Investigation of the global asymptotic stability of multidimensional phase systems

Abstract

A general theory of global asymptotic stability of multidimensional dynamical systems with a cylindrical phase space with a countable equilibrium position is created. The boundedness of solutions of multidimensional phase systems and their derivatives is established. Conditions for the fulfillment of which the solution and its derivative have asymptotic properties are found. Conditions for global asymptotic stability of multidimensional phase systems with values of integrals equal to zero in the period from the components of periodic nonlinearities are obtained. Conditions for global asymptotic stability of phase systems with nonzero values of the integrals of the components of nonlinear periodic functions are obtained. The asymptotic properties of solutions of dynamical systems with a countable equilibrium position are investigated in the general case when some of the components of nonlinear periodic functions have values of the integrals in the period equal to zero, and for other components the values of the integrals in the period are not equal to zero. A distinctive feature of the proposed method for investigating multidimensional phase systems from known methods is that it is applicable to systems of any order with any number of nonlinear periodic functions, and are not involved in research periodic Lyapunov functions and frequency theorems. It is noteworthy, that the proposed conditions for global asymptotic stability, which are easily verified in comparison with the frequency conditions and conditions obtained with the help of periodic Lyapunov functions.