Stability of sign patterns from a system of second order ODEs

The stability and inertia of sign pattern matrices with entries in {+,−,0} associated with dynamical systems of second-order ordinary differential equations x¨=Ax˙+Bx are studied, where A and B are real matrices of order n. An equivalent system of first-order differential equations has coefficient matrix C=[ABIO] of order 2n, and eigenvalue properties are considered for sign patterns C=[ABDO] of order 2n, where A,B are the sign patterns of A,B respectively, and D is a positive diagonal sign pattern. For given sign patterns A and B where one of them is a negative diagonal sign pattern, results are determined concerning the potential stability and sign stability of C, as well as the refined inertia of C. Applications include the stability of such dynamical systems in which only the signs rather than the magnitudes of entries of the matrices A and B are known.