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Order and chaos

  • Part I Galactic Dynamics
  • Conference paper
  • First Online:
Galactic Dynamics and N-Body Simulations

Part of the book series: Lecture Notes in Physics ((LNP,volume 433))

Abstract

We review integrable and chaotic systems and the transition from Order to Chaos. We give the various types of orbits in Stiickel potentials of 2 and 3 degrees of freedom. Then we discuss integrable systems in general. In particular we consider Li ouville systems and the Painlevé test of integrability. Formal integrals (like the “third” integral) are an important tool in galactic dynamics. We discuss their construction, and the limits of their applicability by means of Nehoroshev's theory. Then we enumerate the various routes to chaos in dissipative and conservative systems. The creation of large degree of chaos follows the destruction of the last KAM (Kolmogorov, Arnold, Moser) torus and the formation of cantori. We discuss -the structure of the chaotic regions, that are filled with the asymptotic curves of the unstable periodic orbits. In systems of 3 degrees of freedom we have 3 main new phenomena: Complex instability, Collisions of bifurcations, and Arnold diffusion. Finally we discuss applications of the theory of fractals on the forms of the orbits.

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References

  • Ablowitz, M.J., Ramani, A. and Segur, H.: J. Math. Phys. 21 715 (1980)

    Google Scholar 

  • Arnold, V.I.: Russian Math. Surveys 18 No 5 9 and No 6 85 (1963)

    Google Scholar 

  • Arnold, V.I.: Sov. Math. Dokl. 5 581 (1964)

    Google Scholar 

  • Arnold, V.I. and Avez, A.: Ergodic Problems of Classical Mechanics, N. York: Benjamin (1968)

    Google Scholar 

  • Bishop, J. L.: Astrophys. J. 322 618 (1987)

    Google Scholar 

  • Benettin, G., Cercignani, C., Galgani, L. and Giorgilli, A.: Lett. Nuovo Cim. 29 1 (1980)

    Google Scholar 

  • Bleher, S. Grebogi, C., Ott, E. and Brown, R.: Phys. Rev. A38 930 (1988)

    Google Scholar 

  • Bountis, A., Segur, H. and Vivaldi, F.: Phys. Rev. A25 1257 (1982)

    Google Scholar 

  • Celetti, A. and Chierchia, L.: Commun. Math. Phys. 118 119 (1988)

    Google Scholar 

  • Charlier, C.L.: Die Mechanik des Himmels (Leipzig-von Veit) (1902)

    Google Scholar 

  • Chernikov, A.A., Sagdeev, R.Z. and Zaslavsky, G.M.: Physica D33 65 (1988)

    Google Scholar 

  • Chirikov, B.V.Z: Phys. Rep. 52 263 (1979)

    Google Scholar 

  • Chirikov, B.V.: Chaos, Solitons and Fractals 1 79 (1991)

    Google Scholar 

  • Contopoulos, G.: Z. Astrophys. 49 273 (1960)

    Google Scholar 

  • Contopoulos, G.: Astron. J. 68 1 (1963a)

    Google Scholar 

  • Contopoulos, G.: Astron. J. 68 763 (1963b)

    Google Scholar 

  • Contopoulos, G.: In “Les Nouvelles Méthodes de la Dynamique Stellaire”, ed. by Hénon, M., and Nahon, F., CNRS, Paris, and Bull. Astron., 3e Ser. 2 Fasc. 1, 223 (1967)

    Google Scholar 

  • Contopoulos, G.: Astrophys. J. 160 113 (1970)

    Google Scholar 

  • Contopoulos, G.: Astron. J. 76 147 (1971)

    Google Scholar 

  • Contopoulos, G.: Astrophys. J. 181 657 (1973)

    Google Scholar 

  • Contopoulos, G.: Astron. Astrophys. 81 198 (1980)

    Google Scholar 

  • Contopoulos, G.: Astrophys. J. 275 511; Physica D8 142 (1983a)

    Google Scholar 

  • Contopoulos, G.: Lett. Nuovo Cim. 37 149 (1983b)

    Google Scholar 

  • Contopoulos, G.: Celest. Mech. 38 1 (1986a)

    Google Scholar 

  • Contopoulos, G.: Astron. Astrophys. 161 244 (1986b)

    Google Scholar 

  • Contopoulos, G.: Celest. Mech. 42 239 (1988)

    Google Scholar 

  • Contopoulos, G.: Astron. Astrophys. 231 41 (1990a)

    Google Scholar 

  • Contopoulos, G.: Proc. R. Soc. Lond. A431 183 (1990b)

    Google Scholar 

  • Contopoulos, G.: Proc. R. Soc. Lond. A435 951 (1991)

    Google Scholar 

  • Contopoulos, G.: Celest. Mech. Dyn. Astron. 50 291 (1991)

    Google Scholar 

  • Contopoulos, G. and Barbanis, B.: Astron. Astrophys. 227 329 (1989)

    Google Scholar 

  • Contopoulos, G. and Giorgilli, A.: Meccanica 23 19 (1988)

    Google Scholar 

  • Contopoulos, G. and Kaufmann, D.: Astron. Astrophys. 253 379 (1992)

    Google Scholar 

  • Contopoulos, G. and Magnenat, D.: Celest. Mech. 37 387 (1985)

    Google Scholar 

  • Contopoulos, G. and Moutsoulas, M.: Astron. J. 70 817 (1965)

    Google Scholar 

  • Contopoulos, G. and Moutsoulas, M.: Astron. J. 71 687 (1966)

    Google Scholar 

  • Contopoulos, G. and Polymilis, C.: Physica D24 328 (1987)

    Google Scholar 

  • Contopoulos, G. and Polymilis, C.: Phys. Rev. E47 1546 (1993)

    Google Scholar 

  • Contopoulos, G. and Seimenis, J.: Astron. Astrophys. 227 49 (1990)

    Google Scholar 

  • Contopoulos, G. and Vandervoort, P.: Astrophys. J. 389 118 (1992)

    Google Scholar 

  • Contopoulos, G. and Zikides, M.: Astron. Astrophys. 90 198 (1980)

    Google Scholar 

  • Contopoulos, G., Kandrup, H.E. and Kaufmann, D.: Physica D64 310 (1993)

    Google Scholar 

  • Contopoulos, G., Varvoglis, H. and Barbanis, B.: Astron. Astrophys. 172 55 (1987)

    Google Scholar 

  • Coullet, P. and Tresser, C.: J. de Physique 39 C5 (1978)

    Google Scholar 

  • de Zeeuw, P.T.: Mon. Not. R. astr. Soc. 216 273 (1985)

    Google Scholar 

  • de Zeeuw, P.T.: Ann. N. York Acad. Sci. 536 15 (1988)

    Google Scholar 

  • de Zeeuw, P.T. and Hunter, C.: Astrophys. J. 356 365 (1990)

    Google Scholar 

  • de Zeeuw, P.T. and Lynden-Bell, D.: Mon. Not. R. astr. Soc. 215 713 (1985)

    Google Scholar 

  • de Zeeuw, P.T., Hunter, C. and Schwarzchild, M.: Astrophys. J. 317 607 (1987)

    Google Scholar 

  • Dumas, H.S. and Laskar, J.: Phys. Rev. Lett. 70 2975 (1993)

    Google Scholar 

  • Eckmann, J.P.: Rev. Mod. Phys. 53 643 (1981)

    Google Scholar 

  • Eckmann, J.P. and Ruelle, D.: Rev. Mod. Phys. 57 617 (1985)

    Google Scholar 

  • Eddington, A.S.: Mon. Not. R. astr. Soc. 76 37 (1915)

    Google Scholar 

  • Feigenbaum, M.: J. Stat. Phys. 19 25 (1978)

    Google Scholar 

  • Flashka, H.: Phys. Rev. B9 1924 (1974)

    Google Scholar 

  • Freeman, K.C.: Mon. Not. R. astr. Soc. 133 47; 134 1;134 15 (1966)

    Google Scholar 

  • Gaspard, P. and Rice, S.A.: J. Chem. Phys. 90 2225 (1989)

    Google Scholar 

  • Giorgilli, A.: Computer Phys. Comm. 16 331 (1979)

    Google Scholar 

  • Giorgilli, A.: Ann. Inst. H. Poincaré 48 No 4, 423 (1988)

    Google Scholar 

  • Giorgilli, A.: In “Les Méthodes Modernes de la Mécanique Céleste”, ed. by Froeschlé, C. and Benest, D., Ed. Frontiéres (1990)

    Google Scholar 

  • Giorgilli, A., Casati, D., Sironi, L. and Galgani, L.: Phys. Lett. A115 202 (1986)

    Google Scholar 

  • Grassberger, P.: Phys. Lett. A97 224 (1983)

    Google Scholar 

  • Grassberger, P. and Procaccia, I.: Physica D9 189 (1983)

    Google Scholar 

  • Greene, J.M.: J. Math. Phys. 20 1183 (1969)

    Google Scholar 

  • Greene, J.M., MacKay, R.S., Vivaldi, F. and Feigenbaum, M.J.: Physica D3 468 (1981)

    Google Scholar 

  • Grossman, S. and Thomae, S.: Z. Naturforshung A32 1353 (1977)

    Google Scholar 

  • Gustavson, F.G.: Astron. J. 71 670 (1966)

    Google Scholar 

  • Heggie, D.C.: Celest. Mech. 35 357 (1985)

    Google Scholar 

  • Hénon, M.: Ann. Astroph. 28 992 (1965)

    Google Scholar 

  • Hénon, M.: Phys. Rev. B9 1921 (1974)

    Google Scholar 

  • Hénon, M. and Heiles, C.: Astron. J. 69 73 (1964)

    Google Scholar 

  • Henrard, J.: Celest. Mech. Dyn. Astr. 49 43 (1990)

    Google Scholar 

  • Hietarinta, J.: Phys. Rep. 147 87 (1987)

    Google Scholar 

  • Hunter, C.: Ann. N. York Acad. Sci. 536 25 (1988)

    Google Scholar 

  • Krein, M.G.: Dokl. Akad. Nauk. SSSR 73 445 (1950)

    Google Scholar 

  • Kuzmin, G.G.: Astron. Zh. 33 27 (1956)

    Google Scholar 

  • Li, W. and Bak, P.: Phys. Rev. Lett. 57 655 (1986)

    Google Scholar 

  • Lichtenberg, A.J. and Lieberman, M.A.: Regular and Stochastic Motion, New York: Springer Verlag (1983)

    Google Scholar 

  • Laskar, J.: Physica D67 257 (1993)

    Google Scholar 

  • Laskar, J., Froeschlé, C. and Celetti, A.: Physica D56 252 (1992)

    Google Scholar 

  • Lynden-Bell, D.: Mon. Not. R. astr. Soc. 123 447 (1961)

    Google Scholar 

  • Lynden-Bell, D.: Mon. Not. R. astr. Soc. 124 95 (1962)

    Google Scholar 

  • MacKay, R.S.: J. Phys. (Lett.) A20 559 (1987)

    Google Scholar 

  • Mandelbrot, B.B.: Fractals: Form, Chance and Dimension, San Francisco: Freeman (1977)

    Google Scholar 

  • Mandelbrot, B.B.: The Fractal Geometry of Nature, San Fransisco: Freeman (1982)

    Google Scholar 

  • Martinet, L. and Pfenniger, D.: Astron. Astrophys. 173 81 (1987)

    Google Scholar 

  • Morbidelli, A. and Henrard, J.: Celest. Mech. Dyn. Astr. 51 131 (1991)

    Google Scholar 

  • Morbidelli, A.: Celest. Mech. Dyn. Astr. 55 101 (1993)

    Google Scholar 

  • Morbidelli, A. and Giorgilli, A.: Celest. Mech. Dyn. Astr. 55 131 (1993)

    Google Scholar 

  • Moser, J.: Comm. Pure Appl. Math. 11 81 (1958)

    Google Scholar 

  • Moser, J.: Nachr. Akad. Wiss. Gottingen, No 1 (1962)

    Google Scholar 

  • Nehoroshev, N.N.: Russian Math. Surveys 32 No 6, 1 (1977)

    Google Scholar 

  • Newhouse, S., Ruelle, D. and Takens, F.: Comm. Math. Phys. 64 35 (1978)

    Google Scholar 

  • Papadaki, H., Contopoulos, G. and Polymilis, C.: In “From Newton to Chaos”, ed. by Roy, A., N. York: Plenum Press, (in press) (1993)

    Google Scholar 

  • Patsis, P.A. and Zachilas, L.: Astron. Astrophys. 227 37 (1990)

    Google Scholar 

  • Pfenniger, D.: Astron. Astrophys. 150 97, 112 (1985)

    Google Scholar 

  • Pinotsis, A.: In “Long Term Behaviour of Natural and Artificial N-Body Systems”, ed. by Roy, A. Dordrecht; Kluwer, p. 465 (1991)

    Google Scholar 

  • Poincaré, H.: Les Méthodes Nouvelles de la Mécanique Céleste, Vols 1, 2, 3, Paris: Gauthier Villars (1892)

    Google Scholar 

  • Pomeau, Y. and Manneville, P.: Comm. Math. Phys. 74 189 (1980)

    Google Scholar 

  • Prendergast, K.H.: Lecture Notes in Mathematics 925, Berlin: Springer Verlag (1982)

    Google Scholar 

  • Ramani, A., Grammaticos, B. and Bountis, T.: Phys. Rep. 180 159 (1989)

    Google Scholar 

  • Rosenbluth, M.N., Sagdeev, R.A., Taylor, J.B. and Zaslavskii, M.: Nucl. Fusion 6 217 (1966)

    Google Scholar 

  • Russel, D.A., Hanson, J.D. and Ott, E.: Phys. Rev. Lett. 45 1175 (1980)

    Google Scholar 

  • Shirts, R.B. and Reinhardt, W.P.: J. Chem. Phys. 77 5204 (1982)

    Google Scholar 

  • Shenker, S. J. and Kadanoff, L.P.: J. Stat. Phys. 27 631 (1982)

    Google Scholar 

  • Stäckel, P.: Math. Ann. 35 91 (1890)

    Google Scholar 

  • Stäckel, P.: Math. Ann. 42 537 (1893)

    Google Scholar 

  • Statler, T.S.: Astrophys. J. 321 13 (1987)

    Google Scholar 

  • Tel, T.: Z. Naturforshung A43 1154 (1988)

    Google Scholar 

  • Vandervoort, P.: Astrophys. J. 232 91 (1979)

    Google Scholar 

  • Vandervoort, P.: Astrophys. J. 287 475 (1984)

    Google Scholar 

  • Weinacht, J.: Mon. Not. R. astr. Soc. 91 279 (1924)

    Google Scholar 

  • Whittaker, E.T.: A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, Cambridge: Cambridge Univ. Press, (Dover Publications 1944) (1904)

    Google Scholar 

  • Yoshida, H.: Comm. Math. Phys. 116 529 (1988)

    Google Scholar 

  • Zachilas, L.: Astron. Astrophys. Suppl. 97 549 (1993)

    Google Scholar 

  • Zaslavskii, G.M.: Chaos in Dynamic Systems, New York: Harwood (1985)

    Google Scholar 

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Authors and Affiliations

  1. Astronomy Department, University of Athens, Greece

    G. Contopoulos

  2. Astronomy Department, University of Florida, Gainesville, Fl., USA

    G. Contopoulos

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  1. G. Contopoulos
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G. Contopoulos N. K. Spyrou L. Vlahos

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© 1994 Springer-Verlag

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Contopoulos, G. (1994). Order and chaos. In: Contopoulos, G., Spyrou, N.K., Vlahos, L. (eds) Galactic Dynamics and N-Body Simulations. Lecture Notes in Physics, vol 433. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-57983-4_18

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  • DOI: https://doi.org/10.1007/3-540-57983-4_18

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  • Print ISBN: 978-3-540-57983-0

  • Online ISBN: 978-3-540-48409-7

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