Cusp and Tangential Bifurcations Associated with the Limit Cycles of Autonomous Systems

Two distinct dynamic bifurcation phenomena associated with the limit cycles of nonlinear, autonomous, lumped parameter systems are analyzed in general terms in a two-dimensional state space. In both cases Hopf´s transversality condition is violated. In the first case it is assumed that a pair of complex conjugate eigenvalues crosses the imaginary axis in such a way that certain derivatives of the real part with respect to the system paramter vanish at the crossing point. In the second case it is assumed that the loci of the eigenvalues touch the imaginary axis, so that the curvature evaluated at this point is zero and crossing does not take place. In both problems, the asymptotic equations describing the limit cycles, the bifurcating paths, and the frequency-parameter relationships are derived. These results are obtained explicitly via a recently introduced intrinsic method of harmonic balancing technique and they are presented in the form of formula-type expressions, which are ready to be used in the analysis of specific problems.