Signature of on-off intermittency in measured signals

Abstract

On-off intermittency is a phase-space mechanism that allows dynamical systems to undergo bursting. As its name suggests, bursting is a phenomenon in which episodes of high activity are alternated with periods of inactivity. Here we attempt to see whether we can tell from the output of a signal when an observed bursting behavior is caused by the presence of on-off intermittency, using the example of a forced logistic map. The results of our study indicate that on-off intermittency can be readily distinguished from other mechanisms for bursting we know of, except for one. Many statistical properties of finite-length signals generated by on-off intermittency can in fact be mimicked by the output of a nonlinearly filtered, linear autoregressive random process.

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“…In the intermittent system under discusssion, both the signal amplitude and the power spectrum resulted [26] in power-law distributions (with low frequencies predominating in the power-spectrum)…”

“…A is further specified below). The system has the invariant manifold X = 0 and the level of its activity is measured by X t [26]. In order to have the on-off mechanism in action, we specialize to the case of a noise-driven logistic map, with…”

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“…Under the assumption that r is independent of N t (the skew product structure), we have to choose a way of determining r at each time step. Heagy et al (1994) and Toniolo et al (2002)'s works indicate that the method of determining r does not influence on-off intermittency. However, without specific physical constraints, we choose Gaussian white noise as our driving method with appeal to the central limit theorem.…”

“…Here, r t is the parameter determining the stability of the invariant manifold X t = 0 and hence r t = A 0 + σξ t , where A 0 is the average of r t and ξ t is uniformly distributed noise between 0 and 1. Depending on the choice of A 0 and σ, we can generate an on-off intermittency signal in X (Toniolo et al 2002). This simple setting for on-off intermittency reveals several important characteristics of an on-off intermittency signal.…”

“…They also found numerically that the probability density function does not depend on the choice of the driving parameters, which hints of the possibility that the −3/2-law could be general for on-off intermittency generated by various dynamical structures. Toniolo et al (2002) sought out various characteristics of on-off intermittency using power spectrum analysis and found a clear negative slope in a log-log scale plot. These results can be used to identify the on-off intermittency in the time series obtained from observation or experiments.…”