Global dynamics in the simplest models of three-dimensional water-wave patterns

Abstract

We explore the idea that periodic or chaotic finite motions corresponding to attractors in the simplest models of resonant wave interactions might shed light on the problem of pattern formation. First we identify those dynamical regimes of interest which imply certain specific relations between physically observable variables, e.g. between amplitudes and phases of Fourier harmonics comprising the pattern. To be of relevance to reality, the regimes must be robust. The issue of structural stability of low-dimensional dynamical models is central to our work. We show that the classical model of three-wave resonant interactions in a non-conservative medium is structurally unstable with respect to small cubic interactions. The structural instability is found to be due to the presence of certain extremely sensitive points in the unperturbed system attractors. The model describing the horse-shoe pattern formation due to non-conservative quintet interactions [11] is also analyzed and a rich family of attractors is mapped. The absence of such sensitive points in the found attractors thus indicates the robustness of the regimes of interest. Applicability of these models to the problem of 3-D water wave patterns is discussed. Our general conclusion is that extreme caution is necessary in applying the dynamical system approach, based upon low-dimensional models, to the problem of water wave pattern formation.