The periodic solution has been shown to be globally asymptotically stable by use of the Floquet theorem
and small-amplitude perturbations, if p > max([r.sub.1][k.sub.0] TM/[[lambda].sub.1] [k.sub.1], [r.sub.2] [k.sub.2] TM/u[[lambda].sub.2] [k.sub.3]) and p exp(-MT)/ (1 - exp(-MT)) > max([r.sub.1] (1 + [h.sub.1] [[lambda].sub.1] M+u[h.sub.2] [[lambda].sub.2] M)/[[lambda].sub.1], [r.sub.2](1 + [h.sub.1] [[lambda].sub.1] M+u[h.sub.2] [[lambda].sub.2]M)/u[[lambda].sub.2]).
The Floquet-modes are the eigenmodes in periodic structures, and the Floquet theorem asserts that the fields in the structure can be expressed by superposition of the Floquet-modes .
Because of the pseudo-periodicity of the transformed fields, the conventional grating theory based on the Floquet theorem becomes possible to be applied for the scattering problem of imperfectly periodic structures.
On the basis of Floquet theorem
[11, 12], the formulation of scalar Floquet modes  is given by:
First, with the digitized RF fields in a single period obtained from the electromagnetic simulation software (such as HFSS ), the RF field profile in the whole tube can be provided by means of the Floquet theorem. Second, the RF field equations are developed from the Law of Energy Conservation.
The RF field profiles [e.sub.n](x) and [h.sub.n](x) can be generally expressed by the Floquet theorem as
Employing the Floquet theorem
, we factor the field solution in the structure according to