Analysis of a Generalized Lorenz–Stenflo Equation

Although the globally attractive sets of a hyperchaotic system have important applications in the fields of engineering, science, and technology, it is often a difficult task for the researchers to obtain the globally attractive set of the hyperchaotic systems due to the complexity of the hyperchaotic systems. Therefore, we will study the globally attractive set of a generalized hyperchaotic Lorenz–Stenflo system describing the evolution of finite amplitude acoustic gravity waves in a rotating atmosphere in this paper. Based on Lyapunov-like functional approach combining some simple inequalities, we derive the globally attractive set of this system with its parameters. The effectiveness of the proposed methods is illustrated via numerical examples.

In this paper, all the simulations are carried out by using fourth-order Runge-Kutta Method with time-step ℎ = 0.005.
The rest of this paper is organized as follows.In Section 2, the globally attractive set for the chaotic attractors in (2) is studied using Lyapunov stability theory.To validate the ultimate bound estimation, numerical simulations are also provided.Finally, the conclusions are drawn in Section 3.
Proof.Define the following functions: then we can get Construct the Lyapunov-like function Differentiating the above Lyapunov-like function  , () in (11) with respect to time  along the trajectory of system (2) yields Thus, we have which clearly shows that Ω , = { |  , () ≤  2 , } is the globally exponential attractive set of system (2).
The proof is complete.
Let ((), (), (), ()) be an arbitrary solution of system (2) and Then the estimation holds for system (2), and thus is the globally exponential attractive set and positive invariant set of system (2); that is, (ii) Let us take  1 = 0,  3 = 0,  = 1,  = 1; then we can get as the globally exponential attractive set and positive invariant set of system (2) according to Theorem 2.

Conclusions
In this paper, we have investigated some global dynamics of a generalized Lorenz-Stenflo system describing the evolution of finite amplitude acoustic gravity waves in a rotating atmosphere.Based on the Lyapunov method, the globally attractive sets were formulated combining simple inequalities.Finally, numerical examples were presented to show the effectiveness of the proposed method.