Chaos or noise: Difficulties of a distinction

Abstract

In experiments, the dynamical behavior of systems is reflected in time series. Due to the finiteness of the observational data set, it is not possible to reconstruct the invariant measure up to an arbitrarily fine resolution and an arbitrarily high embedding dimension. These restrictions limit our ability to distinguish between signals generated by different systems, such as regular, chaotic, or stochastic ones, when analyzed from a time series point of view. We propose to classify the signal behavior, without referring to any specific model, as stochastic or deterministic on a certain scale of the resolution epsilon, according to the dependence of the (epsilon,tau) entropy, h(epsilon, tau), and the finite size Lyapunov exponent lambda(epsilon) on epsilon.

How random is your heart beat?

“…The method makes use of the properties and theorems of deterministic dynamical systems and chaos theory. If noise is added to the trajectory of a deterministic dynamical system (measurement noise) or if noise is present in the equation of motion (dynamical noise affecting the dynamics of the system) then the complexity measure called coarse-grained correlation entropy K 2 [21] increases. Knowing the analytical dependence of this entropy on the standard deviation of the uncorrelated noise σ [20], we can estimate the noise level from the calculation of the entropy K 2 .…”

Intrinsic chaos and external noise in population dynamics

“…As pointed out by Cencini et al, [37], all these methods have in common that one has to choose certain length scale ǫ and a particular embedding dimension m. Thus the scenarios discussed in [36,37] can be very useful in all the investigations aimed at the distinction between chaos and noise.…”

“…Several limitations have been found for the usual methods that are based on the calculation of the Lyapunov exponent and the Kolmogorov-Sinai entropy [37]. Many of the practical problems are related to the fact that these quantities are defined as infinite time averages taken in the limit of arbitrary fine resolution.…”

“…We should say here that recently several important papers have been dedicated to the question of distinguishing between generic deterministic chaos and noise [33,34,35,36,37]. Several limitations have been found for the usual methods that are based on the calculation of the Lyapunov exponent and the Kolmogorov-Sinai entropy [37].…”

“…[37] this problem is solved by introducing the (ǫ, τ ) entropy (h(ǫ, τ )), which is a generalization of the Kolmogorov-Sinai entropy with finite resolution ǫ, and where the time is discretized by using a time interval τ .…”