# Symmetric Strange Attractors: A Review of Symmetry and Conditional Symmetry

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## Abstract

**:**

## 1. Introduction

## 2. Symmetric Strange Attractors and Symmetric Pairs of Coexisting Attractors

#### 2.1. Various Regimes of Symmetry

_{h}; meanwhile a negative sign on the right-hand side could be obtained by many approaches, including the offset boosting of a variable or from a function.

#### 2.2. Multiple Modes of Coexisting Attractors in a Symmetric System

#### 2.3. Diversities of Stability in Symmetric Systems

## 3. Offset Boosting for Symmetric Pairs of Strange Attractors

- (I)
- System (6) is one of symmetry according to the dimension of ${x}_{i}$;
- (II)
- System (6) has 2 m coexisting attractors ${O}_{11},{O}_{12},\cdots ,{O}_{1m},{O}_{21},{O}_{22},\cdots ,{O}_{2m}$;
- (III)
- All the attractors ${O}_{kj},k=1,2,j=1,2,\dots ,m,$ in system (6) share the same structure with the ones in system (5), and all the equilibria in system (6) have the same stabilities with those of system (5).

_{KY}= 2.0132. Replace the variable z with an absolute function |z| − d,

_{1}, |y| − d

_{2}, |z| − d

_{3}, the derived system is one of inversion symmetry, and when the variables x, y, and z are polarity inversed, the derived system keeps the same equation, as written in system (9),

## 4. Coexisting Strange Attractors of Conditional Symmetry

_{KY}) are included. Those systems could be systems of symmetry or asymmetry or jerk systems. Readers could check these systems to see if they have polarity balance. Here, the factors for polarity balance are from the system variables and are also recovered by the absolute functions. The broken polarity originates from the polarity reversal of some of the system variables, and the polarity balance is reconstructed by the internal polarity reversals of the absolute-value function. A one absolute-value function introduces a one-time polarity reversal by offset boosting; multiple absolute-value functions drive multiple instances of a polarity jump due to offset boosting. Those chaotic systems with amplitude control [77,78,79] or other coexisting attractors [80,81,82] may preserve this property since the polarity reversal induced by the offset boosting does not fundamentally change the system structure.

## 5. Symmetry and Elegance in Simple Chaotic Circuits

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Attractor of rotational symmetry in system (1) when σ = 0.279, r = 0, b = −0.3 and IC = (−0.1, 0.1, −2): (

**a**) x–y, (

**b**) x–z.

**Figure 4.**Symmetric pair of attractors with rotational symmetry in system (1) under the parameters σ = 0.256, r = 0, b = −0.3 and IC

_{1}= (−0.1, 0.1, −2) (left), IC

_{2}= (0.1, −0.1, −2) (right): (

**a**) x–y, (

**b**) x–z.

**Figure 5.**Coexisting strange attractors of system (1) with rotational symmetry when σ = 0.279, r = 0, b = −0.3 and IC

_{1}= (−0.1, 0.1, −2) (blue), IC

_{2}= (−0.1, 0.1, −14) (cyan), IC

_{3}= (0.1, −0.1, −14) (purple): (

**a**) x–y, (

**b**) x–z.

**Figure 6.**Symmetric attractor of inversion symmetry in system (2) when a = 1.7, b = 2 and IC = (1, 0, 0): (

**a**) x–y, (

**b**) x–z.

**Figure 7.**A symmetric pair of coexisting attractors in system (2) when a = 2.1, b = 2 and IC

_{1}= (−1, 0, 0) (left), IC

_{2}= (1, 0, 0) (right): (

**a**) x–y, (

**b**) x–z.

**Figure 8.**Attractor of reflection invariant system (3) with a = 0.35 and IC = (−2, 2, 0): (

**a**) x–y, (

**b**) x–z.

**Figure 9.**A symmetric pair of attractors in a reflection invariant system (3) with a = 0.7 and IC

_{1}= (−2, 2, 0) (cyan), IC

_{2}= (2, 2, 0) (purple): (

**a**) x–y, (

**b**) x–z.

**Figure 10.**Quasi-periodic torus coexisting with a symmetric pair of chaotic attractors at a = 6, b = 0.1 (red and blue attractors correspond to two symmetric initial conditions under IC = (0, ±4, 0, ±5), cyan is for symmetric torus under IC = (1, −1, 1, −1)): (

**a**) x–z, (

**b**) y–z.

**Figure 11.**Symmetric attractors of system (8) with a = b = 0.2, c = 6.5, d = 0 and IC

_{1}= (−9, 0, 2) (up), IC

_{2}= (−9, 0, −2) (down): (

**a**) x–z, (

**b**) y–z.

**Figure 12.**Symmetric attractors of system (8) with a = b = 0.2, c = 6.5, d = 12 and IC

_{1}= (−9, 0, 12) (up), IC

_{2}= (−9, 0, −12) (down): (

**a**) x–z, (

**b**) y–z.

**Figure 13.**Eight coexisting attractors of system (9) with a = b = 0.2, c = 6.5, d

_{1}= 11, d

_{2}= 13, d

_{3}= 12.

**Figure 15.**Coexisting conditional reflection symmetric attractors of system (10) with IC

_{1}= (3, −1.5, −2) (red), IC

_{2}= (3, −1.5, 1) (green): (

**a**) x–z, (

**b**) y–z.

**Figure 16.**Coexisting conditional rotational symmetric attractors of system (11) with IC

_{1}= (3, 1, 0.5) (red), IC

_{2}= (−3, 1, 0.5) (green): (

**a**) x–y, (

**b**) x–z.

**Figure 17.**Coexisting attractors in system (12) by 2D offset boosting, when a = 0.22 and IC

_{1}= (2, 6, −1) (red), IC

_{2}= (−1, 1, −1) (green): (

**a**) x–z, (

**b**) y–z.

**Figure 18.**Coexisting conditional rotational symmetric attractors of system (13) with a = 0.35, IC

_{1}= (0, 0.4, 6) (red), IC

_{2}= (0, 0.4, −5) (green): (

**a**) x–z, (

**b**) y–z.

**Figure 19.**Coexisting attractors in conditional symmetric system (14) with a = 0.4, b = 1, IC

_{1}= (0, 14, 17) (red), IC

_{2}= (0, −6, −7) (green): (

**a**) x–y, (

**b**) x–z.

**Figure 20.**Coexisting attractors in symmetric system (15) with IC

_{1}= (1, 5, 5.5) (red), IC

_{2}= (1, −5, −4.5) (green): (

**a**) x–z, (

**b**) y–z.

**Figure 21.**Coexisting repellors with conditional symmetry in system (16): (

**a**) y–x, (

**b**) z–x. (IC = (0, 0.96, 0) is red, IC = (6, 2, 2) is yellow, IC = (0, −0.96, 0) is green, IC = (−6, −2, −2) is blue).

**Figure 22.**Coexisting attractors in system (17): (

**a**) x–y, (

**b**) x–z. (IC = (1, 0, 0) is green, IC = (1 + 1.25π, 0, 0) is pink, IC = (1 + 2.5π, 0, 0) is red, IC = (1−1.25π, 0, 0) is cyan, IC = (1−2.5π, 0, 0) is blue).

**Figure 25.**Symmetric attractor in system (18) with α = 1, ω = 1 and IC = (4.1, 0.7, 5): (

**a**) x–y, (

**b**) y–z.

**Figure 26.**Conditional symmetric chaotic attractors of system (19) with a = 0.6, b = 1, c = 2, (

**a**) z–y, (

**b**) z–x. (IC = (2, 0, −1) is up, IC = (−2, 0, 1) is down).

**Figure 28.**Conditional symmetric chaotic attractors of system (19) with a = 0.6, b = 1, c = 2 observed in oscilloscope, (

**a**) z–y, (

**b**) z–x. (IC = (2, 0, −1) is green, IC = (−2, 0, 1) is brown).

**Figure 29.**Chaotic attractors in system (20) with a = 0.6, b = 1, c = 1, d = 4.11 and IC = (1, 1, −1). Here, two coexisting attractors close each other and bond to be a pseudo-double-scroll attractor: (

**a**) x–y, (

**b**) x–z.

**Figure 30.**Coexisting attractors of system (20) with a = 0.6, b = 1, c = 1, d = 8: (

**a**) x–y, (

**b**) x–z. (IC = (−1, 1, −1) is left, IC = (1, 1, −1) is right).

**Figure 31.**Simple circuit operation unit: (

**a**) multiplier current constraint under external resistance, (

**b**) the resistor-capacitor coupling realizes current control.

Equations | Parameters | Equilibria | Eigenvalues |
---|---|---|---|

$\left\{\begin{array}{l}\dot{x}=y\\ \dot{y}=z\\ \dot{z}=-z-ay-{x}^{3}+bx\end{array}\right.$ [51] | a = 2.1 b = 2 | (0, 0, 0) ($\sqrt{2}$, 0, 0) ($-\sqrt{2}$, 0, 0) | (0.6366, $-$0.8183 $\pm $ 1.5723i) ($-$1.4516, 0.2258 $\pm $ 1.6446i) |

$\left\{\begin{array}{l}\dot{x}=a(y-x)\\ \dot{y}=-xz+cx-y\\ \dot{z}=xy-bz\end{array}\right.$ [51] | a = 0.279 b = $-$0.3 c = 0 | (0, 0, 0) ($\sqrt{30}/10$, $\sqrt{30}/10$, $-$1) ($-\sqrt{30}/10$, $-\sqrt{30}/10$, $-$1) | ($-$1, $-$0.279, 0.3) ($-$1.1722, 0.0966 $\pm $ 0.3653i) |

$\left\{\begin{array}{l}\dot{x}=y-x\\ \dot{y}=-xz\\ \dot{z}=xy-a\end{array}\right.$ [51] | a = 4.7 | ($\sqrt{470}/10$, $\sqrt{470}/10$, 0) ($-\sqrt{470}/10$, $-\sqrt{470}/10$, 0) | ($-$1.6369, 0.3185 $\pm $ 2.3751i) |

$\left\{\begin{array}{l}\dot{x}=y-bx\\ \dot{y}=yz-x\\ \dot{z}=a-{y}^{2}\end{array}\right.$ [51] | a = 1 b = 1.06 | (50/53, 1, 50/53) ($-$50/53, $-$1, 50/53) | ($-$0.8218, 0.3526 $\pm $ 1.5669i) |

$\left\{\begin{array}{l}\dot{x}=x-xy\\ \dot{y}=z\\ \dot{z}=-y-az+{x}^{2}\end{array}\right.$ [51] | a = 0.7 | (0, 0, 0) (1, 1, 0) ($-$1, 1, 0) | (1, $-$0.35 $\mp $ 0.9368i) ($-$1.2216, 0.2608 $\pm $ 1.2527i) |

$\left\{\begin{array}{l}\dot{x}=y-x\\ \dot{y}=-axz+u\\ \dot{z}=xy-1\\ \dot{u}=-by\end{array}\right.$ [52] | a = 6 b = 0.1 | None | None |

Equations | Parameters | (x_{0}, y_{0}, z_{0}) | LEs | D_{KY} |
---|---|---|---|---|

$\left\{\begin{array}{l}\dot{x}={y}^{2}-a{z}^{2}\\ \dot{y}=-{z}^{2}-by+c\\ \dot{z}=yz+\mathrm{F}\left(x\right)\end{array}\right.$ [69] F(x) = |x| $-$ 3 | a = 0.4 b = 1.75 c = 3 | (3, $-$1.5, $-$2) (3, $-$1.5, 1) | 0.1191, 0, $-$1.2500 | 2.0953 |

$\left\{\begin{array}{l}\dot{x}={y}^{2}-a\\ \dot{y}=bz\\ \dot{z}=-y-z+\mathrm{F}\left(x\right)\end{array}\right.$ [69] F(x) = |x| $-$ 3 | a = 1.22 b = 8.48 | (3, 1, 0.5) ($-$3, 1, 0.5) | 0.2335, 0, $-$1.2335 | 2.1893 |

$\left\{\begin{array}{l}\dot{x}=\mathrm{F}\left(y\right)\\ \dot{y}=z\\ \dot{z}=-{x}^{2}-az+b{\left(\mathrm{F}\right(y\left)\right)}^{2}+1\end{array}\right.$ [69] F(y) = |y| $-$ 4 | a = 2.6 b = 2 | (0.5, 4, $-$1) (0.5, $-$4, $-$1) | 0.0463, 0, $-$2.6463 | 2.0175 |

$\left\{\begin{array}{l}\dot{x}=y\\ \dot{y}=\mathrm{F}\left(z\right)\\ \dot{z}={x}^{2}-a{y}^{2}+bxy+x\mathrm{F}\left(z\right)\end{array}\right.$ [69] F(z) = |z| $-$ 8 | a = 1.24 b = 1 | (4, 0.8, $-$2) (4, 0.8, 14) | 0.0645, 0, $-$1.2582 | 2.0513 |

$\left\{\begin{array}{l}\dot{x}=1-\mathrm{G}\left(y\right)z\\ \dot{y}=a{z}^{2}-\mathrm{G}\left(y\right)z\\ \dot{z}=\mathrm{F}\left(x\right)\end{array}\right.$ [69] F(x) = |x| $-$ 3 G(y) = |y| $-$ 5 | a = 0.22 | ($-$1, 1, $-$1) (2, 6, $-$1) | 0.0729, 0, $-$1.6732 | 2.0436 |

$\left\{\begin{array}{l}\dot{x}=\mathrm{F}\left(y\right)\\ \dot{y}=\mathrm{G}\left(z\right)x\\ \dot{z}=-ax\mathrm{F}\left(y\right)-bx\mathrm{G}\left(z\right)-{x}^{2}+({\mathrm{F}\left(y\right))}^{2}\end{array}\right.$ [69] F(y) = |y| $-$ 5 G(z) = |z| $-$ 5 | a = 3 b = 1.2 | (0, $-$6, $-$6) (0, 6, 6) | 0.0506, 0, $-$0.2904 | 2.1735 |

$\left\{\begin{array}{l}\dot{x}=-y\\ \dot{y}=x+\mathrm{F}\left(z\right)\\ \dot{z}=2{y}^{2}+x\mathrm{F}\left(z\right)-a\end{array}\right.$ [70] F(z) = |z| $-$ 5 | a = 0.35 | (0, 0.4, 6) (0, 0.4, $-$5) | 0.0776, 0, $-$1.5008 | 2.0517 |

$\left\{\begin{array}{l}\dot{x}=y\\ \dot{y}=-x+\mathrm{F}\left(z\right)\\ \dot{z}=-0.8{z}^{2}+{\mathrm{F}\left(z\right)}^{2}+a\end{array}\right.$ [70] F(z) = |z| $-$ 12 | a = 2.0 | (0, 2.3, 12) (0, $-$2.3, $-$12) | 0.0252, 0, $-$6.8521 | 2.0037 |

$\left\{\begin{array}{l}\dot{x}=\mathrm{F}\left(z\right)\\ \dot{y}=x-y\\ \dot{z}=-4{x}^{2}+8xy+y\mathrm{F}\left(z\right)+a\end{array}\right.$ [70] F(z) = |z| $-$ 12 | a = 0.1 | (0.5, 0, 11) (0.5, 0, $-$13) | 0.0665, 0, $-$2.0410 | 2.0326 |

$\left\{\begin{array}{l}\dot{x}=-y\\ \dot{y}=x+\mathrm{F}\left(z\right)\\ \dot{z}=xy+x\mathrm{F}\left(z\right)+0.2y\mathrm{F}\left(z\right)-a\end{array}\right.$ [70] F(z) = |z| $-$ 15 | a = 0.4 | (2.5, 0, 15) ($-$2.5, 0, $-$15) | 0.1026, 0, $-$2.1275 | 2.0482 |

$\left\{\begin{array}{l}\dot{x}=y\\ \dot{y}=\mathrm{F}\left(z\right)\\ \dot{z}={x}^{2}-{y}^{2}+ax\mathrm{F}\left(z\right)+y\mathrm{F}\left(z\right)\end{array}\right.$ [70] F(z) = |z| $-$ 15 | a = 2.0 | (1, 0, 11) (1, 0, $-$19) | 0.0538, 0, $-$11.8591 | 2.0045 |

$\left\{\begin{array}{l}\dot{x}=y\\ \dot{y}=\mathrm{F}\left(z\right)\\ \dot{z}={x}^{2}-{y}^{2}+xy+0.4x\mathrm{F}\left(z\right)+a\end{array}\right.$ [70] F(z) = |z| $-$ 30 | a = 1.0 | (0, 1, 26.1) (0, 1, $-$32.9) | 0.1105, 0, $-$1.3882 | 2.0796 |

$\left\{\begin{array}{l}\dot{x}=z\\ \dot{y}=-ay-z\mathrm{F}\left(x\right)\\ \dot{z}=z-b{z}^{2}+y\mathrm{F}\left(x\right)\end{array}\right.$ [70] F(x) = |x| $-$ 9 | a = 1.62 b = 0.2 | (9, 1, 0.8) ($-$9, 1, 0.8) | 0.0645, 0, $-$0.6845 | 2.0943 |

$\left\{\begin{array}{l}\dot{x}=\mathrm{G}\left(y\right)\\ \dot{y}=a{\mathrm{G}\left(y\right)}^{2}-x\mathrm{F}\left(z\right)\\ \dot{z}={x}^{2}+x\mathrm{G}\left(y\right)-bx\mathrm{F}\left(z\right)\end{array}\right.$ [70] G(y) = |y| $-$ 10 F(z) = |z| $-$ 12 | a = 0.4 b = 1 | (0, 14, 17) (0, $-$6, $-$7) | 0.0749, 0, $-$0.7390 | 2.1013 |

$\left\{\begin{array}{l}\dot{x}=y\\ \dot{y}=c\left|y\right|\mathrm{F}\left(z\right)+0.1y\left|y\right|-x\\ \dot{z}=0.5{x}^{2}-axy-bx\mathrm{F}\left(z\right)\end{array}\right.$ [70] F(z) = |z| $-$ 4 | a = 0.8 b = 0.5 c = 0.5 | (1, 0.5, 5) ($-$1, $-$0.5, $-$4) | 0.0177, 0, $-$0.4092 | 2.0433 |

$\left\{\begin{array}{l}\dot{x}=ay-ax+e\mathrm{F}\left(z\right)\\ \dot{y}=cx-y-x\left|\mathrm{F}\right(z\left)\right|+u\\ \dot{z}=xy-b\left|\mathrm{F}\right(z\left)\right|\\ \dot{u}=-kx\end{array}\right.$ [71] F(z) = |z| $-$ 50 | a = 10 b = 8/3 c = 28 k = 4 e = 2 | (0, $-$0.3, 2, 0) (0, $-$0.3, 50, 0) | 0.2563, 0.1674, 0, $-$14.0917 | 3.0301 |

$\left\{\begin{array}{l}\dot{x}=1-ayz\\ \dot{y}=z\left|z\right|-z\\ \dot{z}=F\left(x\right)-bz\end{array}\right.$ [72] F(x) = |x| $-$ 3 | a = 3.55 b = 0.6 | ($-$3, 0, $-$1) (3, 0, $-$1) | 0.1455, 0, $-$0.7455 | 2.1952 |

$\left\{\begin{array}{l}\dot{x}={y}^{2}-a{z}^{2}\\ \dot{y}=-{z}^{2}-by+c\\ \dot{z}=yz+\mathrm{F}\left(x\right)\end{array}\right.$ [73] F(x) = |x| $-$ 3 | a = 0.4 b = 1.75 c = 3 | (3, $-$1.5, $-$2) (3, $-$1.5, 1) | 0.1191, 0, $-$1.2500 | 2.0953 |

$\left\{\begin{array}{l}\dot{x}={y}^{2}-a\\ \dot{y}=bz\\ \dot{z}=-y-z+\mathrm{F}\left(x\right)\end{array}\right.$ [73] F(x) = |x| $-$ 3 | a = 1.22 b = 8.48 | (3, 1, 0.5) ($-$3, 1, 0.5) | 0.2335, 0, $-$1.2335 | 2.1893 |

$\left\{\begin{array}{l}\dot{x}=\mathrm{F}\left(y\right)\\ \dot{y}=z\\ \dot{z}=-{x}^{2}-az+b{\left(\mathrm{F}\right(y\left)\right)}^{2}+1\end{array}\right.$ [74] F(y) = |y| $-$ 4 | a = 2.6 b = 2 | (0.5, 4, $-$1) (0.5, $-$4, $-$1) | 0.0463, 0, $-$2.6463 | 2.0175 |

$\left\{\begin{array}{l}\dot{x}=y\\ \dot{y}=\mathrm{F}\left(z\right)\\ \dot{z}={x}^{2}-a{y}^{2}+bxy+x\mathrm{F}\left(z\right)\end{array}\right.$ [74] F(z) = |z| $-$ 8 | a = 1.24 b = 1 | (4, 0.8, $-$2) ($-$4, 0.8, 2) | 0.0645, 0, $-$1.2582 | 2.0513 |

$\left\{\begin{array}{l}\dot{x}=1-\mathrm{G}\left(y\right)z\\ \dot{y}=a{z}^{2}-\mathrm{G}\left(y\right)z\\ \dot{z}=\mathrm{F}\left(x\right)\end{array}\right.$ [74] F(x) = |x| $-$ 3 G(y) = |y| $-$ 5 | a = 0.22 | ($-$1, 1, $-$1) (2, 6, $-$1) | 0.0729, 0, $-$1.6731 | 2.0436 |

$\left\{\begin{array}{l}\dot{x}=\mathrm{F}\left(y\right)\\ \dot{y}=x\mathrm{G}\left(z\right)\\ \dot{z}=-ax\mathrm{F}\left(y\right)-bx\mathrm{G}\left(z\right)-{x}^{2}+{\left(\mathrm{F}\right(y\left)\right)}^{2}\end{array}\right.$ [74] F(y) = |y| $-$ 5 G(z) = |z| $-$ 5 | a = 3 b = 1.2 | (0, $-$6, $-$6) (0, 6, 6) | 0.0506, 0, $-$0.2904 | 2.1742 |

$\left\{\begin{array}{l}\dot{x}=\mathrm{F}\left(y\right)-x\\ \dot{y}=-x\mathrm{G}\left(z\right)\\ \dot{z}=x\mathrm{F}\left(y\right)-a\end{array}\right.$ [75] F(y) = |y| $-$ 5 G(z) = |z| $-$ 5 | a = 1 | (1, 5, 6) (1, $-$5, $-$4) | 0.2102, 0, $-$1.2102 | 2.1737 |

$\left\{\begin{array}{l}\dot{x}=ay\mathrm{G}\left(z\right)\\ \dot{y}=\mathrm{F}\left(x\right)-y\\ \dot{z}=1-y\mathrm{F}\left(x\right)\end{array}\right.$ [75] F(x) = |x| $-$ 7 G(z) = |z| $-$ 7 | a = 1 | (8, 1, 6) ($-$6, 1, $-$8) | 0.2100, 0, $-$1.210 | 2.1736 |

$\left\{\begin{array}{l}\dot{x}=-\mathrm{G}\left(y\right)z\\ \dot{y}={\left(\mathrm{F}\right(x\left)\right)}^{2}-1\\ \dot{z}=\mathrm{F}\left(x\right)-az\end{array}\right.$ [75] F(x) = |x| $-$ 4 G(y) = |y| $-$ 5 | a = 2 | (5, 6, $-$1) ($-$3, $-$4, $-$1) | 0.0568, 0, $-$2.0568 | 2.0276 |

$\left\{\begin{array}{l}\dot{x}=az\\ \dot{y}=-{x}^{2}+1\\ \dot{z}=bx-z-x\left|\mathrm{F}\right(y\left)\right|\end{array}\right.$ [75] F(y) = |y| $-$ 22 | a = 18 b = 1.93 | (1, 23, $-$1) ($-$1, $-$21, 1) | 0.120, 0, $-$1.12 | 2.1071 |

$\left\{\begin{array}{l}\dot{x}=-ax+bx\left|\mathrm{F}\right(y\left)\right|\\ \dot{y}=-{x}^{2}+{\left(\mathrm{G}\right(z\left)\right)}^{2}+\mathrm{F}\left(y\right)\mathrm{G}\left(z\right)\\ \dot{z}=-c{\left(\mathrm{F}\right(y\left)\right)}^{2}-d\mathrm{F}\left(y\right)\mathrm{G}\left(z\right)+1\end{array}\right.$ [75] F(y) = |y| $-$ 5 G(z) = |z| $-$ 12 | a = 0.33 b = 0.75 c = 0.35 d = 0.9 | (2, 1, $-$1) ($-$2, 1, $-$1) | 0.0301, 0, $-$2.0419 | 2.0148 |

$\left\{\begin{array}{l}\dot{x}=-ax\left|\mathrm{F}\right(y\left)\right|+bx\left|z\right|\\ \dot{y}=-{x}^{2}-\mathrm{F}\left(y\right)z+1\\ \dot{z}=c\mathrm{F}\left(y\right)+z\end{array}\right.$ [75] F(y) = |y| $-$ 3 | a = 16.8 b = 2.8 c = 8.25 | ($-$1, 1, $-$1) (1, 1, $-$1) | 0.2985, 0, $-$1.1319 | 2.2638 |

$\left\{\begin{array}{l}\dot{x}=-a\mathrm{F}\left(y\right)z+bx\left|\mathrm{F}\right(y\left)\right|-cx\left|z\right|\\ \dot{y}={x}^{2}-{z}^{2}+d\\ \dot{z}=z+xF\left(y\right)-z\left|\mathrm{F}\right(y\left)\right|\end{array}\right.$ [75] F(y) = |y| $-$ 3 | a = 7.3 b = 6.4 c = 118 d = 0.1 | (1, 4, 1) ($-$1, $-$2, 1) | 0.0708, 0, $-$25.6071 | 2.0028 |

$\left\{\begin{array}{l}\dot{x}=y-y\left|z\right|\\ \dot{y}=-y+y\left|\mathrm{F}\right(x\left)\right|\\ \dot{z}=a\mathrm{F}\left(x\right)\left|y\right|-bz\left|\mathrm{F}\right(x\left)\right|\end{array}\right.$ [75] F(x) = |x| $-$ 4 | a = 1 b = 20 | (5, 1, 3) ($-$3, 1, $-$3) | 0.4543, 0, $-$20.4543 | 2.0222 |

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## Share and Cite

**MDPI and ACS Style**

Li, C.; Li, Z.; Jiang, Y.; Lei, T.; Wang, X.
Symmetric Strange Attractors: A Review of Symmetry and Conditional Symmetry. *Symmetry* **2023**, *15*, 1564.
https://doi.org/10.3390/sym15081564

**AMA Style**

Li C, Li Z, Jiang Y, Lei T, Wang X.
Symmetric Strange Attractors: A Review of Symmetry and Conditional Symmetry. *Symmetry*. 2023; 15(8):1564.
https://doi.org/10.3390/sym15081564

**Chicago/Turabian Style**

Li, Chunbiao, Zhinan Li, Yicheng Jiang, Tengfei Lei, and Xiong Wang.
2023. "Symmetric Strange Attractors: A Review of Symmetry and Conditional Symmetry" *Symmetry* 15, no. 8: 1564.
https://doi.org/10.3390/sym15081564