# Dynamic Analysis of a Novel 3D Chaotic System with Hidden and Coexisting Attractors: Offset Boosting, Synchronization, and Circuit Realization

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model and Its Properties

## 3. Dynamics of Novel Double-Wing Chaotic System

#### 3.1. Bifurcation Diagram and Lyapunov Exponents

#### 3.2. Two-Parameter Lyapunov Exponents Analysis

#### 3.3. Coexisting Attractors and Basins of Attraction

#### 3.4. Impact of Constant Term d

#### 3.5. Offset Boosting Control

## 4. One-Dimensional Symbolic Dynamics for Unstable Cycles Embedded in Hidden Chaotic Attractor

## 5. Circuit Implementation

## 6. Adaptive Synchronization of Novel Three-Dimensional Chaotic System

## 7. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Two-dimensional projections of chaotic attractor onto various planes at time $t=150$: (

**a**) $x-z$, (

**b**) $y-z$, and (

**c**) $x-y$ planes.

**Figure 2.**(

**a**) Bifurcation diagram and (

**b**) Lyapunov exponents spectrum of system (2) vs. a, where $b=3,c=35,$ and $d=10$. 3D view of phase portraits with (

**c**) $a=30$ and (

**d**) $a=50$.

**Figure 3.**Parameter values $(a,c,d)=(35,35,10)$, (

**a**) bifurcation diagram, and (

**b**) Lyapunov exponents spectrum of system (2) for $b\in [0,5]$.

**Figure 4.**Two-dimensional view of different periodic attractors of system (2), $a=c=35$ and $d=10$: (

**a**) $b=0.3$; (

**b**) $b=1$; (

**c**) $b=1.8$.

**Figure 6.**Coexisting hidden chaotic attractor and stable equilibrium state attractors of system (2); $(a,b,c,d)=(35,3,35,10):$ (

**a**) 3D phase portraits; (

**b**) basins of attraction.

**Figure 7.**Two coexisting periodic attractors of system (2); $(a,b,c,d)=(35,0.5,35,10):$ (

**a**) periodic attractor with initial values $(-1,-1,1)$; (

**b**) another periodic attractor with initial values $(1,1,1)$; (

**c**) basins of attraction.

**Figure 8.**Coexisting periodic attractors of system (2) in $(x,z)$ plane: (

**a**) $(a,b,c,d)=(35,0.42,35,10)$ and (

**b**) $(a,b,c,d)=(35,1.5,35,10)$.

**Figure 9.**Coexisting chaotic attractors of system (2) in $(x,z)$ plane; $(a,b,c,d)=(35,0.53,35,10)$: (

**a**) chaotic attractor with initial values $(-1,-1,1)$ and (

**b**) another chaotic attractor with initial values $(1,1,1)$.

**Figure 10.**Riddled basins of attraction in $x\left(0\right)-y\left(0\right)$ initial plane with $z\left(0\right)=35$.

**Figure 11.**Offset boosting of chaotic attractor when varying control parameter w for $(a,b,c,d)=(35,3,35,10)$: (

**a**) in $y-z$ plane and (

**b**) state y with different values of the offset boosting controller $w.$ All computed for initial values $(1,1,1)$.

**Figure 12.**First return map of system (2) under parameters $(a,b,c,d)=(35,3,35,10)$; the Poincaré section is taken as $z=35$.

**Figure 13.**Two simplest periodic orbits as basic building blocks in system (2) for parameters $(a,b,c,d)=(35,3,35,10)$: (

**a**) cycle 0 and (

**b**) cycle 1.

**Figure 14.**Unstable cycles in system (2) under parameters $(a,b,c,d)=(35,3,35,10)$: Cycles (

**a**) 01; (

**b**) 011; (

**c**) 0011; (

**d**) 0111; (e) 00101; (f) 00111.

**Figure 16.**Chaotic behaviors of implemented electronic circuit with initial conditions $\left(X\right(0),Y(0),Z(0\left)\right)=(1V,1V,1V)$ in (

**a**) $X-Z$, (

**b**) $Y-Z$, and (

**c**) $X-Y$ planes.

**Figure 17.**Time evolution sequence diagram of master and slave systems showing results of occurrence of adaptive synchronization. (

**a**) x variable; (

**b**) y variable; (

**c**) z variable.

**Figure 18.**Time evolution of (

**a**) synchronization errors ${e}_{x}$, ${e}_{y}$, and ${e}_{z}$, and (

**b**) parameter estimation errors ${e}_{a},{e}_{b},{e}_{c}$, and ${e}_{d}$.

Systems | Equations | Parameters | Equilibria | Eigenvalues | Lyapunov Exponents | Fractional Dimensions | Attractor Type |
---|---|---|---|---|---|---|---|

This work | $\stackrel{\xb7}{x}=a(y-x)$ | $a=35$ | $(-10.7238,-10.7238,35)$ | $-37.812$ | $1.100$ | $2.0281$ | Hidden |

$\stackrel{\xb7}{y}=cx-xz$ | $b=3$ | $(10.7238,10.7238,35)$ | $-0.094\pm 14.591i$ | 0 | |||

$\stackrel{\xb7}{z}=xy-bz-d$ | $c=35$ | $-39.098$ | |||||

$d=10$ | |||||||

Dong [10] | $\stackrel{\xb7}{x}=a(y-x)+kxz$ | $a=10$ | $(-11.0634,-9.0387,-9.1503)$ | $-18.7413$ | $0.7457$ | $2.0276$ | Hidden |

$\stackrel{\xb7}{y}=-cy-xz$ | $b=100$ | $(11.0634,9.0387,-9.1503)$ | $-0.314\pm 11.424i$ | $-0.0057$ | |||

$\stackrel{\xb7}{z}=-b+xy$ | $c=11.2$ | $-26.8144$ | |||||

$k=-0.2$ | |||||||

$a=10$ | $(-8,-8,0)$ | $-12.8068$ | $1.4456$ | $2.1264$ | Self-excited | ||

$b=64$ | $(8,8,0)$ | $1.4034\pm 9.8983i$ | $0.001$ | ||||

$c=0$ | $-11.4473$ | ||||||

$k=0$ |

This Work | Dong [10] | |
---|---|---|

Establishment of mathematical model | Adding a simple constant term $-d$ to Yang-Chen system | Adding a nonlinear term of cross-product $kxz$ to generalized Lorenz-type system |

Dynamics | Yes | Yes |

Coexisting attractors | Yes | No |

Offset boosting control | Yes | No |

Symbolic dynamics of unstable cycles | Two letters | Four letters for hidden attractor |

Six letters for self-excited attractor | ||

Circuit implementation | Yes | Yes |

Synchronization | Yes | No |

**Table 3.**Unstable cycles embedded in hidden chaotic attractor of system (2) up to topological length 5 for $(a,b,c,d)=(35,3,35,10)$.

Length | Itineraries | Periods | x | y | z |
---|---|---|---|---|---|

1 | 0 | 0.468918 | −10.393417 | −7.216587 | 43.634264 |

1 | 0.468918 | 10.393417 | 7.216587 | 43.634264 | |

2 | 01 | 1.190901 | −15.856545 | −21.285817 | 21.799902 |

3 | 001 | 1.768396 | −1.142202 | 0.192829 | 40.538631 |

011 | 1.768396 | 1.142202 | −0.192829 | 40.538631 | |

4 | 0001 | 2.338366 | −5.390366 | −2.042326 | 44.498047 |

0011 | 2.364638 | 8.016602 | 2.946163 | 47.893544 | |

0111 | 2.338366 | 5.390366 | 2.042326 | 44.498047 | |

5 | 00001 | 2.975663 | −0.259779 | 0.021441 | 36.277845 |

00011 | 2.939762 | −2.797000 | −3.617918 | 20.432365 | |

00101 | 2.962243 | −15.163685 | −7.655255 | 52.919827 | |

00111 | 2.939762 | 2.797000 | 3.617918 | 20.432365 | |

01011 | 2.962243 | 15.163685 | 7.655255 | 52.919827 | |

01111 | 2.975663 | 0.259779 | −0.021441 | 36.277845 |

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**MDPI and ACS Style**

Dong, C.
Dynamic Analysis of a Novel 3D Chaotic System with Hidden and Coexisting Attractors: Offset Boosting, Synchronization, and Circuit Realization. *Fractal Fract.* **2022**, *6*, 547.
https://doi.org/10.3390/fractalfract6100547

**AMA Style**

Dong C.
Dynamic Analysis of a Novel 3D Chaotic System with Hidden and Coexisting Attractors: Offset Boosting, Synchronization, and Circuit Realization. *Fractal and Fractional*. 2022; 6(10):547.
https://doi.org/10.3390/fractalfract6100547

**Chicago/Turabian Style**

Dong, Chengwei.
2022. "Dynamic Analysis of a Novel 3D Chaotic System with Hidden and Coexisting Attractors: Offset Boosting, Synchronization, and Circuit Realization" *Fractal and Fractional* 6, no. 10: 547.
https://doi.org/10.3390/fractalfract6100547