On the number of isolating integrals in Hamiltonian systems

Abstract

In a Hamiltonian system of three degrees of freedom we have found a large stochastic region (the "big sea"), some other stochastic regions, apparently separated from the above ("small seas"), and some ordered regions. In the ordered regions the maximal Lyapunov characteristic number vanishes, while it has finite values in the stochastic regions. However, these values are different in the big sea and the small seas. Three formal integrals were constructed and they were truncated at orders 2,3,...,11. The numerical values of the truncated integrals along several orbits were calculated. The variations of all three integrals decrease with order in the ordered region, while they remain large in the big sea. In a small sea two integrals have large relative variations, while one integral seems to be well conserved. This indicates that in the ordered region there are two integrals, in the big sea none, and in the small seas one integral beyond the energy.