The geometry of scales: chameleon attractors

In 1963 Lorenz discovered what is usually known as “chaos”, that is the sensitive dependence of deterministic chaotic systems upon initial conditions. Since then, this concept has been strictly related to the notion of unpredictability pioneered by Lorenz. However, one of the most interesting and unknown facets of Lorenz ideas is that multiscale fluid flows could spontaneously lose their deterministic nature and become intrinsically random. This effect is radically different from chaos. Turbulent flows are the natural systems when Lorenz ideas can be touched by the hand. They can, indeed, be described via the Navier-Stokes equations, thus conforming to the class of deterministic dissipative systems, as well as, present rich dynamics originating from non-trivial energy fluxes in scale space, non-stationary forcings and geometrical constraints. This complexity appears via non-hyperbolic chaos, randomness, state-dependent persistence and predictability. All these features have prevented a full characterization of the underlying turbulent (stochastic) attractor, which will be the key object to unpin this complexity. 

Here we use a novel formalism to map unstable fixed points to singularities of turbulent flows and to trace the evolution of their structural characteristics when moving from small to large scales and vice versa, providing a full characterization of the attractor. We demonstrate that the properties of the dynamically invariant objects depend on the scale we are focusing on. Our results provide evidence that the large-scale properties of turbulent flows display universal statistical properties that are triggered by, but independent of specific physical properties at small scales. Given the changing nature of such attractors in time, space and scale spaces, we term them chameleon attractors.