Response of Complex Systems to Complex Perturbations: Complexity Matching

We argue that complex systems, defined as non-Poisson renewal process, with complexity index $\mu$, exchange information through complexity matching. We illustrate this property with detailed theoretical and numerical calculations describing a system with complexity index ${\mu }_S$ perturbed by a signal with complexity index ${\mu }_P$. We focus our attention on the case $1.5 \leq \mu_S \leq 2$and$1 \leq \mu_{P} \leq 2$\mathrm{.}Weshowthatfor$\mu_{S} \geq \mu_P$, the system S reproduces the perturbation, and the response intensity increases with increasing ${\mu }_P$. The maximum intensity is realized by the matching condition ${\mu }_P={\mu }_S$. For ${\mu }_P>{\mu }_S$ the response intensity dies out as $1\mathrm{/}{t}^{{\mu }_P-{\mu }_S}$.