Opening Stable Doors: Complexity and Stability in Nonlinear Systems

Abstract

Generic complex systems of many interacting parts can model both natural and artificial systems, and the conditions for their stability are of interest. Two influential papers (Gardner and Ashby, 1970; May, 1972) laid down a mathematical framework suggesting that, without some specific constraints on the interactions, such systems are very likely to be unstable as they increase in size and connectance. We draw attention to a programming error in the first paper and to flaws and omissions in reasoning in the second that discredit such conclusions when applied to nonlinear systems. With nonlinearity the connectance strength of an influence of any one variable upon any other will vary according to context, which May's analysis does not address. Further, in nonlinear systems there can be many equilibria, and global instability requires every relevant local equilibrium to be unstable; neglecting this invalidates the conclusions. We discuss the relevance of ambiguous circuits (Thomas and D'Ari, 1990) and consider simple classes of nonlinear functions that generate these, including the hat shaped viability functions that generate homeostasis in Daisyworld models. We demonstrate that the May results are unreliable even for the simplest families of nonlinear systems that model common biological, physical or artificial systems.