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

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## 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.

### Keywords

hyperchaotic systems stability invariant sets domain of attraction computer simulation## 1 Introduction

*et al*. found the famous Lorenz chaotic system, which can be described by the following autonomous differential equations [1]:

*et al*. constructed a new chaotic system based on the Lorenz chaotic system [22]:

*w*in the first equation, and adding a first-order nonlinear differential state equation with respect to

*w*, one gets a new 4D Lorenz-type chaotic system as follows:

*x*,

*y*,

*z*and

*w*are state variables and

*a*,

*b*,

*c*,

*d*,

*e*,

*h*are positive parameters of system (2).

In this paper, all the simulations are carried out by using the fourth-order Runge-Kutta method with time-step \(h = 0.01\).

The rest of this paper is organized as follows. The invariant sets of chaotic systems (2) and (3) are analyzed in Section 2. In Section 3, ultimate bound sets for the chaotic attractors in (2) and (3) are studied using Lyapunov stability theory. To validate the ultimate bound estimation, numerical simulations are also provided. Finally, the conclusions are drawn in Section 4.

## 2 Invariance analysis for the chaotic attractors in (2) and (3)

The positive *z*-axis is invariant under the flow generated by system (2), that is to say, *z*-axis is positively invariant under the flow generated by system (2). However, this is not the case on the positive *x*-axis, *y*-axis and *w*-axis for system (2). \(x_{1}\)-axis, \(x_{2}\)-axis, \(w_{1}\)-axis and \(w_{2}\)-axis are all not positively invariant under the flow generated by system (3).

## 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 [7, 17, 20, 31]. It is well known that there is a bounded ellipsoid in \(R^{3}\) for the Lorenz system which all orbits of the Lorenz system will eventually enter for all positive parameters [20, 31]. The ultimate bound sets can be used in chaos control and synchronization [32]. Also, 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 [12, 34].

Motivated by the above discussion, we will investigate the bounds of the new 4D Lorenz-type hyperchaotic system (2) and the disk dynamo system (3) in this section. The main results are described by Theorem 1 and Theorem 2.

### 3.1 Ultimate bound sets for the chaotic attractors in system (2)

### Theorem 1

*Suppose that*\(\forall a > 0\), \(h > 0\), \(c > 0\), \(d > 0\).

*Let*\(( x ( t ),y ( t ),z ( t ),w ( t ) )\)

*be an arbitrary solution of system*(2).

*Then the set*

*is the ultimate bound set of chaotic system*(2),

*where*

### Proof

### Remark 1

(ii) From Figure 5, we can see that the bounds estimate for the chaotic attractors of system (2) is conservative, we can get a smaller bound of chaotic attractors of system (2) with the help of the iteration theorem in [32] (see [32] for a detailed discussion of the bounds of chaotic systems).

### 3.2 Bounds for the chaotic attractors in system (3)

### Theorem 2

*Suppose that*\(u_{1} > 0\), \(u_{2} > 0\), \(\varepsilon_{1} > 0\), \(\varepsilon_{2} > 0\),

*and let*

*Then the estimation*

*holds for system*(3),

*and thus*\(\Omega_{\lambda,m} = \{ X \vert V_{\lambda,m} ( X ) \le L_{\lambda,m}^{2} \}\)

*is the globally exponential attractive set and positive invariant set of system*(3),

*i*.

*e*., \(\overline{\lim}_{t \to + \infty} V_{\lambda,m} ( X ( t ) ) \le L_{\lambda,m}^{2}\).

### Proof

*t*along the trajectory of system (3) yields

The proof is complete. □

### Remark 2

(i) In particular, let us take \({m} = 1\) in Theorem 2, we can get the results that obtained in [35]. The results presented in Theorem 2 contain the existing results in [35] as special cases.

(iii) From Figures 6-9, we can see that the bounds estimate for the chaotic attractors of system (3) is conservative, we can get a smaller bound of chaotic attractors of system (3) with the help of the iteration theorem in [32] (see [32] for a detailed discussion of the bounds of chaotic systems).

## 4 Conclusions

This paper presents a new 4D autonomous hyperchaotic system based on Lorenz chaotic system and another coupled dynamo chaotic system. By means of Lyapunov stability theory as well as optimization theory, the bounds of the new 4D autonomous hyperchaotic system and the coupled dynamo chaotic system are estimated. To show the ultimate bound region, numerical simulations are provided.

## Notes

### Acknowledgements

Fuchen Zhang is supported by National Natural Science Foundation of China (Grant No. 11501064), the Scientific and Technological Research Program of Chongqing Municipal Education Commission (Grant No. KJ1500605), the Research Fund of Chongqing Technology and Business University (Grant No. 2014-56-11), China Postdoctoral Science Foundation (Grant No. 2016M590850) and the Program for University Innovation Team of Chongqing (Grant No. CXTDX201601026). Xiaofeng Liao is supported by the National Key Research and Development Program of China (Grant No. 2016YFB0800601), in part by the National Nature Science Foundation of China (Grant No. 61472331). Da Lin is supported by the National Natural Science Foundation of China (Grant No. 61640223), the Open Project Program of the State Key Laboratory of Management and Control for Complex Systems (Grant No. 20160106), the Natural Science Foundation of Sichuan Province (Grant No. 2016JY0179), the Innovation Group Build Plan for the Universities of Sichuan Province (Grant No. 15TD0024), the Youth Science and Technology Innovation Group of Sichuan Provincial (Grant No. 2015TD0022), the High-level Innovative Talents Plan of Sichuan University of Science and Engineering (2014), the Key project of Artificial Intelligence Key Laboratory of Sichuan Province (Grant No. 2016RZJ02) and the Talents Project of Sichuan University of Science and Engineering (Grant No. 2015RC50). We thank professors Min Xiao in the College of Automation, Nanjing University of Posts and Telecommunications and Gaoxiang Yang at the Department of Mathematics and Statistics of Ankang University for their help with us. The authors wish to thank the editors and reviewers for their conscientious reading of this paper and their numerous comments for improvement which were extremely useful and helpful in modifying the paper.

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