Numerical Investigation of the Galactic Problem by Computing Families of Periodic Solutions
Authors: Goudas, C.; Papadakis, K.; Katsiaris, G.
Source: Astrophysics and Space Science, Volume 301, Numbers 1-4, January 2006 , pp. 97-116(20)
Abstract:The galactic dynamical system expressed by a third-order axisymmetric polynomial potential is investigated numerically by computing periodic solutions. We define as Sthe compact set of initial conditions generating bounded motions, and as Sp, with Sp⊂ S, the countable set of all initial conditions generating periodic solutions. Then, we consider the subsets Sspand Sapof Sp, where Ssp∪ Sap= Sp, Ssp Sap= Ø, the first of which corresponds to symmetric periodic solutions, and the second to asymmetric solutions. Then, we approximate the set Ssp, leaving treatment of the set Sapof asymmetric solutions for a future publication. The set Sspis known to be dense in S (‘Last Geometric Theorem of Poincar;’, Birkhoff, 1913). Using a computer programme capable to locate all elements of the set Sspthat generate symmetric periodic solutions that re-enter after intersecting the axis of symmetry from 1 to ntimes. The results of the approximation of Sspin the total domain and in the sample sub-domains of zooming, we present in graphical form as family curves in the (x, C) plane. The solutions located with the largest periods re-enter after 440 galaxy revolutions while the families calculated fully (initial conditions, period, energy, stability co-efficient) include solutions that re-enter after 340 galaxy revolutions. To advance further the approximation of the set Sspthus obtained, we applied the same procedure inside eight sub-domains of the domain Sinto which we ‘zoomed’ through selection of finer search steps and double maximum periods. The family curves thus calculated presented in the (x, C) plane do not intersect anywhere in some sub-domains and their pattern resembles that of laminar flow. In other sub-domains, however, we found family curves from which branching families emanate. The concepts of completeand non-completeapproximation of Sspin sub-domains of laminar and sub-domains with branching family curves, respectively, is introduced. Also, the concept of basic family of order1, 2, ..., n, are defined. The morphology of individual periodic solutions of all families is investigated, and the types of envelopes found are described. The approximate set Sspwas also checked by computing Poincar; sections for energy values corresponding to the mean energy range of the eight sub-domains of zooming mentioned above. These sections show that most parts of the compact domain in Sgenerating non-periodic but bounded solutions correspond to with well-shaped tori that intersect the x-axis, a fact that implies that dominant to exclusive type of periodic solutions are the symmetric ones with two normal crossings of this axis. The presence of non-symmetric periodic solutions as well as of chaotic regions is encountered. All calculations reported here were performed using the variable step R-K 8th-order direct integration and setting the allowable energy variation Δ C= |Cstart− Cend| < 10−13. The output, consisting of many thousands of families and their properties (initial conditions, morphology, stability, etc.), is stored in a directory entitled ‘Atlas of the Symmetric Periodic Solution of the Galactic Motion Problem’.
Keywords: Axisymmetric potential; Dynamical system; Equilibrium points; Numerical integration; Path envelopes; Periodic family curves; Periodic solutions; Poincar; Solution path morphology; Stability; Stable/unstable periodic family arcs; Zooming into space of solutions; surface of sections
Document Type: Research Article
Publication date: January 1, 2006