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Singularly perturbed and nonlocal modulation equations for systems with interacting instability mechanisms. (English) Zbl 0885.58075
From the authors’ abstract: “Two related systems of coupled modulation equations are studied and compared in this paper. The modulation equations are derived for a certain class of basic systems which are subject to two distinct, interacting, destabilizing mechanisms...It is found that spatially periodic stationary solutions of the nonlocal system exist under the same conditions as spatially periodic stationary solutions of the singularly perturbed system. Thus, the “Landau reduction” to the nonlocal system has no significant influence on the stationary quasi-periodic patterns. However, a large variety of intricate heteroclinic and homoclinic connections is found for the singularly perturbed system...None of these patterns can be described by the nonlocal system. So, one may conclude that the reduction to the nonlocal system destroys a rich and important set of patterns”.
37J40Perturbations, normal forms, small divisors, KAM theory, Arnol’d diffusion
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