foliations. Ann. of Math. 115 (1982) 169-200
    • (1982) Ann. of Math. , vol.115 , pp. 169-200
    • Masur, H.1
  • 21
    • 84956089807 scopus 로고
    • The Hamiltonian formalism and many-valued analogs of the Morse theory
    • Novikov S.P. The Hamiltonian formalism and many-valued analogs of the Morse theory. Russian Math. Surveys 37 5 (1982) 1-56
    • (1982) Russian Math. Surveys , vol.37 , Issue.5 , pp. 1-56
    • Novikov, S.P.1
  • 23
    • 33845761080 scopus 로고
    • Complex coordinates and quantum mechanics
    • Strocchi F. Complex coordinates and quantum mechanics. Rev. Modern Phys. 38 (1966) 36-40
    • (1966) Rev. Modern Phys. , vol.38 , pp. 36-40
    • Strocchi, F.1
  • 24
    • 0000894545 scopus 로고
    • Analytic structure of the Lorenz system
    • Tabor M., and Weiss J. Analytic structure of the Lorenz system. Phys. Rev. A 24 (1981) 2157
    • (1981) Phys. Rev. A , vol.24 , pp. 2157
    • Tabor, M.1    Weiss, J.2
  • 25
    • 45349111974 scopus 로고
    • The Painlevé property and singularity analysis of integrable and non-integrable systems
    • Ramani A., Grammaticos B., and Bountis T. The Painlevé property and singularity analysis of integrable and non-integrable systems. Phys. Rep. 180 (1989) 159-245
    • (1989) Phys. Rep. , vol.180 , pp. 159-245
    • Ramani, A.1    Grammaticos, B.2    Bountis, T.3
  • 26
    • 0039199410 scopus 로고
    • Non-integrable systems with algebraic singularities in complex time
    • Bountis T., Drossos L., and Percival I. Non-integrable systems with algebraic singularities in complex time. J. Phys. A 23 (1991) 3217-3236
    • (1991) J. Phys. A , vol.23 , pp. 3217-3236
    • Bountis, T.1    Drossos, L.2    Percival, I.3
  • 27
    • 0039577818 scopus 로고
    • Investigating non-integrability and chaos in complex time
    • Bountis T. Investigating non-integrability and chaos in complex time. Physica D 86 (1995) 256-267
    • (1995) Physica D , vol.86 , pp. 256-267
    • Bountis, T.1
  • 28
    • 0030164513 scopus 로고    scopus 로고
    • Order and the ubiquitous occurrence of chaos
    • Fokas A.S., and Bountis T. Order and the ubiquitous occurrence of chaos. Physica A 288 (1996) 236-244
    • (1996) Physica A , vol.288 , pp. 236-244
    • Fokas, A.S.1    Bountis, T.2
  • 29
    • 0041077766 scopus 로고    scopus 로고
    • On the connection between hyperelliptic separability and Painlevé integrability
    • Abenda S., Marinakis V., and Bountis T. On the connection between hyperelliptic separability and Painlevé integrability. J. Phys. A 34 (2001) 3521-3539
    • (2001) J. Phys. A , vol.34 , pp. 3521-3539
    • Abenda, S.1    Marinakis, V.2    Bountis, T.3
  • 30
    • 5044237364 scopus 로고    scopus 로고
    • An example of jump from chaos to integrability in magnetic geodesic flows
    • Taimanov I.A. An example of jump from chaos to integrability in magnetic geodesic flows. Math. Notes 76 (2004) 587-589
    • (2004) Math. Notes , vol.76 , pp. 587-589
    • Taimanov, I.A.1
  • 31
    • 0000742776 scopus 로고
    • Gauss measures for transformations on the space of interval exchange maps
    • Veech W.A. Gauss measures for transformations on the space of interval exchange maps. Ann. of Math. 115 (1982) 201-242
    • (1982) Ann. of Math. , vol.115 , pp. 201-242
    • Veech, W.A.1
  • 32
    • 34250146236 scopus 로고
    • Necessary condition for the existence of algebraic first integrals: I, II
    • 381-399
    • Yoshida H. Necessary condition for the existence of algebraic first integrals: I, II. Celes. Mech. 31 (1983) 363-379 381-399
    • (1983) Celes. Mech. , vol.31 , pp. 363-379
    • Yoshida, H.1
  • 33
    • 0041020915 scopus 로고
    • Branching of solutions and nonexistence of first integrals in Hamiltonian mechanics; I, II
    • Ziglin S. Branching of solutions and nonexistence of first integrals in Hamiltonian mechanics; I, II. Funct. Anal. Appl. 16 (1983) 181-189
    • (1983) Funct. Anal. Appl. , vol.16 , pp. 181-189
    • Ziglin, S.1
  • 34
    • 32244444336 scopus 로고
    • Self-intersection of the complex separatrices and the nonexistence of the integrals in the Hamiltonian systems with one-and-half degrees of freedom
    • Ziglin S. Self-intersection of the complex separatrices and the nonexistence of the integrals in the Hamiltonian systems with one-and-half degrees of freedom. J. Appl. Math. 45 (1982) 411-413
    • (1982) J. Appl. Math. , vol.45 , pp. 411-413
    • Ziglin, S.1
  • 35
    • 84988600853 scopus 로고
    • The Painlevé-Kowalewski and poly-Painlevé tests for integrability
    • Kruskal M.D., and Clarkson P. The Painlevé-Kowalewski and poly-Painlevé tests for integrability. Stud. Appl. Math. 86 (1992) 87-165
    • (1992) Stud. Appl. Math. , vol.86 , pp. 87-165
    • Kruskal, M.D.1    Clarkson, P.2
  • 36
    • 0242308939 scopus 로고    scopus 로고
    • Nonintegrability criteria for a class of differential equations with two regular singular points
    • Costin R.D., and Kruskal M.D. Nonintegrability criteria for a class of differential equations with two regular singular points. Nonlinearity 16 (2003) 1295-1317
    • (2003) Nonlinearity , vol.16 , pp. 1295-1317
    • Costin, R.D.1    Kruskal, M.D.2
  • 37
    • 0242281477 scopus 로고    scopus 로고
    • Integrability properties of a generalized Lamé equation; applications to the Hénon-Heiles system
    • Costin R.D. Integrability properties of a generalized Lamé equation; applications to the Hénon-Heiles system. Methods Appl. Anal. 4 (1997) 113-123
    • (1997) Methods Appl. Anal. , vol.4 , pp. 113-123
    • Costin, R.D.1
  • 38
    • 33745630475 scopus 로고    scopus 로고
    • Flat surfaces. Frontiers in number theory, physics and geometry
    • Springer-Verlag
    • Zorich A. Flat surfaces. Frontiers in number theory, physics and geometry. Proceedings of Les Houches Winter School-2003 (2006), Springer-Verlag 439-586
    • (2006) Proceedings of Les Houches Winter School-2003 , pp. 439-586
    • Zorich, A.1
  • 39
    • 0040601079 scopus 로고    scopus 로고
    • Deviation for interval exchange transformations
    • Zorich A. Deviation for interval exchange transformations. Ergodic Theory Dynam. Syst. 17 (1997) 1477-1499
    • (1997) Ergodic Theory Dynam. Syst. , vol.17 , pp. 1477-1499
    • Zorich, A.1
  • 40
    • 0009914833 scopus 로고
    • World Scientific Pb. Co.
    • Zorich A. Asymptotic Flag of an Orientable Measured Foliation on a Surface. Geometric Study of Foliations (1994), World Scientific Pb. Co. 479-498
    • (1994) Geometric Study of Foliations , pp. 479-498
    • Zorich, A.1
  • 41
    • 34249973241 scopus 로고
    • Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards
    • Veech W.A. Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards. Invent. Math. 97 3 (1989) 553-583
    • (1989) Invent. Math. , vol.97 , Issue.3 , pp. 553-583
    • Veech, W.A.1


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