Algebraic techniques and perturbation methods to approach frequency response curves

The Algebra is exploited to study approximated responses of nonlinear dynamical systems leading to tracing solutions of approximated bifurcation diagrams associated with polynomial equations resulting from search of approximated periodic solutions of nonlinear ordinary differential equations. In detail via using the Gröbner basis, a polynomial with the smallest degree in term of the approximated amplitude of the systems, here the L2 norm of coefficients of truncated Fourier series, is extracted where its coefficients are parameters of the systems such as the frequency. The presented methodology permits to detect maximal number of solutions even those which belong to isola of the frequency response curves of the system. •The Algebra is exploited to study approximated periodic responses of nonlinear dynamical systems.•Searching these responses leads to polynomial equations.•Gröbner bases are built from the set of polynomial equations.•A polynomial with the smallest degree in term of the approximated amplitude of the response is extracted.