Superpersistent chaotic transients

Superpersistent chaotic transients are characterized by the following scaling law for its average lifetime: τ∼exp [C(p - p_c) ^-χ], where C > 0 and χ > 0 are constants, p ≥ p _c is a bifurcation parameter, and p_c is its critical value. As p approaches p_c from above, the exponent in the exponential dependence diverges, leading to an extremely long transient lifetime. Historically the possibility of such transient raised the question of whether asymptotic attractors are relevant to turbulence.Superpersistent chaotic transients were first discovered by Grebogi et al. in 1983. In their seminal work, unstable - unstable pair bifurcation was identified as the dynamical mechanism for the transients. In this Review this bifurcation and how it leads to superpersistent chaotic transients will be described. The occurrence of the transients in spatially extended dynamical systems will then be exemplified. Superpersistent chaotic transients associated with the riddling bifurcation that creates a riddled basin of attraction will be discussed, and the effect of noise on the transient lifetimes will be addressed. Finally, application to a physical problem, advection of finite-size particles in open hydrodynamical flows, will be demonstrated.