Distinguishing chaos from noise by scale-dependent Lyapunov exponent

Time series from complex systems with interacting nonlinear and stochastic subsystems and hierarchical regulations are often multiscaled. In devising measures characterizing such complex time series, it is most desirable to incorporate explicitly the concept of scale in the measures. While excellent scale-dependent measures such as $ϵ$ entropy and the finite size Lyapunov exponent (FSLE) have been proposed, simple algorithms have not been developed to reliably compute them from short noisy time series. To promote widespread application of these concepts, we propose an efficient algorithm to compute a variant of the FSLE, the scale-dependent Lyapunov exponent (SDLE). We show that with our algorithm, the SDLE can be accurately computed from short noisy time series and readily classify various types of motions, including truly low-dimensional chaos, noisy chaos, noise-induced chaos, random $1∕f^{\alpha }$ and $\alpha$-stable Levy processes, stochastic oscillations, and complex motions with chaotic behavior on small scales but diffusive behavior on large scales. To our knowledge, no other measures are able to accurately characterize all these different types of motions. Based on the distinctive behaviors of the SDLE for different types of motions, we propose a scheme to distinguish chaos from noise.