Stability analysis of polynomials with an approach of differential topology

In this paper, we study the Hurwitz stability of polynomials. By considering that a 2m-$$ 2m- $$degree Hurwitz polynomial has its corresponding Markov parameters, we define the set Hankel2m$$ {Hankel}_{2m} $$ in Section 3, we also define Hankel2m-1$$ {Hankel}_{2m-1} $$. Based on properties of the Hankel matrices and the stability test, as well as by using ideas of differential topology, we show that Hankel2m$$ {Hankel}_{2m} $$ is a fiber bundle with a Hankel2m-1$$ {Hankel}_{2m-1} $$ base. This result allows us to obtain an interesting application: Given a Hurwitz polynomial, we can generate two families of positive definite Hankel matrices.