Distinguishing chaos from noise by scale-dependent Lyapunov exponent

Abstract

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 ⑀ entropy and the finite size Lyapunov exponent ͑FSLE͒ 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 ͑SDLE͒. 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 ␣ and ␣-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.

Characterizing heart rate variability by scale-dependent Lyapunov exponent

“…16 For a stochastic process, which is infinite dimensional, the embedding procedure transforms a self-affine stochastic process to a self-similar process in a phase space, and often m = 2 is not only sufficient but also best illustrates a nonchaotic scaling behavior from a finite data set. 16,17 We now become more concrete. Denote the initial distance between two nearby trajectories by 0 and their average distances at time t and t + ⌬t, respectively, by t and t+⌬t , where ⌬t is small.…”

“…SDLE as a multiscale complexity measure SDLE is defined in a phase space through consideration of an ensemble of trajectories. 16,17 In the case of a scalar time series x͑1͒ , x͑2͒ , ... ,x͑n͒, a suitable phase space may be obtained by using time delay embedding [18][19][20] to construct vectors of the form V i = ͓x͑i͒,x͑i + L͒, ... ,x͑i + ͑m − 1͒L͔͒, ͑1͒…”

“…Denote the initial distance between two nearby trajectories by 0 and their average distances at time t and t + ⌬t, respectively, by t and t+⌬t , where ⌬t is small. The SDLE ͑ t ͒ is defined by 16,17 …”

“…The debate can hardly be settled if one does not go beyond the standard theories of chaos and random fractals, since the foundations for the two theories are different: chaos theory is mainly concerned about apparently irregular behaviors in a complex system that are generated by nonlinear deterministic interactions with only a few degrees of freedom, where noise or intrinsic randomness does not play an important role, while random fractal theory assumes that the dynamics of the system are inherently random. 16 To shed new light on the problem, here we employ a new multiscale complexity measure, the SDLE, 16,17 to characterize HRV, especially the relative importance of nonlinear, chaotic, and stochastic dynamics in HRV of healthy, congestive heart failure ͑CHF͒, and atrial fibrillation ͑AF͒ subjects.…”

A nonlinear dynamics model for simulating long range correlations of cognitive bistability

“…The chaotic behavior of idealized technical systems with time-delayed feedback and zero noise as described by our perception state equation (1) was investigated in detail both theoretically (Ikeda and Matsumoto 1987) and experimentally with electro-optical devices (Derstine et al 1987), so that we refer to these publications for further details. An advanced method for distinguishing chaos from noise by the scale-dependent Lyapunov exponent which is suitable also for short time series was presented by Gao et al (2006c). In our case, it would be useful, e.g., for analysis of stationary perception states with noise term L t = 0 which might become relevant if we include in the simulation feedforward processing and the loop via SC and LGN for modeling (fixational) eye movement (see Fig.…”