Stability of the Lorentz System

Abstract

In this paper, a variational method is proposed for obtaining the necessary (and sufficient) stability conditions for perturbed solutions of the system of Lorentz equations. This method allows us to establish the necessary conditions for Lyapunov stability. It is effective even in cases when the application of the classical Lyapunov method causes difficulties associated with the construction of the Lyapunov function or with inaccuracies of Taylor linearization, which is typical for dynamical systems of large dimension. In some cases, this method can be used to find regions of phase variables in which the necessary stability conditions coincide with sufficient stability conditions (asymptotic stability) according to Lyapunov. In these cases, the metric function itself, as shown in this paper, can play the role of the Lyapunov function to obtain sufficient stability conditions.

Keywords

• the lorentz system
• nonlinear dynamic system
• necessary conditions for stability
• partial derivatives of the first and second orders
• the attractor
• andronov–hopf bifurcation