|Abstract:||We consider the problem of constructing a quadratically nonlinear ordinary differential equation (ODE) system from time series, with the latter obtained either from experiment, or from computational simulation using a higher-dimensional mathematical model (e.g., one or more partial differential equations). Previous approaches iteratively seek a solution of the nonlinear problem of finding a set of coefficients for the quadratically nonlinear right-hand side providing the best agreement between the time series and numerical solutions of the ODE system. The present approach has several advantages compare to these iterative approaches. First, our approach involves solution only of linear algebraic equations, and avoids the problems associated with the iterative solution of a nonlinear algebraic equation system, namely the possibility of multiple solutions and failure of the iteration to converge. Second, our approach finds the ODE system which is best satisfied (in a least-square sense) by the time series, rather than attempting to find the ODE system whose solution best matches the time series. Among other advantages, this avoids the sensitive dependence on initial conditions encountered for ODE systems having solutions that exhibit chaotic behavior. The approach is illustrated with numerical examples demonstrating its utility for a variety of nonchaotic and chaotic time series, including systems where the time series is corrupted by multiplicative noise, and for cases where a given ODE system has two qualitatively different solutions for different initial conditions.