The Dynamics of Cross-coupled, Self-referential Linear Systems

This article presents the central ideas behind cross-coupled, self-referential linear systems—dual systems where each system in the pair provides the other system with reference or control signals using a form of state feedback. Such systems are ubiquitous in nature, the most noteworthy being the mammalian brain. Although complex systems and feedback mechanisms have several decades worth of literature, the fundamental aspects of simple cross-coupled linear systems have apparently not been fully explored or articulated. This surprising fact and the simplicity of the concepts involved provide a backdrop in which the fundamental nature of cross-coupled systems are investigated by examination of linear iterative maps. The cross-coupling effect in iterative maps is shown to reduce the magnitude of the eigenvalues of the linear system when the two inputs to the system are unequal. Applications to the solution of linear systems are also presented and shown to enlarge the applicability of the Gauss-Seidel iterative method. Self-similarity and scaling properties are also examined in which cross-coupled systems are cross-coupled. The eigenvalues in such systems have multiplicities described by Pascal’s Triangle. Future research areas such as neural networks, control systems, and Markov Decision Processes are discussed including ideas on how such cross-coupled systems can serve as a model for autonomous control systems and even for human consciousness.