Application of new dynamical spectra of orbits in Hamiltonian systems
Tóm tắt
In the present article, we investigate the properties of motion in Hamiltonian systems of two and three degrees of freedom, using the distribution of the values of two new dynamical parameters. The distribution functions of the new parameters define the S(g) and the S(w) dynamical spectra. The first spectrum definition that is the S(g) spectrum will be applied in a Hamiltonian system of two degrees of freedom (2D), while the S(w) dynamical spectrum will be deployed in a Hamiltonian system of three degrees of freedom (3D). Both Hamiltonian systems, describe a very interesting dynamical system which displays a large variety of resonant orbits, different chaotic components, and also several sticky regions. We test and prove the efficiency and the reliability of these new dynamical spectra, in detecting tiny ordered domains embedded in the chaotic sea, corresponding to complicated resonant orbits of higher multiplicity. The results of our extensive numerical calculations suggest that both dynamical spectra are fast and reliable discriminants between different types of orbits in Hamiltonian systems, while requiring very short computation time in order to provide solid and conclusive evidence regarding the nature of an orbit. Furthermore, we establish numerical criteria in order to quantify the results obtained from our new dynamical spectra. A comparison to other previously used dynamical indicators, reveals the leading role of the new spectra.
Tài liệu tham khảo
Binney, J., Spergel, D.: Spectral stellar dynamics. Astrophys. J. 252, 308–321 (1982)
Binney, J., Spergel, D.: Spectral stellar dynamics. II—The action integrals. Mon. Not. R. Astron. Soc. 206, 159–177 (1984)
Caranicolas, N.D., Papadopoulos, N.J.: The S(c) spectrum machine to visualize the motion in galaxies. Astron. Nachr. 328(6), 556–561 (2007)
Caranicolas, N.D., Zotos, E.E.: Chaotic orbits in a 3D galactic dynamical model with a double nucleus. Mech. Res. Commun. 36, 875–881 (2009)
Caranicolas, N.D., Zotos, E.E.: Using the S(c) spectrum to distinguish between order and chaos in a 3D galactic potential. New Astron. 15, 427–432 (2010)
Caranicolas, N.D., Zotos, E.E.: Investigating the properties of motion in 3D perturbed elliptic oscillators displaying exact periodic orbits. Nonlinear Dyn. (2012) doi:10.1007/s11071-012-0386-2
Carpintero, D., Aguilar, L.: Orbit classification in arbitrary 2D and 3D potentials. Mon. Not. R. Astron. Soc. 298, 1–21 (1998)
Contopoulos, G., Magnenat, P., Martinet, L.: Invariant surfaces and orbital behavior in dynamical systems of 3 degrees of freedom II. Physica D 6, 123–136 (1982)
Contopoulos, G., Grousousakou, E., Voglis, N.: Invariant spectra in Hamiltonian systems. Astron. Astrophys. 304, 374–380 (1995)
Contopoulos, G., Voglis, N.: Spectra of stretching numbers and helicity angles in dynamical systems. Celest. Mech. Dyn. Astron. 64, 1–20 (1996)
Contopoulos, G., Voglis, N.: In: Buta, R., Crocker, D.A., Elmegreen, B.G. (eds.) Barred Galaxies. ASP Conf. Ser., vol. 91, p. 321 (1996)
Contopoulos, G., Voglis, N.: A fast method for distinguishing between ordered and chaotic orbits. Astron. Astrophys. 317, 73–81 (1997)
Contopoulos, G., Voglis, N., Efthymiopoulos, C., Froeschlé, C., Gonczi, R., Lega, E., Dvorak, R., Lohinger, E.: Transition spectra of dynamical systems. Celest. Mech. Dyn. Astron. 67, 293–317 (1997)
Froeschlé, Ch.: Numerical study of dynamical systems with three degrees of freedom. Astron. Astrophys. 4, 115–128 (1970)
Froeschlé, Ch.: Numerical study of a four-dimensional mapping. Astron. Astrophys. 16, 172–189 (1972)
Froeschlé, C., Froeschlé, Ch., Lohinger, E.: Generalized Lyapunov characteristic indicators and corresponding Kolmogorov like entropy of the standard mapping. Celest. Mech. Dyn. Astron. 56, 307–314 (1993)
Froeschlé, Ch., Lega, E., Gonczi, R.: Fast Lyapunov indicators. Application to asteroidal motion. Celest. Mech. Dyn. Astron. 67, 41–62 (1997)
Hénon, M., Heiles, C.: The applicability of the third integral of motion: some numerical experiments. Astron. J. 69, 73–79 (1964)
Karanis, G.I., Caranicolas, N.D.: Transition from regular motion to chaos in a logarithmic potential. Astron. Astrophys. 367, 443–448 (2001)
Karanis, G.I., Caranicolas, N.D.: A new dynamical spectrum for galactic potentials. Astron. Nachr. 323(1), 3–11 (2002)
Laskar, J.: Frequency analysis for multi-dimensional systems. Global dynamics and diffusion. Physica D 67, 257–281 (1993)
Laskar, J., Froeschlé, C., Celleti, A.: The measure of chaos by the numerical analysis of the fundamental frequencies. Application to the standard mapping. Physica D 56, 253–269 (1992)
Lichtenberg, A.J., Lieberman, M.A.: Regular and Chaotic Dynamics. Springer, Berlin (1992)
Magnenat, P.: Numerical study of periodic orbit properties in a dynamical system with three degrees of freedom. Celest. Mech. 28, 319–343 (1982)
Martinet, L., Magnenat, P.: Invariant surfaces and orbital behavior in dynamical systems with 3 degrees of freedom. Astron. Astrophys. 96, 68–77 (1981)
Papaphilippou, Y., Laskar, J.: Frequency map analysis and global dynamics in a galactic potential with two degrees of freedom. Astron. Astrophys. 307, 427–449 (1996)
Patsis, P.A., Zachilas, L.: Using color and rotation for visualizing four-dimensional Poincaré cross-sections: with applications to the orbital behavior of a three-dimensional Hamiltonian system. Int. J. Bifurc. Chaos 4, 1399–1424 (1994)
Patsis, P.A., Efthymiopoulos, C., Contopoulos, G., Voglis, N.: Dynamical spectra of barred galaxies. Astron. Astrophys. 326, 493–500 (1997)
Pfenniger, D.: The 3D dynamics of barred galaxies. Astron. Astrophys. 134, 373–386 (1984)
Pfenniger, D., Friedli, D.: Structure and dynamics of 3D N-body barred galaxies. Astron. Astrophys. 252, 75–93 (1991)
Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes in FORTRAN. Cambridge University Press, Cambridge (1992)
Revaz, Y., Pfenniger, D.: Periodic orbits in warped disks. Astron. Astrophys. 372, 784–792 (2001)
Skokos, Ch.: Alignment indices: a new, simple method for determining the ordered or chaotic nature of orbits. J. Phys. A 34, 10029–10043 (2001)
Skokos, Ch., Contopoulos, G., Polymilis, C.: Structures in the phase space of a four dimensional symplectic map. Celest. Mech. Dyn. Astron. 65, 223–251 (1997)
Skokos, Ch., Contopoulos, G., Polymilis, C.: Numerical study of the phase space of a four dimensional symplectic map. In: Simó, C. (ed.) Hamiltonian Systems with Three or More Degrees of Freedom, pp. 583–587. Plenum, New York (1999)
Skokos, Ch., Bountis, T.C., Antonopoulos, Ch.: Geometrical properties of local dynamics in Hamiltonian systems: the generalized alignment index (GALI) method. Physica D 231, 30–54 (2007)
Voglis, N., Contopoulos, G.: Invariant spectra of orbits in dynamical systems. J. Phys. A 27, 4899–4912 (1994)
Zotos, E.E.: A new dynamical parameter for the study of sticky orbits in a 3D galactic model. Balt. Astron., 20, 339–354 (2011). Paper I
Zotos, E.E.: A new dynamical model for the study of galactic structure. New Astron. 16, 391–401 (2011). Paper II
Zotos, E.E.: Disks controlling chaos in a 3D dynamical model for elliptical galaxies. Balt. Astron., 20, 77–90 (2011). Paper III