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Abstract: Systems of N equations in N unknowns are ubiquitous in mathematical modeling. Thesesystems, often nonlinear, are used to identify equilibria of dynamical systemsin ecology, genomics, control, and many other areas. Structured systems, wherethe variables that are allowed to appear in each equation are pre-specified,are especially common. For modeling purposes, there is a great interest indetermining circumstances under which physical solutions exist, even if thecoefficients in the model equations are only approximately known.

Thestructure of a system of equations can be described by a directed graph G that reflects the dependence of one variable onanother, and we can consider the family F(G) of systems that respect G.

We define a solution X of F(X)=0 to be robust if for each continuous FÃ¢ÂÂ sufficiently close to F, a solution XÃ¢ÂÂ exists. Robust solutions are those that are expected tobe found in real systems. There is a useful concept in graph theory called"cycle-coverable". We show that if G is cycle-coverable, then for "almost every" FÃ¢ÂÂF(G) in the sense of prevalence, every solutionis robust. Conversely, when G fails to be cycle-coverable, each system FÃ¢ÂÂF(G) has no robust solutions.

Failureto be cycle-coverable happens precisely when there is a configuration of nodesthat we call a "bottleneck," a criterion that can be verified fromthe graph. A "bottleneck" is a direct extension of what ecologistscall the Competitive Exclusion Principle, but we apply this idea from theequations of nature to describe the nature of almost all structured systems.Sana Jahedi, Timothy Sauer, James A. YorkePosted on arXiv Jan 2021 A recording of the talk can be found here: https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=7a19d4c3-c70b-4e34-bbd3-acc5015366ab