  External flows of energy, entropy, and matter can cause sudden transitions in
the stability of biological and industrial systems, fundamentally altering
their dynamical function. How might we control and design these transitions in
chemical reaction networks? Here, we analyze transitions giving rise to complex
behavior in random reaction networks subject to external driving forces. In the
absence of driving, we characterize the uniqueness of the steady state and
identify the percolation of a giant connected component in these networks as
the number of reactions increases. When subject to chemical driving (influx and
outflux of chemical species), the steady state can undergo bifurcations,
leading to multistability or oscillatory dynamics. By quantifying the
prevalence of these bifurcations, we show how chemical driving and network
sparsity tend to promote the emergence of these complex dynamics and increased
rates of entropy production. We show that catalysis also plays an important
role in the emergence of complexity, strongly correlating with the prevalence
of bifurcations. Our results suggest that coupling a minimal number of chemical
signatures with external driving can lead to features present in biochemical
processes and abiogenesis.
