# A New Chaotic System with Positive Topological Entropy

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## Abstract

**:**

## 1. Introduction

- (a)
- The system has a simple algebraic structure including one constant term, two linear terms and two nonlinear terms.
- (b)
- With some typical parameters, the Lyapunov dimension of the considered system is greater than other known 3D chaotic systems
- (c)
- The divergence of flow of the proposed system is not a constant but is always less than zero, while it’s a negative constant for other known 3D chaotic systems. The bigger the divergence is, the more scattered the phase trajectory is.
- (d)
- The proposed chaotic attractor has a compound structure which can be demonstrated using a half-image operation to obtain the left or the right half-image attractors.
- (e)
- The system exhibits chaotic behavior over a large range of parameters.

## 2. The New Chaotic System and Its Basic Properties

#### 2.1. System Description

#### 2.2. Non-generalized Lorenz System

#### 2.3. Equilibria

**Proposition 1.**

**Proof.**

**Remark 1.**

## 3. Observation and Analysis of the New System

#### 3.1. Fixing $c=6,d=80$ and Varying a

- (1)
- When $a\in [1,1.25)\cup [3.54,3.55)\cup [3.84,3.86)\cup [3.94,3.98)\cup [4.18,4.19)\cup [4.58,4.91)\cup [4.99,6]$, we can obtain ${L}_{1}=0$, and both ${L}_{2}$ and ${L}_{3}$ are less than 0, and the trajectories started from different initial conditions approach a closed orbit surrounding two equilibria. For example, the periodic orbits for $a=3.54,4.18,4.7$ and $4.99$ are depicted in Figure 3c,e,h, respectively. When $a\in [4.6,4.85]$, there is a vertical white region, that’s because the system (3) has a 2-periodic orbit within this range. The low density of trajectories leads to the low density of the points in the bifurcation diagram.
- (2)
- When a belongs to $[1.25,3.54)\cup [3.55,3.84)\cup [3.86,3.94)\cup [3.98,4.18)\cup [4.19,4.58)\cup [4.91,4.99)$, ${L}_{1}>0,{L}_{2}=0$ and ${L}_{3}<0$, and strange chaotic attractors will appear. When $a=1.25,3,3.98,4.19$ and $4.91$, some double-scroll chaotic attractors are shown in Figure 3a,b,d,f,g, respectively.

**Figure 3.**Phase portraits of the system(3) projected on y − z plane for different a. (

**a**) a = 1.25; (

**b**) a = 3; (

**c**) a = 3.54; (

**d**) a = 3.98;(

**e**) a = 4.18; (

**f**) a = 4.19; (

**g**) a = 4.7; (

**h**) a = 4.91; (

**I**) a = 4.99.

#### 3.2. Fixing $a=3,c=6$ and Varying d

**Figure 4.**The Lyapunov exponent spectrum of system (3) versus the parameter $d\u220a[0,1000]$ with $a=3,c=6$.

**Figure 5.**Phase portraits with ($a=3,c=6,d=15$) (

**a**) 3D view; (

**b**) Projection on x − y plane; (

**c**) Projection on x − z plane; (

**d**) Projection on y − z plane.

**Figure 7.**Poincaré map projected on y − z plane for different d. (

**a**) $d=15$ (

**b**) $d=80$ (

**c**) $d=500$ (

**d**) $d=1000$.

#### 3.3. A Dissipative System

**Figure 8.**(

**a**) The change of $\text{\u2207}V$ versus a with $c=6,d=80$; (

**b**) The change of $\text{\u2207}V$ versus d with $a=3,c=6$;

#### 3.4. Compound Structures

**Figure 9.**(

**a**) A left half-image chaotic attractor for the y − z plane at $k=1.95$; (

**b**) A right half-image chaotic attractor for the y − z plane at $k=-1.95$; (

**c**) A superposition of (

**a**) and (

**b**); (

**d**) A chaotic attractor for y − z plane at $k=0$.

- (1)
- When $\left|k\right|\ge 2.26$, the system (14) has limit cycles. For example, Figure 11a shows a limit cycle at $k=2.4$.
- (2)
- When $2.1<\left|k\right|<2.26$, there are period-doubling bifurcations. For example, Figure 11b,c show such period-doubling bifurcations at $k=2.2$ and $2.12$, respectively.
- (3)
- When $1.92\le \left|k\right|\le 2.1$, the system (14) becomes a left (or a right) half-image attractor as shown in Figure 9a,b.
- (4)
- When $0.5\le \left|k\right|<1.92$, the system demonstrates partial attractors, which are bounded. For example, Figure 11d shows a partially-right, dominantly-left, attractor at $k=0.9$.
- (5)
- For $\left|k\right|<0.5$, the system exhibits a complete attractor. For example, Figure 5 shows the complete attractor at $k=0$.

**Figure 11.**Phase portraits of the system (14) at (a) $k=2.4$, (

**b**) $k=2.2$, (

**c**) $k=2.12$, (

**d**) $k=0.9$.

## 4. Topological Horseshoe Analysis for the New Chaotic System

#### 4.1. Review of a Topological Horseshoe Theory

**Proposition 2.**

- (i)
- $\sigma \left({\sum}_{m}\right)={\sum}_{m}$;
- (ii)
- σ is continuous;
- (iii)
- σ has countable many periodic orbits;
- (iv)
- σ has uncountable many aperiodic orbits;
- (v)
- σ has a dense orbit.

**Definition 1.**

**Theorem 1.**

**Theorem 2.**

#### 4.2. Horseshoe in the Poincaré Map for the Proposed System

**Proposition 3.**

**Proof.**

**Figure 13.**(

**a**) ${D}_{1},{D}_{2},{D}_{3}$ and the image of ${D}_{1}$; (

**b**) ${D}_{1},{D}_{2},{D}_{3}$ and the image of ${D}_{2}$; (

**c**) ${D}_{1},{D}_{2},{D}_{3}$ and the image of ${D}_{3}$.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Wang, Z.; Ma, J.; Chen, Z.; Zhang, Q.
A New Chaotic System with Positive Topological Entropy. *Entropy* **2015**, *17*, 5561-5579.
https://doi.org/10.3390/e17085561

**AMA Style**

Wang Z, Ma J, Chen Z, Zhang Q.
A New Chaotic System with Positive Topological Entropy. *Entropy*. 2015; 17(8):5561-5579.
https://doi.org/10.3390/e17085561

**Chicago/Turabian Style**

Wang, Zhonglin, Jian Ma, Zengqiang Chen, and Qing Zhang.
2015. "A New Chaotic System with Positive Topological Entropy" *Entropy* 17, no. 8: 5561-5579.
https://doi.org/10.3390/e17085561