On the application of Simple Feedback Diagrams in nature and society: daisy topology and tipping points (part I)

Abstract. Feedback loops are common in all kinds of complex systems and are a way to understand these systems’ large-scale behaviour. Feedback diagrams are often used to illustrate such positive or negative loops qualitatively. The theory presented here allows these feedbacks to be quantified and their stability properties computed, in order to make the diagrams more widely applicable. In particular, the theory derived here can be used to describe the dynamics beyond the tipping point of a system and thereby to estimate fluctuations and extremes during the unfolding of the tipping process. To this end we derive simple rules for computing the Lyapunov exponents that determine stability from simple estimates of the processes that form a feedback. We exploit the simplification that follows from a loop structure, which is much more tractable than a situation in which every variable depends on every other variable. To make the diagrams useful across applications we (a) show that loops with multiple processes can be collapsed into two-process loops, (b) derive the relevant Lyapunov exponent from a loop’s feedback strength and timescale, (c) derive rules for combining multiple loops into a system of loops sharing one central variable – a daisy topology and (d) derive rule to combine multiple daisy topologies. We show the existence of oscillatory instabilites in systems with at least 3 variables. The analysis allows the identification and quantification of thresholds, so-called tipping points, and the computation of their stability properties. It requires a minimum of information about the underlying system and is thereby applicable in physical, chemical and biological systems as well as in economic and societal ones. We illustrate the approach with examples from climate – the melt-elevation feedback of ice sheets and the water-vapour greenhouse feedback – and from society, using the feedback between wages and prices and a known business-cycle model. Finally, we provide a cookbook for a feedback and tipping-point analysis, with or without known governing equations.