ASYMPTOTIC ANALYSIS OF A CLASS OF 3-DEGREE-OF-FREEDOM HAMILTONIAN-SYSTEMS NEAR STABLE EQUILIBRIA

A conservative near-integrable Hamilton dynamical system is examined, which to leading order consists of three uncoupled harmonic oscillator s with constant frequencies in the ratio 1:2 alpha for certain rationa l alpha. Formally, the problem considered can arise by perturbing any three-degree-of-freedom Hamiltonian near a stable equilibrium point, s o that the Hamiltonian consists of a power series expansion in a small parameter, where successive terms are homogeneous polynomials of incr easing degree in the coordinates and the momenta. The special case of two exact simultaneous resonances, one in the first perturbation term and one in the second, is examined and explicit asymptotic solutions a re obtained. The solution procedure involves reducing the original Ham iltonian to two degree of freedom using one integral of the motion; th en transforming to standard form to find two additional adiabatic inva riants by near-identity averaging canonical transformations. A specifi c example is studied numerically to verify the asymptotic validity of the results over long times.