# Symmetry Evolution in Chaotic System

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Conditional Symmetry from Asymmetry

## 3. Constructing Conditional Symmetry from Symmetry

_{KY}= 2.1736 under initial conditions (−1, 0, −1). In this work, for obtaining representative Lyapunov exponents rather than absolute ones [23,24,25], all the finite-time Lyapunov exponents (LEs) are computed for the time interval [0, 10

^{7}] for the initial points on the attractor based on the Wolf algorithm. It is a simple matter to determine the Kaplan–Yorke dimension from the spectrum of Lyapunov exponents by k + (LE

_{1}+ … + LE

_{k})/|LE

_{k+}

_{1}| (here LE

_{1}+ … + LE

_{k}≥ 0, and LE

_{1}+ … + LE

_{k+1}≤ 0). System (1) is of rotational symmetry since the system is invariant under the transformation (x, y, z) → (−x, −y, z) when m = n = 0, corresponding to a 180° rotation about the z-axis. In this case, system (1) has a symmetric oscillation or a symmetric pairs of twin attractors under different initial condition (IC), as shown in Figure 3.

_{1}, z + c

_{2}) (c

_{1}, c

_{2}stand for calling a polarity reverse from the absolute value function). We can compare these twin attractors; each one is symmetrically different from the above cases.

## 4. Recovering Conditional Symmetry from Destroyed Symmetry

#### 4.1. Symmetry Destroyed by the Constant Planting

#### 4.2. Symmetry Evolution Induced by the Dimension Growth

_{KY}= 3.2019, is shown in Figure 10.

_{1}, z+c

_{2}, –u) (c

_{1}, c

_{2}stand for calling a polarity reverse from the absolute value function).

_{KY}= 2.3989 under initial conditions (1, 1, 1, 2). Interestingly, this time the variable u is positive, and therefore the absolute value symbol of u can be introduced for hatching coexisting attractors, as shown in Figure 12.

_{1}, z + c

_{2}, u + c

_{3}) (c

_{1}, c

_{2}, c

_{3}stand for calling a polarity reverse from the absolute value function). The mechanism of the coexistence of attractors hides in the same balance ability from the structure (6).

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**Coexisting twin attractors in chaotic systems in Table 1: (

**a**) CS1, (

**b**) CS2, (

**c**) CS3, (

**d**) CS4, (

**e**) CS5, (

**f**) CS6.

**Figure 3.**Symmetric attractor or symmetric pairs of attractors of system (1) with m = n = 0, IC = (1, 1, 1) is red and IC = (1, −1, 1) is green: (

**a**) R =1, (

**b**) R =4.9, (

**c**) R = 5.2, (

**d**) R = 5.4.

**Figure 4.**Coexisting twin attractors of system (2) with $F\left(y\right)=\left|y\right|-6$, $G\left(z\right)=\left|z\right|-8$, m = n = 0, R = 1, IC = (1, 7, 9) is red, and IC = (−1, −6, −7) is green.

**Figure 5.**Dynamical evolvement in system (1) with n = 0, R = 1 and initial conditions (1, 1, 1): (

**a**) Lyapunov exponents (LEs), and (

**b**) bifurcation diagram.

**Figure 6.**Dynamical evolvement in system (1) with m = 0, R = 1 and initial conditions (1, 1, 1): (

**a**) Lyapunov exponents, and (

**b**) bifurcation diagram.

**Figure 7.**Dynamical evolvement in system (2) with n = 0, R = 1: (

**a**) Lyapunov exponents, and (

**b**) bifurcation diagram.

**Figure 8.**Conditional symmetric pairs of attractors in system (2) with n = 0, R = 1, IC = (1, 7, 9) is red and (−1, −6, −7) is green: (

**a**) m = 0.25, (

**b**) m = 0.45, (

**c**) m = 0.55, (

**d**) m = 0.7.

**Figure 9.**Conditional symmetric pairs of signals in system (2) with n = 0, R = 1, IC = (1, 7, 9) is red and (−1, −6, −7) is green: (

**a**) m = 0.25, (

**b**) m = 0.45.

**Figure 10.**Symmetric attractor of system (3) with a = 0.5, b = 0.1, R = 3, IC = (1, 1, 1, 2) is red, IC = (−1, −1, 1, −2) is green: (

**a**) x-y-z space, (

**b**) x-z-u space.

**Figure 11.**Coexisting conditional symmetric attractors in system (4) with $F\left(y\right)=\left|y\right|-15$, $G\left(z\right)=\left|z\right|-15$, a = 0.5, b = 0.1, R = 3, IC = (1, 16, 16, 2) is red, IC = (−1, −14, −14, −2) is green.

**Figure 12.**Symmetric attractor of system (6) with a = 0.5, b = 0.1, R = 3, IC = (1, 1, 1, 2) is red, IC = (−1, −1, 1, −2) is green.

**Figure 13.**Coexisting attractors in systems (7) with $F\left(y\right)=\left|y\right|-15$, $G\left(z\right)=\left|z\right|-15$,$H\left(u\right)=\left|u\right|-10$, a = b = 0.1, R = 3, IC = (1, 16, 16, 11) is red, IC = (−1, −14, −14, −10) is green.

Cases | System Equations | Parameters | Initial Condition | Lyapunov Exponents |
---|---|---|---|---|

CS1 | $\{\begin{array}{l}\dot{x}={y}^{2}-a{z}^{2},\\ \dot{y}=-{z}^{2}-by+c,\\ \dot{z}=yz+F(x),\end{array}$ $F\left(x\right)=\left|x\right|-3$ | a = 0.4, b = 1.75, c = 3 | (3, −1.5, −2) (3, −1.5, 1) | 0.1191, 0, −1.2500 |

CS2 | $\{\begin{array}{l}\dot{x}={y}^{2}-a,\\ \dot{y}=bz,\\ \dot{z}=-y-z+F(x),\end{array}$ $F\left(x\right)=\left|x\right|-3$ | a = 1.22, b = 8.48 | (3, 1, 0.5) (−3, 1, 0.5) | 0.2335, 0, −1.2335 |

CS3 | $\{\begin{array}{l}\dot{x}=F(y),\\ \dot{y}=z,\\ \dot{z}=-{x}^{2}-az+b{(F(y))}^{2}+1,\end{array}$ $F\left(y\right)=\left|y\right|-4$ | a = 2.6, b = 2 | (0.5, 4, −1) (0.5, −4, −1) | 0.0463, 0, −2.6463 |

CS4 | $\{\begin{array}{l}\dot{x}=y,\\ \dot{y}=F(z),\\ \dot{z}={x}^{2}-a{y}^{2}+bxy+xF(z),\end{array}$ $F\left(z\right)=\left|z\right|-8$ | a = 1.24, b = 1 | (4, 0.8, −2) (−4, 0.8, 2) | 0.0645, 0, −1.2582 |

CS5 | $\{\begin{array}{l}\dot{x}=1-G(y)z,\\ \dot{y}=a{z}^{2}-G(y)z,\\ \dot{z}=F(x),\end{array}$ $F\left(x\right)=\left|x\right|-3$ $G\left(y\right)=\left|y\right|-5$ | a = 0.22 | (−1, 1, −1) (2, 6, −1) | 0.0729, 0, −1.6732 |

CS6 | $\{\begin{array}{l}\dot{x}=F(y),\\ \dot{y}=xG(z),\\ \dot{z}=-axF(y)-bxG(z)-{x}^{2}+{(F(y))}^{2},\end{array}$ $F\left(y\right)=\left|y\right|-5$ $G\left(z\right)=\left|z\right|-5$ | a = 3, b = 1.2 | (0, −6 −6) (0, 6, 6) | 0.0506, 0, −0.2904 |

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**MDPI and ACS Style**

Li, C.; Sun, J.; Lu, T.; Lei, T.
Symmetry Evolution in Chaotic System. *Symmetry* **2020**, *12*, 574.
https://doi.org/10.3390/sym12040574

**AMA Style**

Li C, Sun J, Lu T, Lei T.
Symmetry Evolution in Chaotic System. *Symmetry*. 2020; 12(4):574.
https://doi.org/10.3390/sym12040574

**Chicago/Turabian Style**

Li, Chunbiao, Jiayu Sun, Tianai Lu, and Tengfei Lei.
2020. "Symmetry Evolution in Chaotic System" *Symmetry* 12, no. 4: 574.
https://doi.org/10.3390/sym12040574