arXiv:2610.17604v1 Announce Type: new Abstract: We develop analytical sufficient conditions for certifying the local asymptotic stability of nonlinear autonomous systems based on Lyapunov functions derived from their finite-order Taylor polynomial approximations. This approach translates the stability analysis problem from the original nonlinear dynamics to a more computationally tractable polynomial domain. The derived conditions allow for the explicit calculation of a guaranteed inner approximation of the region of attraction ($\mathrm{ROA}$). We show how sum-of-squares (SOS) programming and grid search can be used to systematically search for polynomial Lyapunov functions that yield a large certified threshold radius and extract a certified invariant set inner approximation of the $\mathrm{ROA}$ of the nonpolynomial original system. The framework is then applied to a control tuning problem: finding reduced-effort Linear Quadratic Regulator (LQR) controllers that ensure that a given initial state of interest belongs to the certified $\mathrm{ROA}$ of the nonpolynomial closed loop system. The effectiveness of the overall methodology is demonstrated using a pendulum with nonlinear damping and a nonlinear control system.