On the subject of influence of dissipative and constant of moments on the stability of uniform rotations of non-free two elastically connected gyroscopes of Lagrange

The conditions for asymptotic stability of uniform rotations in a resisting medium of two heavy Lagrange gyroscopes connected by an elastic spherical hinge are obtained in the form of a system of three inequalities. The bottom gyroscope has a fixed point. The rotation of the gyroscopes is maintained by constant moments in the inertial coordinate system. The influence of the elasticity of the hinge on the stability conditions is estimated. It is shown that for a sufficiently high rigidity of the hinge, the asymptotic stability conditions are determined by only one inequality, which coincides with the inequality obtained for the case of a cylindrical hinge. When the angular velocities of the gyroscopes' own rotations coincide, this inequality coincides with the well--known condition for one gyroscope. Cases of degeneration of an elastic spherical hinge into a spherical inelastic, cylindrical and universal elastic hinge (Hooke's hinge) are considered. For the Hooke hinge, it is shown that there is no asymptotic stability at a sufficiently high angular velocity of gyroscopes rotation.

Mean Square Asymptotic Boundedness of Stochastic Complex Networks via Impulsive Control

In this paper, the mean square asymptotic boundedness of a class of stochastic complex systems with different dynamic nodes represented by Ito stochastic differential equations is studied. By using the Lyapunov function and Ito formula, the mean square asymptotic boundedness and mean square asymptotic stability conditions of stochastic complex systems with different dynamic nodes are obtained.

Study of Samuelson’s General Model of National Economy and Definition of Asymptotic-Stability Conditions

In this particular paper we firstly deal with Samuelson’s model of national economy. We create a difference equation which reflects Samuelson’s model for the national income of a country taking into consideration the expenditure and the investments of the two previous years and not only the immediately previous one. Later we find the saddle-point and deal with its stability giving conditions concerning the coefficient of the difference equation and which are able (sufficient) and necessary in order for the saddle-point to be stable.

On a three term linear difference equation with complex coefficients

The paper discusses some known asymptotic stability conditions for the difference equation xn = αxn-1 + βxn-k, n = 0, 1, 2... with complex numbers As the main result, we show that the system of necessary and sufficient stability conditions, derived in a previous work, can be significantly simplified.