Space–time complexity and multifractal predictability

Time complexity is associated with sensitive dependence on initial conditions and severe intrinsic predictability limits, in particular, the ‘butterfly effect’ paradigm: an exponential error growth and a corresponding characteristic predictability time. This was believed to be the universal long-time asymptotic predictability limit of complex systems. However, systems that are complex both in space and time (e.g. turbulence and geophysics) have rather different predictability limits: a limited uncertainty on initial and/or boundary conditions over a given subrange of time and space scales, grows across the scales and there is no characteristic predictability time. The relative symmetry between time and space yields scaling (i.e., power-law) decays of predictability. Furthermore, intermittency plays a fundamental role; the loss of information occurs by intermittent puffs. Therefore, contrary to the prediction of homogeneous turbulence theory its description should depend on an infinite hierarchy of exponents, not on a unique one. However, we show that for a large class of space–time multifractal processes this hierarchy is defined in a straightforward manner. We point out a few initial consequences of this result.