Numerical Simulation of Dynamical Evolution of the Solar System over 5 Billion Years

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Planetary motions are said to be chaotic in the sense that the maximum Lyapunov exponent is positive.So far the longest integration up to now, over 845 million years (42 Pluto's Lyapunov times) , does not show any indication of a gross instability in the motion of Pluto.We carried out the numerical integration of Pluto over the age of the solar system (5.5 billion years both towards the past and the future) . This integration also did not give any indication of chaotic evolution of Pluto.The divergences of Keplerian elements of a nearby trajectory at first grow limearly with the time and then start to increase exponentially.The exponential divergences stop at about 420 million years and saturate.The exponential divergneces are suppressed by the following three resoncnaces that Pluto has :1) Pulto is in the 3 : 2 mean motion resonance with Neptune.2) The argument of perihelion librates around 90 degrees and its period is 3.8 Myr.The dominant periodic variations of the eccentricity and the inclination are synchronized with the libration of the argument of perihelion.3) Moreover the longitude of Pluto's node referred to the longitude of Neptune's node circulates and the period of this circulation is equal to the period of the libration of the argument of Perihelion (a secondary resonance) .In order to investigate how Pluto evolves to the present stable state of three resonance lockings, we have to take account of a non-conservative mechanism in the earlsy stage of the solar system.