The dynamics of three-planet systems: An approach from a dynamical system

Bungo Shikita*, Hiroko Koyama, Shoichi Yamada

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

6 Citations (Scopus)


We study in detail the motions of three planets interacting with each other under the influence of a central star. It is known that the system with more than two planets becomes unstable after remaining quasi-stable for long times, leading to highly eccentric orbital motions or ejections of some of the planets. In this paper, we are concerned with the underlying physics for this quasi-stability as well as the subsequent instability and advocate the so-called stagnant motion in the phase space, which has been explored in the field of a dynamical system. We employ the Lyapunov exponent, the power spectra of orbital elements, and the distribution of the durations of quasi-stable motions to analyze the phase-space structure of the three-planet system, the simplest and hopefully representative one that shows the instability. We find from the Lyapunov exponent that the system is almost non-chaotic in the initial quasi-stable state whereas it becomes intermittently chaotic thereafter. The non-chaotic motions produce the horizontal dense band in the action-angle plot whereas the voids correspond to the chaotic motions. We obtain power laws for the power spectra of orbital eccentricities. Power-law distributions are also found for the durations of quasi-stable states. With all these results combined together, we may reach the following picture: the phase space consists of the so-called KAM tori surrounded by satellite tori and imbedded in the chaotic sea. The satellite tori have a self-similar distribution and are responsible for the scale-free power-law distributions of the duration times. The system is trapped around one of the KAM torus and the satellites for a long time (the stagnant motion) and moves to another KAM torus with its own satellites from time to time, corresponding to the intermittent chaotic behaviors.

Original languageEnglish
Pages (from-to)819-832
Number of pages14
JournalAstrophysical Journal
Issue number2
Publication statusPublished - 2010


  • Celestial mechanics
  • Chaos
  • Planets and satellites: dynamical evolution and stability

ASJC Scopus subject areas

  • Astronomy and Astrophysics
  • Space and Planetary Science


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