The Phase Transition of Covariant Lyapunov Vector Precisely Locates a Stability Reversal of Quasi-Periodic Response

Abstract

Assessing the stability of quasi-periodic (QP) response is crucial, as the bifurcation of QP response is usually accompanied by a stability reversal. The largest Lyapunov exponent (LLE), as an important indicator for chaotic motion, can also be used for the stability analysis of QP response. The precise location of a stability reversal, however, is tough to achieve as a poor convergence rate would be usually encountered when solving the LLE. Herein a straightforward and precise approach is suggested to identify the critical point when a stability reversal happens. Our approach is based on an explicit differential equation that provides the LLE straightforwardly via numerical integration, and the corresponding covariant Lyapunov vector is simultaneously obtained. The major finding consists in the phase transition of the covariant Lyapunov vector, which can happen much early before the LLE reaches a relatively convergent value. More importantly, the phase transition can serve as a strong indicator to locate a stability reversal of the QP response qualitatively. Numerical examples are provided to verify of the effectiveness and wide applicability the presented approach.