Response of Complex Systems to Complex Perturbations: the Complexity Matching Effect

Abstract

The dynamical emergence (and subsequent intermittent breakdown) of collective behavior in complex systems is described as a non-Poisson renewal process, characterized by a waiting-time distribution density $ψ(τ)$ for the time intervals between successively recorded breakdowns. In the intermittent case $ψ(t)\sim t^{-μ}$, with complexity index $μ$. We show that two systems can exchange information through complexity matching and present theoretical and numerical calculations describing a system with complexity index $μ_{S}$ perturbed by a signal with complexity index $μ_{P}$. The analysis focuses on the non-ergodic (non-stationary) case $μ\leq 2$ showing that for $μ_{S}\geq μ_{P}$, the system $S$ statistically inherits the correlation function of the perturbation $P$. The condition $μ_{P}=μ_{S}$ is a resonant maximum for correlation information exchange.