A generic intermittency model and its 1-D meta-map: Power laws, invariants and the succession of laminar sequences

Intermittent behaviour has been found in many systems able to switch between two different dynamic states, e.g. between long laminar phases and short chaotic bursts. Despite the apparently high-dimensional complexity, certain one-dimensional (1-D) maps are known to mimic properties of such dynamics. To these belongs the iterative map x(n+1i) = (x(n,i) + (x(n,i))(z) + epsilon) mod 1, giving rise to long laminar lengths. The statistics of the laminar lengths are of special interest. Starting from this map, we are interested in the values of x(0,1) which arise after passing through the module operation. These determine the laminar lengths uniquely. A 1-D meta-map x(0,i) = f(x(0,i-1)) is derived heuristically. It is used to calculate statistical properties of the laminar phases. Our results show an improvement in the behaviour of short and very long laminar phases as compared to earlier analytical results. Introducing the concept of the generic starting value, we find laminar phases not to be strictly independent of their predecessors.