Observational heterarchy enhancing active coupling

Heterarchy and the agent of self-organization are ones of most challengeable topics in complex systems. When the notion of heterarchy is generalized and dilated to general phenomena and is replaced by observational heterarchy, the agent of self-organization (i.e., internal observer carrying measurement) is found even in non-hierarchical system. A usual dynamics in the sense of complex system is articulated into two kinds of dynamics, Intent and Extent dynamics, and the interaction between them, where Intent corresponds to an attribute to a given phenomenon and Extent corresponds to a collection of objects satisfying the phenomenon. We formalize observational heterarchy by introducing pre-equivalence (pre-adjunction) between Intent and Extent, and apply it to a coupled map system. Since the model inherits evoluvability of dynamics, we call it active coupling to distinguish usual (passive) coupling from it. We show that perturbation can modify the dynamics itself and enhance robust behavior opened to emergence in the system of active coupling, and discuss the significance of observational heterarchy in complex systems.