Generalized eigendecomposition, which extracts the generalized eigenvector from a matrix pencil, is a powerful tool and has been widely used in many fields, such as data classification and blind source separation. First, to extract the minor generalized eigenvector (MGE), we propose a deterministic discrete-time (DDT) system. Unlike some existing systems, the proposed DDT system does not need to normalize the weight vector in each iteration, since the weight vectors in the proposed DDT system are self-stabilizing. Second, we propose an adaptive algorithm corresponding to the proposed DDT system. Moreover, we study the dynamic behavior and convergence properties of the proposed DDT system and prove that the weight vector must converge to the direction of the MGE of a matrix pencil under some mild conditions. Numerical simulations show that the proposed algorithm has a better performance in terms of convergence speed and estimation accuracy than some existing algorithms. Finally, we conduct two experiments on real data sets to demonstrate its practicability.