Topology and bifurcations of the invariant level sets of a Fokker-Planck Hamiltonian through two coupled anisotropic quartic anharmonic oscillators

Abstract

Integrable Hamiltonians with velocity-dependent potentials, including those of Fokker-Planck Hamiltonians H = ½(px2 + py2) + kxpx + kypy, are constructed from integrable Hamiltonians of type H = ½(px2 + py2) + V(x,y). In order to carry out the analytical investigations, we convert the problem into that of two coupled anisotropic quartic anharmonic oscillators using certain canonical transformations; afterwards we give a complete description of the real phase space topology of the system. We give also an explicit periodic solution for singular common-level sets of the first integrals. All generic bifurcations of Liouville tori were determined analytically and numerically.