Simple algebraic necessary and sufficient conditions for Lyapunov stability of a Chen system and their applications

In this paper, we study the Lyapunov stability problem of a Chen chaotic system. Because of the positive elements of the main diagonal of a linearized Chen system, compared to the coefficient of a linearized Lorenz system which are all negative, it is more difficult to deal with the stability analysis. Since it has the properties of invariance and symmetry, different Lyapunov functions in different regions are constructed to solve stability problems with geometric and algebraic methods. Then, simple algebraic necessary and sufficient conditions of global exponential stability, global asymptotic stability and global instability of equilibrium S0(0, 0, 0). are proposed. We obtain the relevant expression of corresponding parameters for local exponential stability, local asymptotic stability and local instability of equilibria S +/-(+/-root b(2c - a), +/-root b(2c - a, (2c - a)). Furthermore, the smallest conservative linear feedback controllers are used to globally exponentially stabilize equilibria.