On the Calculation of the Equilibrium Points of a Nonlinear System via Exact Quadratization

We show that, for quadratizable dynamic nonlinear systems, all the equilibrium points satisfy an augmented system of nonlinear equations obtained by applying exact quadratization to a suitably modified dynamic system. For autonomous polynomial dynamic systems (i.e. with no input) this method can be viewed as a general method of solving systems of polynomial equations, which has a lower computational complexity with respect to the classical one consisting of the Buchberger’s algorithm (that searches for a Groebner basis) followed by variable elimination. As a matter, we show that the augmented system of polynomials obtained by quadratization is always a Groebner basis for the ideal associated to the originary problem. This allows skipping the computationally heavy Buchberger’s algorithm, and applying directly elimination theory in the solution-searching algorithm.