Abstract
We use probability density functions (pdfs) of sums of orbit coordinates, over time intervals of the order of one Hubble time, to distinguish weakly from strongly chaotic orbits in a barred galaxy model. We find that, in the weakly chaotic case, quasi-stationary states arise, whose pdfs are well approximated by q-Gaussian functions (with 1 < q < 3), while strong chaos is identified by pdfs which quickly tend to Gaussians (q = 1). Typical examples of weakly chaotic orbits are those that “stick” to islands of ordered motion. Their presence in rotating galaxy models has been investigated thoroughly in recent years due to their ability to support galaxy structures for relatively long time scales. In this paper, we demonstrate, on specific orbits of 2 and 3 degree of freedom barred galaxy models, that the proposed statistical approach can distinguish weakly from strongly chaotic motion accurately and efficiently, especially in cases where Lyapunov exponents and other local dynamic indicators appear to be inconclusive.
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Antonopoulos Ch., Bountis T., Basios V.: Quasi-stationary chaotic states of multidimensional Hamiltonian systems. Phys. A 390, 3290–3307 (2011)
Arnold, V.I., Avez, A.: Problèmes Ergodiques de la Mécanique Classique. Gauthier-Villars, Paris & Benjamin, New York, 1968 (1967)
Athanassoula E., Romero-Gómez M., Masdemont J.J.: Rings and spirals in barred galaxies—I. Building blocks. Mon. Not. R. Astron. Soc. 394, 67–81 (2009a)
Athanassoula E., Romero-Gómez M., Bosma A., Masdemont J.J.: Rings and spirals in barred galaxies—II. Ring and spiral morphology. Mon. Not. R. Astron. Soc. 400, 1706–1720 (2009b)
Athanassoula E., Romero-Gómez M., Bosma A., Masdemont J.J.: Rings and spirals in barred galaxies—III. Further comparisons and links to observations. Mon. Not. R. Astron. Soc. 407, 1433–1448 (2010)
Baldovin F., Brigatti E., Tsallis C.: Quasistationary states in low-dimensional Hamiltonian systems. Phys. Lett. A 320, 254–260 (2004a)
Baldovin F., Moyano M.G., Majtey A.P., Robledo A., Tsallis C.: Ubiquity of metastable to stable crossover in weakly chaotic dynamical systems. Phys. A 340, 205–218 (2004b)
Benettin G., Galgani L., Giorgilli A., Strelcyn J.M.: Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems: A method for computing all of them. Part 1: theory. Meccanica 15, 9–20 (1980a)
Benettin G., Galgani L., Giorgilli A., Strelcyn J.M.: Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems: A method for computing all of them. Part 2: Numerical application. Meccanica 15, 21–30 (1980b)
Cachucho F., Cincotta P.M., Ferraz-Mello S.: Chirikov diffusion in the asteroidal three-body resonance (5, −2, −2). Celest. Mech. Dyn. Astron. 108, 35–58 (2010)
Cincotta P.M., Giordano C.M.: Nonlinear Phenomena Research. Nova Science Inc, Hauppauge, New York (2008)
Contopoulos G.: Order and Chaos in Dynamical Astronomy. Springer, Berlin (2002)
Contopoulos G., Harsoula M.: Stickiness in Chaos. Int. J. Bif. Chaos 18, 2929–2949 (2008)
Contopoulos G., Harsoula M.: Stickiness effects in chaos. Celest. Mech. Dyn. Astron. 107, 77–92 (2010)
Eckmann J.P., Ruelle D.: Ergodic theory of chaos and strange attractors. Rev. Mod. Phys. 57, 617–656 (1985)
Ferrers N.M.: Q. J. Pure Appl. Math. 14, 1 (1877)
Giordano C.M., Cincotta P.M.: Chaotic diffusion of orbits in systems with divided phase space. Astron. Astrophys. 423, 745–753 (2004)
Harsoula M., Kalapotharakos C.: Orbital structure in N-body models of barred-spiral galaxies. Mon. Not. R. Astron. Soc. 394, 1605–1619 (2009)
Kaufmann D.E., Contopoulos G.: Self-consistent models of barred spiral galaxies. Astron. Astrophys. 309, 381–402 (1996)
Katok A.: Liapunov exponents, entropy and periodic orbits for diffeomorphisms. Publications Mathématiques de l’IHÉS 51, 137–173 (1980)
Katsanikas M., Patsis P.A.: The structure of invariant tori in a 3D galactic potential. Int. J. Bif. Chaos 21, 467–496 (2011)
Manos, T.: PhD Thesis, Université de Provence (Aix-Marseille I), France (2008)
Manos T., Skokos Ch., Athanassoula E., Bountis T.: Studying the global dynamics of conservative dynamical systems using the SALI chaos detection method. Nonlinear Phenom. Complex Syst. 11, 171–176 (2008)
Manos T., Athanassoula E.: Dynamical study of 2D and 3D barred galaxy models. In: Contopoulos, G., Patsis, P. (eds) Chaos in Galaxies., pp. 115–122. Springer, Berlin (ASSP) (2009)
Manos T., Athanassoula E.: Regular and chaotic orbits in barsolid black galaxies - I. Applying the SALI/GALI method to explore their distribution in several models. Mon. Not. R. Astron. Soc. 415, 629–642 (2011a)
Manos, T., Athanassoula, E.: Regular and chaotic orbits in bar galaxies—II. Observable chaos. (in preparation) (2011b)
Miyamoto M., Nagai R.: Three-dimensional models for the distribution of mass in galaxies. Astron. Soc. Jpn. 27(4), 533–543 (1975)
Patsis P.A., Athanassoula E., Quillen A.C.: Orbits in the Bar of NGC 4314. Astron. Astrophys. 483, 731–744 (1997)
Patsis P.A., Skokos Ch., Athanassoula E.: Orbital dynamics of three-dimensional bars—III. Boxy/peanut edge-on profiles. Mon. Not. R. Astron. Soc. 337, 578–596 (2002)
Patsis P.A., Skokos Ch., Athanassoula E.: Orbital dynamics of three-dimensional bars—IV. Boxy isophotes in face-on views. Mon. Not. R. Astron. Soc. 342, 69–78 (2003a)
Patsis P.A., Skokos Ch., Athanassoula E.: On the 3D dynamics and morphology of inner rings. Mon. Not. R. Astron. Soc. 346, 1031–1040 (2003b)
Patsis P.A.: The stellar dynamics of spiral arms in barred spiral galaxies. Mon. Not. R. Astron. Soc. 369, L56–L60 (2006)
Pesin Y.B.: Invariant manifold families which correspond to nonvanishing characteristic exponents. Izv. Akad. Nauk. SSSR Ser. Mat. 40, 1332–1379 (1976)
Pfenniger D.: The 3D dynamics of barsolid black galaxies. Astron. Astrophys. 134, 373–386 (1984)
Plummer H.C.: On the problem of distribution in globular star clusters. Mon. Not. R. Astron. Soc. 71, 460–470 (1911)
Press W.H., Teukolsky S.A., Vetterling W.T., Flannery B.P.: Numerical Recipes in Fortran 77: The Art of Scientific Computing. Cambridge University Press, Cambridge (1986)
Rice J.: Mathematical Statistics and Data Analysis. Duxbury Press, Belmont (1995)
Romero-Gómez M., Masdemont J.J., Athanassoula E., García-Gómez C.: The origin of rR1 ring structures in barred galaxies. Astron. Astrophys. 453, 39–45 (2006)
Romero-Gómez M., Athanassoula E., Masdemont J.J., García-Gómez C.: The formation of spiral arms and rings in barred galaxies. Astron. Astrophys. 472, 63–75 (2007)
Ruelle D.: Ergodic theory of differentiable dynamical systems. Phys. Math. IHES 50, 275–306 (1979)
Tsallis C.: Introduction to Nonextensive Statistical Mechanics: Approaching a Complex World. Springer, New York (2009)
Tsallis C., Tirnakli U.: Nonadditive entropy and nonextensive statistical mechanics—some central concepts and recent applications. J. Phys. Conf. Ser. 201, 012001 (2010)
Sinai Y.G.: Gibbs measures in ergodic theory. Uspekhi Matematicheskikh Nauk 27, 21–64 (1972)
Skokos Ch.: Alignment indices: a new, simple method for determining the ordered or chaotic nature of orbits. J. Phys. A Math. Gen. 34, 10029–10043 (2001)
Skokos Ch., Patsis P.A., Athanassoula E.: Orbital dynamics of three-dimensional bars—I. The backbone of three-dimensional bars. A fiducial case. Mon. Not. R. Astron. Soc. 333, 847–860 (2002a)
Skokos Ch., Patsis P.A., Athanassoula E.: Orbital dynamics of three-dimensional bars—II. Investigation of the parameter space. Mon. Not. R. Astron. Soc. 333, 861–870 (2002b)
Skokos Ch., Bountis T., Antonopoulos Ch.: Geometrical properties of local dynamics in Hamiltonian systems: the Generalized Alignment Index (GALI) method. Phys. D 231, 30–54 (2007)
Skokos Ch.: The Lyapunov characteristic exponents and their computation. Lect. Notes Phys. 790, 63–135 (2010)
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Bountis, T., Manos, T. & Antonopoulos, C. Complex statistics in Hamiltonian barred galaxy models. Celest Mech Dyn Astr 113, 63–80 (2012). https://doi.org/10.1007/s10569-011-9392-9
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DOI: https://doi.org/10.1007/s10569-011-9392-9