On the stability of third order differential equations

In this paper, we study the problem of stability of the equilibrium of nonlinear ordinary differential equations by the method of semi-definite Lyapunov’s functions. We have identified nonlinear third order differential equations of general form for which the choice of a semi-definite function does not present difficulties. For such equations, sufficient conditions of stability and asymptotic stability (local and global) are obtained. The results of asymptotic stability of the equilibrium coincide with necessary and sufficient conditions in the corresponding linear case. Consequently, they meet generally accepted requirements. The conducted studies show that the use of semi-defined positive functions can give advantages in comparison with the classical method of application of Lyapunov’s definite positive functions.