1. Introduction

2. Chaotic Systems

• Sine map:

  $${\mathrm{x}}_{\mathrm{t}+1}=\mathrm{S}\left(\mathrm{x}\right)=\mathsf{\epsilon }\sin (\mathsf{\pi }{\mathrm{x}}_{\mathrm{t}})$$

  (1)

  where $\mathsf{\epsilon }\in [0,1]$ is the control parameter, ${\mathrm{x}}_{\mathrm{t}}\in [0, 1]$ is the status value of the system at time $\mathrm{t}$.
• Tent map:

  $${\mathrm{x}}_{\mathrm{t}+1}=\mathrm{T}(\mathrm{x})=\left\{\begin{array}{l}2\mathsf{\epsilon }{\mathrm{x}}_{\mathrm{t}}, {\mathrm{x}}_{\mathrm{t}}<0.5\\ 2\mathsf{\epsilon }(1-{\mathrm{x}}_{\mathrm{t}}), {\mathrm{x}}_{\mathrm{t}}\ge 0.5\end{array}\right.$$

  (2)
• Logistic map:

  $${\mathrm{x}}_{\mathrm{t}+1}=\mathrm{L}\left(\mathrm{x})=4\mathsf{\epsilon }{\mathrm{x}}_{\mathrm{t}}(1-{\mathrm{x}}_{\mathrm{t}}\right)$$

  (3)
• PWLCM map:

  $${\mathrm{x}}_{\mathrm{t}+1}=\left\{\begin{array}{l}\frac{{\mathrm{x}}_{\mathrm{t}}}{\mathsf{\epsilon }}, {\mathrm{x}}_{\mathrm{t}}\in [0,\mathsf{\epsilon })\\ \frac{1-{\mathrm{x}}_{\mathrm{t}}}{1-\mathsf{\epsilon }}, {\mathrm{x}}_{\mathrm{t}}\in (\mathsf{\epsilon },1]\end{array}\right.$$

  (4)

5.

Chen system:

$$\left\{\begin{array}{l}\dot{\mathrm{x}}=\mathrm{a}(\mathrm{y}-\mathrm{x})\\ \dot{\mathrm{y}}=(\mathrm{c}-\mathrm{a})\mathrm{x}-\mathrm{x}\mathrm{z}+\mathrm{c}\mathrm{y}\\ \dot{\mathrm{z}}=\mathrm{x}\mathrm{y}-\mathrm{b}\mathrm{z}\end{array}\right.$$

(5)

where $\mathrm{a}$=35, $\mathrm{b}$=3, and $\mathrm{c}\in [20, 28.4]$. The $\mathrm{x}$, $\mathrm{y}$, and $\mathrm{z}$ are the status values of the system at time $\mathrm{t}$. The $\dot{\mathrm{x}}$, $\dot{\mathrm{y}}$, and $\dot{\mathrm{z}}$ represent the derivative of the independent variable $\mathrm{x}$, $\mathrm{y}$, and $\mathrm{z}$ at time $\mathrm{t}$.

6.

hyper-chaotic Chen system:

$$\left\{\begin{array}{l}\dot{\mathrm{x}}=\mathrm{a}(\mathrm{y}-\mathrm{x})\\ \dot{\mathrm{y}}=-\mathrm{x}\mathrm{z}+\mathrm{d}\mathrm{x}+\mathrm{c}\mathrm{y}-\mathrm{w}\\ \dot{\mathrm{z}}=\mathrm{x}\mathrm{y}-\mathrm{b}\mathrm{z}\\ \dot{\mathrm{w}}=\mathrm{x}+\mathrm{k}\end{array}\right.$$

(6)

where $\mathrm{a}$=36, $\mathrm{b}$=3, $\mathrm{c}$=28, $\mathrm{d}$=-16, and $-0.7\le \mathrm{k}\le 0.7$. The $\mathrm{w}$ is the status value of the system at time $\mathrm{t}$. The $\dot{\mathrm{w}}$ represents the derivative of the independent variable $\mathrm{w}$ at time $\mathrm{t}$.

7.

Lorenz system:

$$\left\{\begin{array}{l}\dot{\mathrm{x}}=-\mathrm{a}(\mathrm{x}-\mathrm{y})\\ \dot{\mathrm{y}}=-\mathrm{x}\mathrm{z}+\mathrm{b}\mathrm{x}-\mathrm{y}\\ \dot{\mathrm{z}}=\mathrm{x}\mathrm{y}-\mathrm{c}\mathrm{z}\end{array}\right.$$

(7)

where $\mathrm{a}$=10, $\mathrm{b}$=28, and $\mathrm{c}$=8/3.

8.

hyper-chaotic Lorenz system:

$$\left\{\begin{array}{l}\dot{\mathrm{x}}=\mathrm{a}(\mathrm{y}-\mathrm{x})+\mathrm{w}\\ \dot{\mathrm{y}}=\mathrm{c}\mathrm{x}-\mathrm{y}-\mathrm{x}\mathrm{z}\\ \dot{\mathrm{z}}=\mathrm{x}\mathrm{y}-\mathrm{b}\mathrm{z}\\ \dot{\mathrm{w}}=-\mathrm{y}\mathrm{z}+\mathrm{k}\mathrm{w}\end{array}\right.$$

(8)

where $\mathrm{a}$=10, $\mathrm{b}$=8/3, c=28, $\mathrm{d}$=-16, and $-1.52\le \mathrm{k}\le -0.06$.

9.

Rossler system:

$$\left\{\begin{array}{l}\dot{\mathrm{x}}=-\mathrm{a}\mathrm{y}-\mathrm{z}\\ \dot{\mathrm{y}}=\mathrm{a}\mathrm{x}+\mathrm{b}\mathrm{y}\\ \dot{\mathrm{z}}=\mathrm{c}+\mathrm{z}(\mathrm{x}-\mathrm{d})\end{array}\right.$$

(9)

where $\mathrm{a}$=1.0, $\mathrm{b}$=0.165, $\mathrm{c}$=0.2, and $\mathrm{d}$=10.

10.

hyper-chaotic Rossler system:

$$\left\{\begin{array}{l}\dot{\mathrm{x}}=-\mathrm{y}-\mathrm{z}\\ \dot{\mathrm{y}}=\mathrm{x}+\mathrm{a}\mathrm{x}+\mathrm{w}\\ \dot{\mathrm{z}}=\mathrm{b}+\mathrm{x}\mathrm{z}\\ \dot{\mathrm{w}}=-\mathrm{c}\mathrm{z}+\mathrm{d}\mathrm{w}\end{array}\right.$$

(10)

where $\mathrm{a}$=0.25, $\mathrm{b}$=3, $\mathrm{c}$=0.5, and $\mathrm{d}$=0.05.

11.

hyper-chaotic Hide-Skeldon-Acheson system [26]:

$$\left\{\begin{array}{l}\dot{\mathrm{x}}=-\mathrm{y}-\mathrm{z}\\ \dot{\mathrm{y}}=\mathrm{x}+\mathrm{a}\mathrm{x}+\mathrm{w}\\ \dot{\mathrm{z}}=\mathrm{b}+\mathrm{x}\mathrm{z}\\ \dot{\mathrm{w}}=-\mathrm{c}\mathrm{z}+\mathrm{d}\mathrm{w}\end{array}\right.$$

(11)

where $\mathrm{a}$=0.01, $\mathrm{b}$=30, $\mathrm{c}$=0, $\mathrm{d}$=2, $\mathrm{e}$=0.001, $\mathrm{f}$=28.5, $\mathrm{g}$=1, $\mathrm{k}$=1.2, $\mathrm{q}$=0, and $\mathrm{h}$=0.

12.

New four-dimensional chaotic system [27]:

$$\left\{\begin{array}{l}\dot{\mathrm{x}}=-\mathrm{y}-\mathrm{z}\\ \dot{\mathrm{y}}=\mathrm{x}+\mathrm{a}\mathrm{x}+\mathrm{w}\\ \dot{\mathrm{z}}=\mathrm{b}+\mathrm{x}\mathrm{z}\\ \dot{\mathrm{w}}=-\mathrm{c}\mathrm{z}+\mathrm{d}\mathrm{w}\end{array}\right.$$

(12)

where $\mathrm{a}$=25, $\mathrm{b}$=3, $\mathrm{c}$=18, $\mathrm{d}$=19, and $\mathrm{e}$=14.

13.

hyper-chaotic Lü system [28]:

$$\left\{\begin{array}{l}\dot{\mathrm{x}}=\mathrm{y}-\mathrm{a}\mathrm{x}\\ \dot{\mathrm{y}}=\mathrm{b}\mathrm{z}-\mathrm{x}\mathrm{z}\\ \dot{\mathrm{z}}=\mathrm{x}\mathrm{y}-\mathrm{x}\mathrm{w}\\ \dot{\mathrm{w}}=-\mathrm{x}+\mathrm{c}\end{array}\right.$$

(13)

where $\mathrm{a}$=10, $\mathrm{b}$=3, and $\mathrm{c}$=12.

3. Algorithm Description

4. Results and Discussion

5. Conclusions

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