Type III intermittency without characteristic relation

Abstract

Chaotic intermittency is a route to chaos when transitions between laminar and chaotic dynamics occur. The main attribute of intermittency is the reinjection mechanism, described by the reinjection probability density (RPD), which maps trajectories from the chaotic region into the laminar one. The RPD classically was taken as a constant. This hypothesis is behind the classically reported characteristic relations, a tool describing how the mean value of the laminar length goes to infinity as the control parameter goes to zero. Recently, a generalized non-uniform RPD has been observed in a wide class of 1D maps; hence, the intermittency theory has been generalized. Consequently, the characteristic relations were also generalized. However, the RPD and the characteristic relations observed in some experimental Poincaré maps still cannot be well explained in the actual intermittency framework. We extend the previous analytical results to deal with the mentioned class of maps. We found that in the mentioned maps, there is not a well-defined RPD in the sense that its shape drastically changes depending on a small variation of the parameter of the map. Consequently, the characteristic relation classically associated to every type of intermittency is not well defined and, in general, cannot be determined experimentally. We illustrate the results with a 1D map and we develop the analytical expressions for every RPD and its characteristic relations. Moreover, we found a characteristic relation going to a constant value, instead of increasing to infinity. We found a good agreement with the numerical simulation.