Qualitative behaviors of the high-order Lorenz-Stenflo chaotic system arising in mathematical physics describing the atmospheric acoustic-gravity waves

Guangyun Zhang,Min Xiao,Fuchen Zhang

doi:10.1186/s13662-017-1351-7

Abstract

The boundedness of chaotic systems plays an important role in investigating the stability of the equilibrium, estimating the Lyapunov dimension of attractors, the Hausdorff dimension of attractors, the existence of periodic solutions, chaos control, and chaos synchronization. However, as far as the authors know, there are only a few papers dealing with bounds of high-order chaotic systems due to their complex algebraic structure. To sort this out, in this paper, we study the bounds of a high-order Lorenz-Stenflo system arising in mathematical physics. Based on Lyapunov stability theory, we show that there exists a globally exponential attractive set for this system. The innovation of the paper is that we not only prove that this system is globally bounded for all the parameters, but also give a family of mathematical expressions of global exponential attractive sets of this system with respect to its parameters. We also study some other dynamical characteristics of this chaotic system such as invariant sets and chaotic behaviors. To justify the theoretical analysis, we carry out detailed numerical simulations.

Highlights

Chaos phenomena and chaotic systems have been extensively studied by many researchers due to their various applications in the fields of atmospheric dynamics, population dynamics, electric circuits, cryptology, fluid dynamics, lasers, engineering, stock exchanges, chemical reactions, and so on [ – ]
2 Some dynamics of high-order Lorenz-Stenflo system 2.1 Invariance The positive z-axis, u-axis, and ω-axis are invariant under the flow, that is, they are positively invariant under the flow generated by system ( )
3 Conclusions By means of Lyapunov-like functions, we have studied some dynamical behaviors of a high-order Lorenz-Stenflo system using theoretical analysis and numerical simulations

Summary

Introduction

Chaos phenomena and chaotic systems have been extensively studied by many researchers due to their various applications in the fields of atmospheric dynamics, population dynamics, electric circuits, cryptology, fluid dynamics, lasers, engineering, stock exchanges, chemical reactions, and so on [ – ]. The Lorenz-Stenflo system is a four-dimensional continuous-time dynamical system, derived to model atmospheric acoustic-gravity waves in a rotating atmosphere. Many dynamical behaviors such as stability [ ], bifurcation [ , ], periodic solutions [ ] and chaotic behaviors [ ] of the Lorenz-Stenflo equations have been thoroughly studied for decades after Stenflo.

Results

Conclusion