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.

It has recently been emphasized again that the very existence of stationary stable localized structures with short range interactions might allow to store information in non-equilibrium media, opening new perspectives on information storage. We show how to use generalized topological entropies to measure aspects of the quantities of storable and non-storable information. This leads us to introduce a measure of the long term stably storable information. As a first example to illustrate these concepts, we revisit a mechanism for the appearance of stationary stable localized structures that is related to the stabilization of fronts between structured and unstructured states (or between differently structured states). 1It has recently been emphasized again that the very existence of stationary stable localized structures with short range interactions might allow to store information in non-equilibrium media, in a way that would be both local and reversible. This has altogether opened new perspectives on information storage. While the theory of information was put of firm basis by Shannon as far back as 1948, the focus of this theory has been on models for the information itself, for which the concept of information theoretic entropy was proposed, and on transmission channels, for which the concept of channel capacity, later extended to topological entropy, was put forward. We show here that topological entropy, in its generalized form that accommodates multidimensional times as in foliation and tiling theories, characterizes nicely the amount of information that can be stored in non-equilibrium media with local stability in time. By measuring also the entropy of non-storable information, we get access to a measure of a fragility of the system. This fragility can be understood as the lack of localization of errors, hereby leading us to introduce a measure of the long term stably storable information. As a first example to illustrate these concepts, we revisit a mechanism for the appearance of stationary stable localized structures that is related to the stabilization of fronts between structured and unstructured states (or between differently structured states). The theory is far less complete for media of dimension greater that one, but the concepts and main effects of dimension one seem to work well for two dimensional media for reasons whose details elude us.

An experimental setting for the polarimetric study of optically induced dynamical behavior in nematic liquid crystal films presented by G. Cipparrone , G. Russo, C. Versace, G. Strangi, and V. Carbone allowed to identify most notably some behavior which was recognized as gluing bifurcations leading to chaos. This analysis of the data used a comparison with a model for the transition to chaos via gluing bifurcations in optically excited nematic liquid crystals previously proposed by G. Demeter and L. Kramer. The model of these last authors, relying on the original model for chaos by cascade of gluing bifurcations proposed by A. Arneodo, P. Coullet, and C. Tresser about twenty years before, does not have the central symmetry which one would expect for minimal dimensional model for chaos in nematics in view of the time series near the gluing bifurcation. What we show here is that the simplest truncated normal forms for gluing with the appropriate symmetry and minimal dimension do exhibit time signals that are embarrassingly similar to the ones that could be found using the above mentioned experimental settings. It so happens that the gluing bifurcation scenario itself is only visible in limited parameter ranges, and that substantial aspect of the chaos that can be observed is due to other factors. First, out of the immediate neighborhood of the homoclinic curve, nonlinearity can produce expansion which easily produces chaos when combined with the recurrence induced by the homoclinic behavior. Also, pairs of symmetric homoclinic orbits create extreme sensitivity to noise, so that when the noiseless approach to attracting homoclinic pairs contains a rich behavior, minute noise can transforms the complex damping into sustained chaos. As Leonid Shil'nikov has taught us, combining global considerations and local spectral analysis near critical points is crucial to understand the phenomenology associated to homoclinic bifurcations in dissipative systems. We see here on an example how this helps construct a phenomenological approach to modeling experiments in nonlinear dissipative contexts.