On the dynamics of new 4D Lorenz-type chaos systems

Guangyun Zhang,Da Lin,Ping Zhou,Xiaofeng Liao,Fuchen Zhang

doi:10.1186/s13662-017-1280-5

Abstract

It is often difficult to obtain the bounds of the hyperchaotic systems due to very complex algebraic structure of the hyperchaotic systems. After an exhaustive research on a new 4D Lorenz-type hyperchaotic system and a coupled dynamo chaotic system, we obtain the bounds of the new 4D Lorenz-type hyperchaotic system and the globally attractive set of the coupled dynamo chaotic system. To validate the ultimate bound estimation, numerical simulations are also investigated. The innovation of this article lies in that the method of constructing Lyapunov-like functions applied to the Lorenz system is not applicable to this 4D Lorenz-type hyperchaotic system; moreover, one Lyapunov-like function cannot estimate the bounds of this 4D Lorenz-type hyperchaos system. To sort this out, we construct three Lyapunov-like functions step by step to estimate the bounds of this new 4D Lorenz-type hyperchaotic system successfully.

Highlights

In, Lorenz et al found the famous Lorenz chaotic system, which can be described by the following autonomous differential equations [ ]: ⎧ ⎪⎪⎨ dx dt = σ (y x), ⎪⎪⎩ dy dt dz dt = =ρx – y – xy – rz.xz, Since chaotic systems have been extensively studied, such as the Rössler system [ ], Chua’s circuit [ ], the Chen system [ ], the Lü system [ – ], the hyperchaos Lorenz system [ ], the Shimizu-Morioka system [ ], the Liu system [ ]
3 Ultimate bound sets for the chaotic attractors in (2) and (3) Recently, ultimate bound estimation of chaotic systems and hyperchaotic systems has been discussed in many research studies [, ]
The ultimate bound sets can be employed for estimation of the fractal dimension of chaotic and hyperchaotic attractors, such as the Hausdorff dimension and the Lyapunov dimension [, ]

Summary

Introduction

Hyperchaotic systems can be obtained by adding one more state variable to a three-dimensional chaotic system [ ]. [ ] and [ ] for a detailed discussion of Lyapunov exponents of strange attractors in dynamical systems). System ( ) has two positive Lyapunov exponents and the strange attractor, which means the new system ( ) can exhibit a variety of interesting and complex chaotic behavior.

Results

Conclusion