Hopf bifurcation from chaos and generalized winding numbers of critical modes

In the study of chaos, Lyapunov exponents have been successfully used in de scribing the expansion and contraction rates of various modes. In this Lett er, generalized winding numbers are defined in association with the corresp onding Lyapunov exponents to characterize the rotation behavior of these mo des during the evolution. A Hopf bifurcation from chaos, namely, a blowout bifurcation with certain finite typical frequency, is revealed. The frequen cy of the motion after the bifurcation is justified to be equal to the gene ralized winding number of the critical transverse mode, for which the Lyapu nov exponent crosses zero at the bifurcation.