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Hopf Bifurcation Generated by Small Nonlinear Terms
Author(s) -
Krasnosel'skii A.,
Mennicken R.,
Rachinskii D.
Publication year - 2000
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.20000801416
Subject(s) - mathematics , degenerate energy levels , monotone polygon , bifurcation , nonlinear system , scalar (mathematics) , bifurcation theory , hopf bifurcation , mathematical analysis , equilibrium point , principal part , contrast (vision) , physics , differential equation , geometry , quantum mechanics , optics
We study the problem of generation of small cycles from the equilibrium in autonomous quasilinear systems depending on a parameter. In contrast to the usual situations, the linearized equation is degenerate for all parameter values (not only for the bifurcation point). Therefore the existence of small cycles is determined by small nonlinear terms. The main example is an equation where the principal degenerate linear part is independent of the parameter. We suggest sufficient conditions for the existence and stability of small cycles for higher‐order scalar equations. The results are based on topological methods and methods of monotone operators.