# Dynamics, Periodic Orbit Analysis, and Circuit Implementation of a New Chaotic System with Hidden Attractor

## Abstract

**:**

## 1. Introduction

## 2. The New System and Its Dynamic Characteristics

## 3. Chaotic and Complex Dynamics in New System

#### 3.1. Fix $a=10,c=11.2,k=-0.2$ and Vary b

#### 3.2. Fix $b=100,c=11.2,k=-0.2$ and Vary a

#### 3.3. Fix $a=10,b=100,k=-0.2$ and Vary c

#### 3.4. Fix $a=10,b=100,c=11.2$ and Vary k

#### 3.5. Fix $a=10,b=100$ and Vary k and c

## 4. Diverse Symbolic Dynamics for Unstable Periodic Orbits

#### 4.1. Variational Method

#### 4.2. Unstable Cycles Embedded in Hidden Chaotic Attractor for $(a,b,c,k)=(10,100,11.2,-0.2)$

#### 4.3. Unstable Periodic Orbits Embedded in Self-Excited Chaotic Attractor for $(a,b,c,k)=(10,64,0,0)$

## 5. Circuit Simulation

## 6. Conclusions and Discussion

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Projections of chaotic attractor onto various planes at time $t=200$: (

**a**) x–z phase space; (

**b**) y–z phase space; (

**c**) x–y phase space; (

**d**) continuous broadband frequency spectrum.

**Figure 3.**(

**a**) 3D phase portrait of system (1) for $(a,b,c,k)=(10,100,11.2,-0.2)$. Initial conditions ${I}_{1}$ lead to hidden chaotic attractor, and initial conditions ${I}_{2},{I}_{3}$ lead to asymptotically converging behaviors to equilibrium point ${E}_{1}$ and ${E}_{2},$ respectively; (

**b**) coexisting time series diagram of $x\left(t\right)$.

**Figure 4.**Basins of attraction for system (1) at $z=-9.1503$. Blue and red basins represent attractors of two stable node-focus points ${E}_{1}$ and ${E}_{2}$, yellow region denotes basin of chaotic attractor, and black stripes denote crossing trajectories of chaotic attractor.

**Figure 5.**Largest Lyapunov exponent spectrum (

**a**) and bifurcation diagram (

**b**) of system (1) versus b, where $a=10,c=11.2,k=-0.2$.

**Figure 6.**3D view of phase portraits of system (1), where $a=10,c=11.2,k=-0.2$: (

**a**) $b=20$, (

**b**) $b=80$, (

**c**) $b=130$.

**Figure 7.**Parameter values $(b,c,k)=(100,11.2,-0.2)$, largest Lyapunov exponent spectrum (

**a**), and bifurcation diagram (

**b**) of system (1) for $a\in [6,20]$.

**Figure 8.**3D view of phase portraits of system (1), $b=100,c=11.2,k=-0.2$: (

**a**) $a=7$; (

**b**) $a=12.2$; (

**c**) $a=13$; (

**d**) $a=20$.

**Figure 9.**Parameter values $(a,b,k)=(10,100,-0.2)$, largest Lyapunov exponent spectrum (

**a**) and bifurcation diagram (

**b**) of system (1) for $c\in [-30,20]$.

**Figure 10.**2D view of different limit cycles of system (1), $a=10,b=100,k=-0.2$: (

**a**) $c=-12$; (

**b**) $c=-7.74$; (

**c**) $c=-7.2$; and (

**d**) $c=3$.

**Figure 11.**Largest Lyapunov exponent spectrum (

**a**) and bifurcation diagram (

**b**) of system (1) versus k, where $a=10,b=100,c=11.2$.

**Figure 12.**Division of parameters k and c with different initial conditions: (

**a**) $({x}_{0},{y}_{0},{z}_{0})=(1,1,1);$ (

**b**) $({x}_{0},{y}_{0},{z}_{0})=(1,10,1)$.

**Figure 13.**First return map of system (1) under different parameters: (

**a**) Poincaré section $z=-9.1503$, $(a,b,c,k)=(10,100,11.2,-0.2)$; (

**b**) Poincaré section $z=0$, $(a,b,c,k)=(10,64,0,0)$.

**Figure 15.**Four basic building blocks in system (1) for parameters $(a,b,c,k)=(10,100,11.2,-0.2)$: (

**a**) cycle 0; (

**b**) cycle 1; (

**c**) cycle 2; and (

**d**) cycle 3.

**Figure 16.**Unstable cycles in system (1) under parameters $(a,b,c,k)=(10,100,11.2,-0.2)$: (

**a**) cycle 12; (

**b**) 03; (

**c**) 01; (

**d**) 23; (

**e**) 001; (

**f**) 112; (

**g**) 023; and (

**h**) 233.

**Figure 17.**Four building blocks in system (1) for parameters $(a,b,c,k)=(10,64,0,0)$: (

**a**) cycle 2; (

**b**) cycle 3; (

**c**) cycle 4; (

**d**) cycle 5.

**Figure 18.**Unstable periodic orbits in system (1) for parameters $(a,b,c,k)=(10,64,0,0)$. Two equilibria are marked with “+”. (

**a**) cycle 24; (

**b**) cycle 25; (

**c**) cycle 45; (

**d**) cycle 033; (

**e**) cycle 021; (

**f**) cycle 132; (

**g**) cycle 324; (

**h**) cycle 255; (

**i**) cycle 335; (

**j**) cycle 325; (

**k**) cycle 225; (

**l**) cycle 254.

**Table 1.**Twenty unstable periodic orbits embedded in hidden chaotic attractor of system (1) for $(a,b,c,k)=(10,100,11.2,-0.2)$, showing topological length, itinerary p, period ${T}_{p}$, and three coordinates of a point on the periodic orbit.

Length | p | ${\mathit{T}}_{\mathit{p}}$ | x | y | z |
---|---|---|---|---|---|

1 | 0 | 0.635920 | −7.028076 | 0.430355 | 1.092913 |

1 | 0.635920 | 7.028076 | −0.430355 | 1.092913 | |

2 | 1.192933 | 2.544434 | 12.123766 | 20.650672 | |

3 | 1.192933 | −2.544434 | −12.123766 | 20.650672 | |

2 | 12 | 1.752388 | 2.807407 | 6.313538 | 12.146665 |

03 | 1.752388 | −2.807407 | −6.313538 | 12.146665 | |

01 | 1.467965 | 0.100280 | 1.271667 | −7.073590 | |

23 | 2.383824 | −14.090307 | 17.922194 | 33.086632 | |

3 | 001 | 2.174153 | −7.214950 | 4.081725 | 7.082844 |

011 | 2.174153 | 7.214950 | −4.081725 | 7.082844 | |

003 | 2.361334 | −1.162938 | −0.669254 | −14.983886 | |

112 | 2.361334 | 1.162938 | 0.669254 | −14.983886 | |

132 | 2.940945 | −2.570910 | −0.291669 | 11.591391 | |

023 | 2.940945 | 2.570910 | 0.291669 | 11.591391 | |

021 | 2.554559 | −0.016908 | −0.016629 | −31.681837 | |

013 | 2.554559 | 0.016908 | 0.016629 | −31.681837 | |

033 | 2.946229 | 0.291076 | 0.540818 | −60.155264 | |

122 | 2.946229 | −0.291076 | −0.540818 | −60.155264 | |

223 | 3.954898 | −5.509519 | −11.111451076 | −71.906347 | |

233 | 3.954898 | 5.509519 | 11.111451076 | −71.906347 |

**Table 2.**Forty-one unstable periodic orbits embedded in self-excited chaotic attractor of system (1) for $(a,b,c,k)=(10,64,0,0)$.

Length | p | ${\mathit{T}}_{\mathit{p}}$ | Self-Linking | Length | p | ${\mathit{T}}_{\mathit{p}}$ | Self-Linking | p | ${\mathit{T}}_{\mathit{p}}$ | Self-Linking |
---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 1.016946 | 0 | 3 | 223 | 2.994130 | 0 | 031 | 2.447450 | 2 |

3 | 1.016946 | 0 | 233 | 2.994130 | 0 | 012 | 2.447450 | 2 | ||

2 | 01 | 1.358438 | 1 | 033 | 2.609712 | 2 | 132 | 2.505368 | 0 | |

23 | 1.965825 | 1 | 122 | 2.609712 | 2 | 023 | 2.505368 | 0 | ||

12 | 1.587528 | 1 | 021 | 2.323226 | 0 | |||||

03 | 1.587528 | 1 | 013 | 2.323226 | 0 | |||||

1 | 4 | 1.312552 | 1 | 445 | 4.235720 | 3 | 354 | 3.955079 | 2 | |

5 | 1.312552 | 1 | 455 | 4.235720 | 3 | 234 | 3.263831 | 1 | ||

2 | 24 | 2.289914 | 2 | 344 | 3.667897 | 1 | 325 | 3.263831 | 1 | |

25 | 2.354458 | 0 | 255 | 3.667897 | 1 | 225 | 3.367249 | 1 | ||

34 | 2.354458 | 0 | 335 | 3.312270 | 1 | 334 | 3.367249 | 1 | ||

35 | 2.289914 | 2 | 224 | 3.312270 | 1 | 254 | 3.606269 | 3 | ||

45 | 2.642183 | 1 | 244 | 3.600833 | 3 | 345 | 3.606269 | 3 | ||

3 | 235 | 3.349139 | 1 | 355 | 3.600833 | 3 | ||||

324 | 3.349139 | 1 | 245 | 3.955079 | 2 |

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Dong, C.
Dynamics, Periodic Orbit Analysis, and Circuit Implementation of a New Chaotic System with Hidden Attractor. *Fractal Fract.* **2022**, *6*, 190.
https://doi.org/10.3390/fractalfract6040190

**AMA Style**

Dong C.
Dynamics, Periodic Orbit Analysis, and Circuit Implementation of a New Chaotic System with Hidden Attractor. *Fractal and Fractional*. 2022; 6(4):190.
https://doi.org/10.3390/fractalfract6040190

**Chicago/Turabian Style**

Dong, Chengwei.
2022. "Dynamics, Periodic Orbit Analysis, and Circuit Implementation of a New Chaotic System with Hidden Attractor" *Fractal and Fractional* 6, no. 4: 190.
https://doi.org/10.3390/fractalfract6040190