increases of eccentricity or inclination in any orbital elements of the planets， which may be revealed by still longer-term numerical integrations. We have also performed a couple of trial integrations including motions of the outer five planets over the duration of ± 5 × 1010 yr. The result indicates that the three major resonances in the Neptune–Pluto system have been maintained over the 1011-yr time-span.
1 Introduction
1.1Definition of the problem
The question of the stability of our Solar system has been debated over several hundred years， since the era of Newton. The problem has attracted many famous mathematicians over the years and has played a central role in the development of non-linear dynamics and chaos theory. However， we do not yet have a definite answer to the question of whether our Solar system is stable or not. This is partly a result of the fact that the definition of the term ‘stability’ is vague when it is used in relation to the problem of p
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