Wall stabilization and the Mathieu–Hill equations

In a recent publication [H. Tasso and G. N. Throumoulopoulos, Phys. Lett. A 271, 413 (2000)] on Lyapunov stability of general mechanical systems, it is shown that “parametric excitations” can be stabilized by dissipation for positive potential energies. Specializing on the damped Mathieu equation permits one to establish its full stability chart. It is then seen that dissipation broadens the regions of stability to the extent that not only the response to parametric excitations is damped, but even “negative-energy” modes are stabilized by the combined action of the parametric excitation and the damping coefficient. The extension of this analysis to the “two-step” Hill’s equation shows that the stability regions become many times larger than those of the Mathieu equation. By analogy, these findings are a strong indication that the “resistive wall mode” could be stabilized by the joint action of a properly tailored time-dependent wall resistivity and a sufficient viscous dissipation in the plasma. Note that within this scheme neither the wall nor the plasma need to be in motion. An extension of this work to include more realistic models is in progress.