Geometrical aspects of stability theory for Hill's equations

Hill's equation with quasi-periodic forcing: resonance tongues, instability pockets and global phenomena

“…Hence they 'cancel' for some small values of b, thereby creating instability pockets. This is similar to what happens in the periodic case, see [5,6]. Figure 3 displays an instability pocket, for p as in (4), where kl = 1, k2 = 1 and c = 0.3.…”

“…The phenomenon of instability pockets and its explanation was quite new for the periodic case [5,6] and again, this part of the theory translates to the quasi-periodic case.…”

“…Indeed, the boundaries of the second resonance tongue, emanating from (a, b) = (1, 0), cross at an asymptotic distance of the order of Icl from the a-axis, thereby creating an instability pocket. In [5,6] this phenomenon is qualitatively explained by singularity theory. Presently, an analogue of such a perturbation is given by the following modification of equation (2): p(t) = cos(t) + cos( t) + c cos((kl + k2 )t),…”

“…Indeed, Hill's equation (1) where the forcing term p is periodic in time, has been widely studied in the literature, compare, e.g., [5,6] and references given there. For related work on nonlinear parametric forcing see, e.g., [2,7].…”

“…Bra~. Mat., VoL 29, N. 2,1998 all information concerning the properties of (1) is contained in the monodromy matrix, i.e., the fundamental matrix which starts at the identity for t = 0 evaluated at t = T, T being the period of p. Equivalently we can use the Floquet matrix, i.e., the logarithm of the monodromy matrix divided by T. Compare, e.g., [5,6].…”