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

We explore the idea that periodic or chaotic finite motions corresponding t o attractors in the simplest models of resonant wave interactions might she d light on the problem of pattern formation. First we identify those dynami cal regimes of interest which imply certain specific relations between phys ically observable variables, e.g. between amplitudes and phases of Fourier harmonics comprising the pattern. To be of relevance to reality, the regime s must be robust. The issue of structural stability of low-dimensional dynamical models is ce ntral to our work. We show that the classical model of three-wave resonant interactions in a non-conservative medium is structurally unstable with res pect to small cubic interactions. The structural instability is found to be due to the presence of certain extremely sensitive points in the unperturb ed system attractors. The model describing the horse-shoe pattern formation due to non-conservative quintet interactions [11] is also analyzed and a r ich family of attractors is mapped. The absence of such sensitive points in the found attractors thus indicates the robustness of the regimes of inter est. Applicability of these models to the problem of 3-D water wave pattern s is discussed. Our general conclusion is that extreme caution is necessary in applying the dynamical system approach, based upon low-dimensional mode ls, to the problem of water wave pattern formation. (C) Elsevier, Paris.