Abstract
In this paper, we show the important role of chaotic transients in Celestial Mechanics through the Sitnikov problem. We compare the two kinds of chaos, permanent and transient, and provide the chaotic saddle of the Sitnikov problem giving also some important quantitative properties of this fractal set. Additionally, we present a link between the stickiness effect of tori and chaotic scattering.
Similar content being viewed by others
References
Alfaro J., Chiralt C.: Invariant rotational curves in Sitnikov’s problem. Celest. Mech. Dyn. Astron. 55, 351–367 (1993)
Alt H., Gräf H.-D., Harney H.L., Hofferbert R., Rehfeld H., Richter A., Schardt P.: Decay of classical chaotic systems: the case of the Bunimovich stadium. Phys. Rev. E 53, 2217–2222 (1996)
Altmann G., Tél T.: Poincaré recurrences from the perspective of transient chaos. Phys. Rev. Lett. 100, id.174101 (2008)
Benet L., Trautmann D., Seligman T.H.: Chaotic scattering in the restricted three-body problem I. The Copenhagen problem. Celest. Mech. Dyn. Astron. 66, 203–228 (1996)
Benet L., Trautmann T.H., Seligman D.: Chaotic scattering in the restricted three-body problem II. Small mass parameters. Celest. Mech. Dyn. Astron. 71, 167–189 (1998)
Benet L., Merlo O.: Phase-space volume of regions of trapped motion: multiple ring components and arcs. Celest. Mech. Dyn. Astron. 103, 209–225 (2009)
Bleher S., Grebogi C., Ott E.: Bifurcation to chaotic scattering. Physica D 46, 87–121 (1990)
Contopoulos G.: Order and chaos in dynamical astronomy, pp. 237–251. Springer, Berlin, Heidelberg, New York (2002)
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)
Contopoulos G., Harsoula N., Voglis N., Dvorak R.: Destruction of islands of stability. J. Phys. A: Math. Gen. 32, 5213–5232 (1999)
Contopoulos G., Efstathiou K.: Escapes and recurrence in a simple hamiltonian system. Celest. Mech. Dyn. Astron. 88, 163–183 (2004)
Contopoulos G., Patsis A.: Outer dynamics and escapes in barred galaxies. Mon. Not. R. Astron. Soc. 369, 1039–1054 (2006)
Cristadoro G., Ketzmerick R.: Universality of algebraic decays in Hamiltonian systems. Phys. Rev. Lett. 100, 184101 (2008)
Dvorak R., Contopoulos G., Efthymiopoulos C., Voglis N.: ‘Stickiness’ in mappings and dynamical systems. Planet. Space Sci. 46, 1567–1578 (1998)
Dvorak R.: Numerical results to the Sitnikov-problem. Celest. Mech. Dyn. Astron. 56, 71–80 (1993)
Dvorak R.: The Sitnikov problem–a complete picture of phase space. Publ. Astron. Dep. Eötvös Univ 19, 129–140 (2007)
Eckhardt B.: Irregular scattering. Physica D 33, 89–98 (1988)
Efthymiopoulos C., Contopoulos G., Voglis N.: Cantori, Islands and asymptotic curves in the stickiness region. Celest. Mech. Dyn. Astron. 73, 221–230 (1999)
Faruque S.B.: Solution of the Sitnikov problem. Celest. Mech. Dyn. Astron. 87, 353–369 (2003)
García F., Gómez G.: A note on weak stability boundaries. Celest. Mech. Dyn. Astron. 97, 87–100 (2007)
Grebogi C., Ott E., Yorke A.J.: Chaotic attractors in crisis. Phys. Rev. Lett. 48, 1507–1510 (1982)
Grebogi C., Ott E., Yorke A.J.: Crisis, sudden changes in chaotic attractors, and transient chaos. Physica D 7, 181–200 (1983)
Hagel J.: A new analytic approach to the Sitnikov problem. Celest. Mech. Dyn. Astron. 53, 267–292 (1992)
Hagel J., Lhotka C.: A high order perturbation analysis of the Sitnikov problem. Celest. Mech. Dyn. Astron. 93, 201–228 (2005)
Hsu G., Ott E., Grebogi C.: Strange saddles and the dimensions of their invariant manifolds. Phys. Lett. A 127, 199–204 (1988)
Jiménez-Lara L., Escalona-Buendía A.: Symmetries and bifurcations in the Sitnikov problem. Celest. Mech. Dyn. Astron. 79, 97–117 (2001)
Jung C., Tél T., Ziemniak E.: Application of scattering chaos to particle transport in a hydrodynamical flow. Chaos: An Interdisciplinary J. Nonlinear Sci. 3, 555–568 (1993)
Jung C., Mejia-Monasterio C., Seligman T.H.: Scattering one step from chaos. Phys. Lett. A 198, 306–314 (1995)
Kantz H., Grassberger P.: Repellers, semi-attractors, and long-lived chaotic transients. Physica D 17, 75–86 (1985)
Kovács T., Érdi B.: The structure of the extended phase space of the Sitnikov problem. Astron. Nachr. 328, 801–804 (2007)
Lai Y., Grebogi C., Blümel R., Ding M.: Algebraic decay and phase-space metamorphoses in microwave ionization of hydrogen Rydberg atoms. Phys. Rev. A 45, 8284–8287 (1992)
Lai Y., Grebogi C., Yorke J.A., Kan I.: How often are the chaotic saddles nonhyperbolic?. Nonlinearity 6, 779–797 (1993)
Liu J., Sun Y.-S.: On the Sitnikov problem. Celest. Mech. Dyn. Astron. 49, 285–302 (1990)
MacKay R.S., Meiss J.D., Percival I.C.: Stochasticity and transport in hamiltonian systems. Phys. Rev. Lett. 52, 697–700 (1984)
MacMillan W.D.: An integrable case in the restricted problem of three bodies. Astron. J. 27, 11–13 (1911)
Meiss J.D., Ott E.: Markov-Tree model of intrinsic transport in hamiltonian systems. Phys. Rev. Lett. 55, 2741–2744 (1985)
Motter A.E, Lai Y.-C.: Dissipative chaotic scattering. Phys. Rev. E 65, 015205 (2002)
Ott E.: Chaos in dynamical systems 2nd Ed, pp. 192–199. Cambridge Univ Press, Cambridge (2002)
Ott E., Tél T.: Chaotic scattering: an introduction. Chaos 3, 417–425 (1993)
Perdios E.A.: The manifolds of families of 3D periodic orbits associated to Sitnikov motions in the restricted three-body problem. Celest. Mech. Dyn. Astron. 99, 85–104 (2007)
Simó C., Vieiro A.: Resonant zones, inner and outer splittings in generic and low order resonances of area preserving maps. Nonlinearity 22, 1191–1245 (2009)
Sitnikov K.: Existence of oscillating motion for the three-body problem. Dokl. Akad. Nauk. USSR 133, 303–306 (1960)
Soulis P., Bountis T., Dvorak R.: Stability of motion in the Sitnikov 3-body problem. Celest. Mech. Dyn. Astron. 99, 129–148 (2007)
Soulis P., Papadakis K., Bountis T.: Periodic orbits and bifurcatiopns in the Sitnikov four-body problem. Celest. Mech. Dyn. Astron. 100, 251–266 (2008)
Sun Y., Zhou L., Zhou J.: The role of hyperbolic invariant sets in stickiness effects. Celest. Mech. Dyn. Astron. 92, 257–272 (2005)
Tél, T., Gruiz, M.: Chaotic dynamics. Cambridge University. Press, Cambridge, pp. 201–202, 338 (2006)
Wodnar K.: The original Sitnikov article–new insights. Celest. Mech. Dyn. Astron. 56, 99–101 (1993)
Zhou J.L., Zhou L.Y., Sun Y.S.: Hyperbolic structure and stickiness effect. Chin. Phys. Lett. 19, 1254–1256 (2002)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kovács, T., Érdi, B. Transient chaos in the Sitnikov problem. Celest Mech Dyn Astr 105, 289–304 (2009). https://doi.org/10.1007/s10569-009-9227-0
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10569-009-9227-0