On the Stability of Isolating Integrals. I. Effect of the Perturbation in the Potential Function

The stability of isolating integrals against the perturbation of potential function was studied by the method of surface of section in the classical dynamical system of two degrees of freedom with the hamiltonian, {aligned} H{=}1/2(ẋ²+ẏ²+x²+y²+2x²y+2/3y³-4/3ε y³), {aligned} in the range -1< ε <1. This system has isolating integrals in this range. Their structure is found to change discontinuously at ε =-1/2 and ε =0.