Poincaré inverse problem and torus construction in phase space

•Construction of invariant phase-space tori for an arbitrary Hamiltonian.•Iterative method with a direct Fourier-series representation of the phase-space variables.•Symplecticity is an optimisation objective.•An algorithm which can probe the phase space and construct tori automatically.

The phase space of an integrable Hamiltonian system is foliated by invariant tori. For an arbitrary Hamiltonian HH such a foliation may not exist, but we can artificially construct one through a parameterised family of surfaces, with the intention of finding, in some sense, the closest integrable approximation to HH. This is the Poincaré inverse problem (PIP). In this paper, we review the available methods of solving the PIP and present a new iterative approach which works well for the often problematic thin orbits.