Hopf Bifurcation from Chaos and Generalized Winding Numbers of Critical Modes

In the study of chaos, Lyapunov exponents have been successfully used in describing the expansion and contraction rates of various modes. In this Letter, generalized winding numbers are defined in association with the corresponding Lyapunov exponents to characterize the rotation behavior of these modes during the evolution. A Hopf bifurcation from chaos, namely, a blowout bifurcation with certain finite typical frequency, is revealed. The frequency of the motion after the bifurcation is justified to be equal to the generalized winding number of the critical transverse mode, for which the Lyapunov exponent crosses zero at the bifurcation.