In an analogy with the problems of non-Hermitian physics, nilpotent 1:1 resonances can originate in the unfolding of the 1:1 semi-simple resonance. This chapter presents a detailed study of this process in the case of general four-dimensional systems. It indicates how the results can be applied to an explicitly given system. The following questions are addressed: Determine the stability domain in parameter space of the original system; Locate the singularities on the boundary of the stability domain; and Identify Hamiltonian and reversible subsystems. The authors find the boundary of the stability domain and list all its singularities including six self-intersections and four “Whitney umbrellas”. They propose an algorithm of approximation of the stability boundary near singularities and apply the results to the study of enhancement of the modulation instability with dissipation as well as to the study of stability of a non-conservative system of rotor dynamics.
|Title of host publication
|Nonlinear Physical Systems: Spectral Analysis, Stability and Bifurcations
|Oleg Kirillov, Dmitry Pelinovsky
|Place of Publication
|Number of pages
|Published - 10 Feb 2014