Positive invariance tests with efficient Hessian matrix eigenvalues bounds

We investigate two simple sufficient criteria for positive invariance of sets in the domain of n-dimensional nonlinear autonomous discrete time systems. These criteria are derived from the exact Taylor expansion with linear and quadratic remainder terms. By a simple example we demonstrate that systems exist for which positive invariance can be established with the second order criterion but not with the first order criterion. Since the second order criterion requires the Hessian matrices of the model equations, this criterion is computationally expensive. We show, however, that the second order criterion can be evaluated at a surprisingly low computational cost. Specifically, we show that the computational complexity is an order of magnitude lower than the calculation of the Hessian matrices.