Sufficient conditions for instability of equilibrium for circulatory and gyroscopic systems

We give a method which generates sufficient conditions for instability of equilibria for circulatory and gyroscopic conservative systems. The method is based on the Gramians of a set of vectors whose coordinates are powers of the roots of the characteristic polynomial for the studied systems. New instability results are obtained for general circulatory and gyroscopic conservative systems. We also apply this method for studying the instability of motion for a charged particle in a stationary electromagnetic field.

► Sufficient conditions for the existence of a complex non-real root for a polynomial.
► Sufficient conditions of instability for circulatory systems.
► Sufficient conditions of instability for gyroscopic conservative systems.
► Instability conditions in terms of norms and traces of the matrices of the system.
► Instability regions for the case of a charged particle acted on by the Lorentz force.