Narrow Results

In the present paper we investigate the nonintegrability of adiabatic one degree of freedom Hamiltonian systems, with the additional assumption that the frozen system possesses an unstable fixed point with two asymmetric homoclinic loops. We prove a criterion for the nonexistence of an integral for such systems, and therefore we prove the nonexistence of a quantity which is conserved in an arbitrarily high order on ε. A specific application is given in the asymmetric quartic oscillator with adiabatic time dependence.

We review a number of methods to prove nonintegrability of Hamiltonian systems and focus on 3 Degrees-of-Freedom (DoF) systems listing the known results for the prominent resonances. Associated with the Hamiltonian systems are the averaged-normal forms that provide us with geometric insight, approximations of orbits and measures of chaos. Symmetries do change the qualitative and quantitative pictures; we illustrate this for the 1:2:1 resonance with discrete symmetry in the 1st and 3rd DoF. In this case, the averaged-normal form is still nonintegrable, but it becomes integrable when adding discrete symmetry in all DoF. Apart from the short-periodic solutions obtained by averaging, we find many periodic solutions. There is numerical evidence of the presence of Šilnikov bifurcation which clarifies the presence of nonintegrability phenomena qualitatively and quantitatively.

The aim of this paper is to investigate a generalized Rikitake system from the integrability point of view. For the integrable case, we derive a family of integrable deformations of the generalized Rikitake system by altering its constants of motion, and give two classes of Hamilton–Poisson structures which implies these integrable deformations, including the generalized Rikitake system, are bi-Hamiltonian and have infinitely many Hamilton–Poisson realizations. By analyzing properties of the differential Galois groups of normal variational equations (NVEs) along certain particular solution, we show that the generalized Rikitake system is not rationally integrable in an extended Liouville sense for almost all parameter values, which is in accord with the fact that this system admits chaotic behaviors for a large range of its parameters. The non-existence of analytic first integrals are also discussed.