Two theorems of instability with application to gimbal walk

Two theorems of instability, different from the traditional Chetaev's insta bility theorem are developed. The theorems determine the instability of aut onomous and non-autonomous dynamical systems by means of investigating the higher-order' derivatives of the Lyapunov function. Using the first theorem of these theorems, the walk of a free gyroscope, which has not been verifi ed by traditional stability theorem, can be solved by our theorem. Furtherm ore, several dynamical systems are presented as examples of application of these two theorems. They are the motion of a symmetric gyroscope with stead y precession, the steady rotations of a rigid body around the intermediate axis of its ellipsoid of inertia, the equilibrium of the equations of Lotka -Volterra model of competition between two species, etc.