Determining the Stability Domain of Perturbed Four-Dimensional Systems in 1:1 Resonance

Igor Hoveijn*, Oleg Kirillov

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review


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.
Original languageEnglish
Title of host publicationNonlinear Physical Systems: Spectral Analysis, Stability and Bifurcations
EditorsOleg Kirillov, Dmitry Pelinovsky
Place of PublicationLondon
Number of pages21
ISBN (Electronic)9781118577608
ISBN (Print)9781848214200
Publication statusPublished - 10 Feb 2014


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