Computation of nonlinear eigenvalues based on the Ore determinant: preliminary results

The concept of eigenvalues has recently been generalized for nonlinear systems, but the method to find them is missing. Unlike the linear case, now one has to deal with non-commutative polynomials from the Ore ring. In the paper, the Ore determinant of a polynomial matrix, describing generic linearization of the state equations, is used instead of the standard definition of determinant of the polynomial matrix with real coefficients. It is shown how to compute the Ore determinant of a polynomial matrix associated with the nonlinear system and conjectured that the eigenvalues can be found from factorization of the Ore determinants of the corresponding system matrix. Moreover, it is proved that such factorization into the first-order polynomials can always be done. Many examples illustrate the computations and concepts throughout the paper.