Abstract: As a key concept in nonlinear science, Lyapunov exponent has been extensively studied in literature. In general, to analyze the stability of a dynamical state, it is the sign of the Lyapunov exponent that is interested. Yet, this approach fails when the state is located exactly at the bifurcation point, where the Lyapunov exponent is equal to zero, suggesting that the perturbations will be keeping unchanged during the system evolution at this point. This prediction, however, is conflicting with the true results, as attractors also exist at the bifurcation points. Here we are able to argue mathematically and demonstrate numerically that at the bifurcation points the perturbation is damped by a power-law scaling to the equilibrium, with the scaling exponent determined by the nonlinear term of the variational equation of the perturbations. That is, the dynamics of the system shows the typical feature of critical slowing down observed at the critical point of many physical systems. The phenomenon of critical-slowing-down is demonstrated in different nonlinear models, and the theoretical predictions are in good agreement with the numerical results in all cases studied.

Key words:  Lyapunov exponent, bifurcation, critical-slowing-down, nonlinear dynamical system, chaos