Control of integrable Hamiltonian systems and degenerate bifurcations

Control of multidimensional integrable Hamiltonian systems

In this paper, we study the controllability of a four-dimensional integrable Hamiltonian system that arises as a low-mode truncation of the nonlinear Schrödinger equation [Bishop, Phys. Lett. A 144, 17 (1990)]. The controller targets a solution of the uncontrolled dynamics. We show that in the limit of small control coupling, a Takens-Bogdanov bifurcation occurs at the control target. These results support our earlier claim that Takens-Bogdanov bifurcations will generically occur when dissipative control is applied to integrable Hamiltonian sytems. The presence of the Takens-Bogdanov bifurcation causes the control to be extremely sensitive to noise. Here, we implement an algorithm first developed in Kulp and Tracy [Phys. Rev. E 70, 016205 (2004)] to extract a subcritical noise threshold for the four-dimensional system.