  Dynamical systems with polynomials on the right-hand side can model a wide
range of physical processes. A subset of such dynamical systems that can model
chemical reactions under mass-action kinetics are called chemical systems. A
central problem in synthetic biology is to map general polynomial dynamical
systems into dynamically similar chemical ones. In this paper, we present a
novel map, called the quasi-chemical map, that can systematically solve this
problem. The quasi-chemical map introduces suitable state-dependent
perturbations into any given polynomial dynamical system which then becomes
chemical under suitably large translation of variables. We prove that this map
preserves robust dynamical features, such as generic equilibria and limit
cycles, as well as temporal properties, such as periods of oscillations.
Furthermore, the resulting chemical systems are of only at most one degree
higher than the original dynamical systems. We demonstrate the quasi-chemical
map by designing relatively simple chemical systems with exotic dynamics and
pre-defined bifurcation structures.
