Novel Versions of D-Stability in Matrices Provide New Insights into ODE Dynamics

It is well-known that a second-order ODE system admits a general matrix form of notation. In this work, we study general relations between system stability/robustness properties and special kinds of matrix stabilities, such as D- and diagonal stability. Basing on these concepts, we provide new stability conditions for second-order dynamical systems and analyze the stability of a parameter-dependent second order model. Next, we study the relations between transient response properties of a second-order ODE system and certain generalizations of D-stability concept obtained in Kushel (SIAM Rev 61(4):643-729, 2019). We provide the conditions when the system has a given minimal decay rate a and when the minimal decay rate of a system is preserved for some variations of positive parameters. A certain way of gyroscopic stabilization of an unstable parameter-dependent system is also considered.