Symmetric and flat bifurcations: An oscillatory phenomenon

Abstract

Two distinct bifurcation phenomena associated with non-linear autonomous lumpedparameter systems are analyzed fully in ann-dimensional state space. The analysis is asymptotic and is performed via an intrinsic harmonic balancing technique. In one case a pair of complex conjugate eigenvalues touches the imaginary axis tangentially without crossing it in violation of the main Hopf condition associated with the so-called Hopf bifurcation. It is shown that under the new condition a bifurcating family of limit cycles may or may not axist. If it exists, the family is represented by two paths in the parameteramplitude space which have finite slopes and are symmetrically located with respect to the amplitude axis. The second case is concerned with aflat Hopf bifurcation, and the limit cycles always exist. This phenomenon occurs when the Hopf condition, that a pair of complex conjugate eigenvalues crosses the imaginary axis with non-zero velocity, is satisfied but another significant coefficient vanishes.