Finding invariant tori in the problem of a periodically corrugated waveguide

Abstract

Some dynamic properties for a light ray suffering specular reflections inside a periodically corrugated waveguide are studied. The dynamics of the model is described in terms of a two dimensional nonlinear area preserving map. We show that the phase space is mixed in the sense that there are KAM islands surrounded by a large chaotic sea that is confined by two invariant spanning curves. We have used a connection with the Standard Mapping near a transition from local to global chaos and found the position of these two invariant spanning curves limiting the size of the chaotic sea as function of the control parameter.

Characterization of multiple reflections and phase space properties for a periodically corrugated waveguide

“…One sees that a mixed structure is present in the sense that periodic islands marked by a resonance (elliptic fixed point), chaotic seas and invariant KAM curves can all be observed. The size of the chaotic sea and the position of the invariant KAM curves depend on the value of the control parameter δ and as discussed in [17], the position of the invariant KAM curve scales with δ 1/2 .…”

“…Mixed structure characterizes the case where either periodic islands, chaotic seas and invariant KAM curves are present in the phase space. The results published in the literature regarding the present model consider so far the so-called simplified version [13,[16][17][18][19] (and citations therein). Therefore, in this paper our main goal is to characterize the description of the model taking into account the whole version.…”

Locating invariant tori for a family of two-dimensional Hamiltonian mappings

“…The phase space is regular and chaos, characterised by the exponential divergence of nearby trajectories, is not observed. The scenario changes for case (ii) where one can observe chaotic seas that eventually surround periodic islands [13][14][15][16][17] and the existence of a set of invariant tori. Generally, the size of the chaotic seas are strongly influenced by the control parameters since they control the intensity of the nonlinearity.…”