# Hidden and Coexisting Attractors in a Novel 4D Hyperchaotic System with No Equilibrium Point

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Novel 4D Hyperchaotic System

## 3. Complex Dynamical Structure of the Proposed Hyperchaotic System

#### 3.1. Lyapunov Exponents, Bifurcation Diagram, and ${C}_{0}$ Complexity Analysis

#### 3.2. Coexisting Attractors

#### 3.2.1. Coexistence of Chaotic and Periodic Attractors

#### 3.2.2. Coexistence of Quasi-Periodic and Periodic Attractors

#### 3.2.3. Coexistence of Chaotic and Quasi-Periodic Attractors

#### 3.2.4. Coexistence of Hidden Periodic Attractors

#### 3.2.5. Coexistence of Hidden Hyperchaotic Attractors

## 4. Analysis of Unstable Cycles for New 4D Hyperchaotic System via Variational Approach

#### 4.1. Variational Method for Calculations

#### 4.2. Extracting Unstable Cycles in a Hidden Hyperchaotic Attractor

#### 4.3. Homotopy Evolution of Cycle Variation with Different Parameters

## 5. Circuit Design and Realization of New System

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Strogatz, S.H. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering; Perseus Books: Reading, MA, USA, 1994. [Google Scholar]
- Cvitanović, P. Universality in Chaos, 2nd ed.; Adam Hilger: Bristol, UK, 1989. [Google Scholar]
- Rössler, O.E. An equation for hyperchaos. Phy. Lett. A
**1979**, 71, 155–156. [Google Scholar] [CrossRef] - Wang, X.; Kuznetsov, N.V.; Chen, G. (Eds.) Chaotic Systems with Multistability and Hidden Attractors; Emergence, Complexity and Computation; Springer: Cham, Switzerland, 2021; Volume 40, pp. 149–150. [Google Scholar]
- Gao, T.; Chen, Z.; Yuan, Z.; Chen, G. A hyperchaos generated from Chen’s system. Int. J. Mod. Phys. C
**2011**, 17, 471–478. [Google Scholar] [CrossRef] - Wang, F.Q.; Liu, C.X. Hyperchaos evolved from the Liu chaotic system. Chin. Phys.
**2006**, 15, 963–968. [Google Scholar] - Wang, X.; Wang, M. A hyperchaos generated from Lorenz system. Phys. A Stat. Mech. Appl.
**2008**, 387, 3751–3758. [Google Scholar] [CrossRef] - Li, Y.; Tang, W.; Chen, G. Hyperchaos evolved from the generalized Lorenz equation. Int. J. Circ. Theor. Appl.
**2005**, 33, 235–251. [Google Scholar] [CrossRef] - Bao, B.; Xu, J.; Liu, Z.; Ma, Z. Hyperchaos from an augmented Lü system. Int. J. Bifurcat. Chaos
**2010**, 20, 3689–3698. [Google Scholar] [CrossRef] - Yang, Q.; Bai, M. A new 5D hyperchaotic system based on modified generalized Lorenz system. Nonlinear Dyn.
**2017**, 88, 189–221. [Google Scholar] [CrossRef] - Shen, C.; Yu, S.; Lü, J.; Chen, G. Generating hyperchaotic systems with multiple positive Lyapunov exponents. In Proceedings of the 9th Asian Control Conference (ASCC), Istanbul, Turkey, 23–26 June 2013; pp. 1–5. [Google Scholar]
- Yang, Q.; Zhu, D.; Yang, L. A New 7D hyperchaotic system with five positive Lyapunov exponents coined. Int. J. Bifurcat. Chaos
**2018**, 28, 1850057. [Google Scholar] [CrossRef] - Leonov, G.A.; Kuznetsov, N.V. Hidden attractors in dynamical systems. From hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits. Int. J. Bifurcat. Chaos
**2013**, 23, 1330002. [Google Scholar] [CrossRef] [Green Version] - Lorenz, E.N. Deterministic nonperiodic flow. J. Atmos. Sci.
**1963**, 20, 130–141. [Google Scholar] [CrossRef] [Green Version] - Chen, G.R.; Ueta, T. Yet another chaotic attractor. Int. J. Bifurcat. Chaos
**1999**, 9, 1465–1466. [Google Scholar] [CrossRef] - Lü, J.; Chen, G. A new chaotic attractor coined. Int. J. Bifurcat. Chaos
**2002**, 12, 659–661. [Google Scholar] [CrossRef] - Sprott, J.C. Some simple chaotic flows. Phys. Rev. E
**1994**, 50, 647–650. [Google Scholar] [CrossRef] - Wei, Z.; Wang, R.; Liu, A. A new finding of the existence of hidden hyperchaotic attractors with no equilibria. Math. Comput. Simulat.
**2014**, 100, 13–23. [Google Scholar] [CrossRef] - Cao, H.Y.; Zhao, L. A new chaotic system with different equilibria and attractors. Eur. Phys. J. Spec. Top.
**2021**, 230, 1905–1914. [Google Scholar] [CrossRef] - Lai, Q.; Wan, Z.; Kuate, P. Modelling and circuit realisation of a new no-equilibrium chaotic system with hidden attractor and coexisting attractors. Electron. Lett.
**2020**, 56, 1044–1046. [Google Scholar] [CrossRef] - Pham, V.T.; Volos, C.; Jafari, S.; Wei, Z.; Wang, X. Constructing a novel no-equilibrium chaotic system. Int. J. Bifurcat. Chaos
**2014**, 24, 1450073. [Google Scholar] [CrossRef] - Azar, A.T.; Volos, C.; Gerodimos, N.A.; Tombras, G.S.; Pham, V.T.; Radwan, A.G.; Vaidyanathan, S.; Ouannas, A.; Munoz-Pacheco, J.M. A novel chaotic system without equilibrium: Dynamics, synchronization, and circuit realization. Complexity
**2017**, 2017, 7871467. [Google Scholar] [CrossRef] - Yang, Q.; Wei, Z.; Chen, G. An unusual 3d autonomous quadratic chaotic system with two stable node-foci. Int. J. Bifurcat. Chaos
**2010**, 20, 1061–1083. [Google Scholar] [CrossRef] - Dong, C. Dynamics, periodic orbit analysis, and circuit implementation of a new chaotic system with hidden attractor. Fractal Fract.
**2022**, 6, 190. [Google Scholar] [CrossRef] - Pham, V.T.; Jafari, S.; Kapitaniak, T. Constructing a chaotic system with an infinite number of equilibrium points. Int. J. Bifurcat. Chaos
**2016**, 26, 1650225. [Google Scholar] [CrossRef] - Wang, X.; Chen, G. Constructing a chaotic system with any number of equilibria. Nonlinear Dyn.
**2013**, 71, 429–436. [Google Scholar] [CrossRef] [Green Version] - Yang, Q.; Qiao, X. Constructing a new 3D chaotic system with any number of equilibria. Int. J. Bifurcat. Chaos
**2019**, 29, 1950060. [Google Scholar] [CrossRef] - Kuznetsov, N.V.; Leonov, G.A.; Vagaitsev, V.I. Analytical-numerical method for attractor localization of generalized Chua’s system. In Proceedings of the IFAC Proceedings Volumes (IFAC-Papers Online), Antalya, Turkey, 26–28 August 2010. [Google Scholar]
- Ren, S.; Panahi, S.; Rajagopal, K.; Akgul, A.; Pham, V.T. A new chaotic flow with hidden attractor: The first hyperjerk system with no equilibrium. Z. Nat. A
**2018**, 73, 239–249. [Google Scholar] [CrossRef] - Wei, Z.; Rajagopal, K.; Zhang, W.; Kingni, S.T.; Akgül, A. Synchronisation, electronic circuit implementation, and fractional-order analysis of 5D ordinary differential equations with hidden hyperchaotic attractors. Pramana–J. Phys.
**2018**, 90, 50. [Google Scholar] [CrossRef] - Yang, Q.; Yang, L.; Ou, B. Hidden hyperchaotic attractors in a new 5D system based on chaotic system with two stable node-foci. Int. J. Bifurcat. Chaos
**2019**, 29, 1950092. [Google Scholar] [CrossRef] - Cui, L.; Luo, W.; Ou, Q. Analysis of basins of attraction of new coupled hidden attractor system. Chaos Soliton. Fract.
**2021**, 146, 110913. [Google Scholar] [CrossRef] - Lai, Q.; Akgul, A.; Li, C.; Xu, G.; Cavusoglu, U. A new chaotic system with multiple attractors: Dynamic analysis, circuit realization and S-Box design. Entropy
**2017**, 20, 12. [Google Scholar] [CrossRef] [Green Version] - Bayani, A.; Rajagopal, K.; Khalaf, A.J.M.; Jafari, S.; Leutcho, G.D.; Kengne, J. Dynamical analysis of a new multistable chaotic system with hidden attractor: Antimonotonicity, coexisting multiple attractors, and offset boosting. Phys. Lett. A
**2019**, 383, 1450–1456. [Google Scholar] [CrossRef] - Nazarimehr, F.; Rajagopal, K.; Kengne, J.; Jafari, S.; Pham, V.T. A new four-dimensional system containing chaotic or hyper-chaotic attractors with no equilibrium, a line of equilibria and unstable equilibria. Chaos Soliton. Fract.
**2018**, 111, 108–118. [Google Scholar] [CrossRef] - Lai, Q.; Chen, C.; Zhao, X.W.; Kengne, J.; Volos, C. Constructing chaotic system with multiple coexisting attractors. IEEE Access
**2019**, 7, 24051–24056. [Google Scholar] [CrossRef] - Ma, C.; Mou, J.; Xiong, L.; Banerjee, S.; Liu, T.; Han, X. Dynamical analysis of a new chaotic system: Asymmetric multistability, offset boosting control and circuit realization. Nonlinear Dyn.
**2021**, 103, 2867–2880. [Google Scholar] [CrossRef] - Lai, Q.; Norouzi, B.; Liu, F. Dynamic analysis, circuit realization, control design and image encryption application of an extended Lü system with coexisting attractors. Chaos Soliton. Fract.
**2018**, 114, 230–245. [Google Scholar] [CrossRef] - Natiq, H.; Said, M.; Al-Saidi, N.; Kilicman, A. Dynamics and complexity of a new 4D chaotic laser system. Entropy
**2019**, 21, 34. [Google Scholar] [CrossRef] [Green Version] - Rajagopal, K.; Akgul, A.; Pham, V.T.; Alsaadi, F.E.; Nazarimehr, F.; Alsaadi, F.E.; Jafari, S. Multistability and coexisting attractors in a new circulant chaotic system. Int. J. Bifurcat. Chaos
**2019**, 29, 1950174. [Google Scholar] [CrossRef] - Lai, Q.; Wan, Z.; Kuate, P.; Fotsin, H. Coexisting attractors, circuit implementation and synchronization control of a new chaotic system evolved from the simplest memristor chaotic circuit. Commun. Nonlinear Sci. Numer. Simul.
**2020**, 89, 105341. [Google Scholar] [CrossRef] - Sprott, J.C. A proposed standard for the publication of new chaotic systems. Int. J. Bifurcat. Chaos
**2011**, 21, 2391–2394. [Google Scholar] [CrossRef] [Green Version] - Li, Y.; Tang, W.K. S; Chen, G.R. Generating hyperchaos via state feedback control. Int. J. Bifurcat. Chaos
**2005**, 15, 3367–3375. [Google Scholar] [CrossRef] - Ramasubramanian, K.; Sriram, M.S. A comparative study of computation of Lyapunov spectra with different algorithms. Phys. D Nonlinear Phenom.
**2000**, 139, 72–86. [Google Scholar] [CrossRef] [Green Version] - Cvitanović, P.; Artuso, R.; Mainieri, R.; Tanner, G.; Vattay, G. Chaos: Classical and Quantum; Niels Bohr Institute: Copenhagen, Denmark, 2012; pp. 131–133. [Google Scholar]
- Lan, Y.; Cvitanović, P. Variational method for finding periodic orbits in a general flow. Phys. Rev. E
**2004**, 69, 016217. [Google Scholar] [CrossRef] [Green Version] - Dong, C.; Jia, L.; Jie, Q.; Li, H. Symbolic encoding of periodic orbits and chaos in the Rucklidge system. Complexity
**2021**, 2021, 4465151. [Google Scholar] [CrossRef] - Dong, C.; Liu, H.; Li, H. Unstable periodic orbits analysis in the generalized Lorenz–type system. J. Stat. Mech.
**2020**, 2020, 073211. [Google Scholar] [CrossRef] - Dong, C.; Lan, Y. Organization of spatially periodic solutions of the steady Kuramoto–Sivashinsky equation. Commun. Nonlinear Sci. Numer. Simul.
**2014**, 19, 2140–2153. [Google Scholar] [CrossRef] - Dong, C.; Liu, H.; Jie, Q.; Li, H. Topological classification of periodic orbits in the generalized Lorenz-type system with diverse symbolic dynamics. Chaos Soliton. Fract.
**2022**, 154, 111686. [Google Scholar] [CrossRef] - Hao, B.L.; Zheng, W.M. Applied Symbolic Dynamics and Chaos; World Scientic: Singapore, 1998; pp. 11–13. [Google Scholar]
- Guckenheimer, J.; Holmes, P. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields; Springer: New York, NY, USA, 1983. [Google Scholar]
- Dong, C. Topological classification of periodic orbits in the Yang-Chen system. EPL Europhys. Lett.
**2018**, 123, 20005. [Google Scholar] [CrossRef] - Zambrano-Serrano, E.; Anzo-Hernández, A. A novel antimonotic hyperjerk system: Analysis, synchronization and circuit design. Physica D Nonlinear Phenom.
**2021**, 424, 132927. [Google Scholar] [CrossRef]

**Figure 1.**Three-dimensional projections of the hyperchaotic attractor of system (2): $(a,b,c,k,m)=(10,100,2.7,-0.2,1)$. (

**a**) x-y-z phase space; (

**b**) x-z-w phase space; (

**c**) x-y-w phase space; (

**d**) y-z-w phase space.

**Figure 3.**Two-dimensional Poincaré maps of the hyperchaotic attractor of system (2); $(a,b,c,k,m)=(10,100,2.7,-0.2,1)$; (

**a**) on section $z=0$; (

**b**) on section $x=0$.

**Figure 4.**Time-sequence diagrams of system (2); $(a,b,c,k,m)=(10,100,2.7,-0.2,1)$: (

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

**b**) $({x}_{0},{y}_{0},{z}_{0},{w}_{0})=(1.001,1,1,1)$; (

**c**) green and brown represent initial values of (

**a**,

**b**), respectively.

**Figure 5.**Dynamics of system (2) versus parameter $b\in [0,120]$ with $(a,c,k,m)=(10,2.7,-0.2,1)$: (

**a**,

**b**) Lyapunov exponent spectrum; (

**c**) bifurcation diagram.

**Figure 6.**Some representative dynamical behaviors of system (2) with parameters $(a,c,k,m)=(10,2.7,-0.2,1)$ and different values of b: (

**a**) $b=10$; (

**b**) $b=20$; (

**c**) $b=42$; (

**d**) $b=120$.

**Figure 7.**${C}_{0}$ complexity curve of the new system (2). (

**a**) Versus b for $a=10,\phantom{\rule{3.33333pt}{0ex}}c=2.7,\phantom{\rule{3.33333pt}{0ex}}k=-0.2,\phantom{\rule{3.33333pt}{0ex}}m=1$; (

**b**) versus k for $a=10,\phantom{\rule{3.33333pt}{0ex}}b=100,\phantom{\rule{3.33333pt}{0ex}}c=2.7,\phantom{\rule{3.33333pt}{0ex}}m=1$; (

**c**) versus m for $a=10,\phantom{\rule{3.33333pt}{0ex}}b=100,\phantom{\rule{3.33333pt}{0ex}}c=2.7,\phantom{\rule{3.33333pt}{0ex}}k=-0.2$. The initial values were set as $(1.67610,-0.37856,3.69140,1.45851)$.

**Figure 8.**Two coexisting hidden attractors of system (2); $(a,b,c,k,m)=(10,12,2.7,-0.2,1)$; (

**a**) chaotic attractor; (

**b**) periodic attractor; (

**c**) coexisting attractors. The yellow line represents chaotic attractor and the black line represents periodic attractor.

**Figure 9.**Two coexisting hidden attractors of system (2); $(a,b,c,k,m)=(10,24,2.7,-0.2,1)$; (

**a**) quasi-periodic attractor; (

**b**) periodic attractor; (

**c**) coexisting attractors. The red line represents quasi-periodic attractor and the blue line represents periodic attractor.

**Figure 10.**Two coexisting hidden attractors of system (2); $(a,b,c,k,m)=(10,40,2.7,-0.2,2)$; (

**a**) chaotic attractor; (

**b**) quasi-periodic attractor; (

**c**) coexisting attractors. The purple line represents chaotic attractor and the yellow line represents quasi-periodic attractor.

**Figure 11.**Two coexisting hidden periodic attractors of system (2); $(a,b,c,k,m)=(10,10,2.7,-0.2,2)$; (

**a**) periodic attractor; (

**b**) another periodic attractor; (

**c**) coexisting periodic attractors. The black line and the green line correspond to the periodic attractor shown in (

**a)**and (

**b**), respectively.

**Figure 12.**Three coexisting hidden hyperchaotic attractors of system (2); $(a,b,c,k,m)=(10,70,2.7,-0.2,5)$; (

**a**) asymmetrical hyperchaotic attractor; (

**b**) the other asymmetrical hyperchaotic attractor; (

**c**) symmetrical hyperchaotic attractor; (

**d**) coexisting hyperchaotic attractors. The green line, the red line and the blue line correspond to the hyperchaotic attractor shown in (

**a**), (

**b**), and (

**c**), respectively.

**Figure 13.**Basins of attraction in the $x\left(0\right)$–$y\left(0\right)$ initial plane with $z\left(0\right)=w\left(0\right)=0$.

**Figure 14.**Two shortest periodic orbits in system (2) for parameters $(a,b,c,k,m)=(10,100,2.7,-0.2,1)$; (

**a**) cycle 2; (

**b**) cycle 3.

**Figure 15.**Four periodic orbits with topological length 2 in system (2) for parameters $(a,b,c,k,m)=(10,100,2.7,-0.2,1)$; (

**a**) cycle 03; (

**b**) 12; (

**c**) 01; (

**d**) 23.

**Figure 16.**Unstable cycles with topological length 3 in system (2) for parameters $(a,b,c,k,m)=(10,100,2.7,-0.2,1)$; (

**a**) cycle 001; (

**b**) 003; (

**c**) 023; (

**d**) 021; (

**e**) 223; (

**f**) 012.

**Figure 17.**Cycle 02130101 with topological length 8 in system (2) for parameters $(a,b,c,k,m)=(10,100,2.7,-0.2,1)$.

**Figure 18.**Homotopy evolution of cycle 2 with respect to different parameters: (

**a**) four a values; (

**b**) b values; (

**c**) c values; (

**d**) k values; (

**e**) m values.

**Figure 20.**Two-dimensional phase portraits of the new system in Multisim of the circuit with $a=10,\phantom{\rule{3.33333pt}{0ex}}b=100,\phantom{\rule{3.33333pt}{0ex}}c=2.7,\phantom{\rule{3.33333pt}{0ex}}k=-0.2,$ and $m=1$: (

**a**) X–Z plane; (

**b**) X–Y plane; (

**c**) Y–W plane.

**Figure 21.**Phase portraits of coexisting attractors in Multisim of the circuit with $a=10,\phantom{\rule{3.33333pt}{0ex}}b=12,$ $c=2.7$, $k=-0.2,$ and $m=1$: (

**a**) hidden chaotic attractor; (

**b**) hidden periodic attractor. Scales of horizontal and vertical axes are 5 and 2 V/div, respectively.

**Table 1.**Lyapunov exponents and Kaplan–Yorke dimension of system (2) with $a=10,\phantom{\rule{3.33333pt}{0ex}}c=2.7$, $k=-0.2$, and $m=1.$

b | ${\mathit{L}}_{1}$ | ${\mathit{L}}_{2}$ | ${\mathit{L}}_{3}$ | ${\mathit{L}}_{4}$ | ${\mathit{D}}_{\mathit{KY}}$ | Dynamics |
---|---|---|---|---|---|---|

10 | 0 | −0.0377 | −0.4173 | −11.6842 | 1.0 | Periodic |

20 | 0.0483 | 0 | −0.2258 | −11.9110 | 2.24 | Chaos |

38 | 0 | −0.0227 | −0.0243 | −12.0242 | 1.0 | Periodic |

42 | 0 | 0 | −0.1340 | −11.9278 | 2.0 | Quasi-periodic |

50 | 0.0182 | 0 | −0.2922 | −11.7656 | 2.06 | Chaos |

120 | 0.9302 | 0.0850 | 0 | −12.8638 | 3.08 | Hyperchaos |

**Table 2.**Eighteen unstable periodic orbits embedded in the hidden hyperchaotic attractor of system (2) for $(a,b,c,k,m)=(10,100,2.7,-0.2,1)$; listed are the topological length, itinerary p, period ${T}_{p}$, and four coordinates of a point on the cycle.

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

1 | 2 | 0.858233 | 0.851259 | 3.599482 | −8.032931 | −39.656931 |

3 | 0.858233 | −0.851259 | −3.599482 | −8.032931 | 39.656931 | |

2 | 03 | 1.362034 | −4.076805 | −1.813737 | 1.109695 | −14.135359 |

12 | 1.362034 | 4.076805 | 1.813737 | 1.109695 | 14.135359 | |

01 | 1.194275 | 5.206540 | 7.525051 | −17.639962 | 1.385740 | |

23 | 1.830597 | 0.626331 | −0.321247 | −4.302274 | 1.490707 | |

3 | 001 | 1.732553 | −5.282481 | 3.245260 | 0.268165 | −34.418329 |

011 | 1.732553 | 5.282481 | −3.245260 | 0.268165 | 34.418329 | |

003 | 1.821191 | −4.653735 | 2.777113 | −2.962048 | −38.837657 | |

112 | 1.821191 | 4.653735 | −2.777113 | −2.962048 | 38.837657 | |

132 | 2.211630 | 11.320228 | 14.639216 | −16.413004 | 25.186818 | |

023 | 2.211630 | −11.320228 | −14.639216 | −16.413004 | −25.186818 | |

021 | 1.968277 | −6.298304 | 3.295041 | 5.572765 | −20.401797 | |

013 | 1.968277 | 6.298304 | −3.295041 | 5.572765 | 20.401797 | |

223 | 2.766255 | 1.453074 | −0.422130 | 2.547336 | 1.463103 | |

233 | 2.766255 | −1.453074 | 0.422130 | 2.547336 | −1.463103 | |

012 | 2.207939 | 4.137109 | 5.676602 | −5.643553 | −9.306008 | |

031 | 2.207939 | −4.137109 | −5.676602 | −5.643553 | 9.306008 |

a | ${\mathit{T}}_{\mathit{p}}$ | b | ${\mathit{T}}_{\mathit{p}}$ | c | ${\mathit{T}}_{\mathit{p}}$ | k | ${\mathit{T}}_{\mathit{p}}$ | m | ${\mathit{T}}_{\mathit{p}}$ |
---|---|---|---|---|---|---|---|---|---|

5 | 1.082797 | 60 | 0.953492 | −2 | 0.880703 | −0.5 | 0.705715 | −40 | 0.996271 |

10 | 0.858233 | 80 | 0.911549 | 0 | 0.873372 | −0.3 | 0.821457 | −20 | 0.968155 |

15 | 0.729400 | 120 | 0.811395 | 2 | 0.864607 | 0.1 | 0.964986 | 10 | 0.799075 |

20 | 0.610639 | 140 | 0.771540 | 4 | 0.833397 | 0.5 | 1.140899 | 30 | 0.611363 |

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**MDPI and ACS Style**

Dong, C.; Wang, J.
Hidden and Coexisting Attractors in a Novel 4D Hyperchaotic System with No Equilibrium Point. *Fractal Fract.* **2022**, *6*, 306.
https://doi.org/10.3390/fractalfract6060306

**AMA Style**

Dong C, Wang J.
Hidden and Coexisting Attractors in a Novel 4D Hyperchaotic System with No Equilibrium Point. *Fractal and Fractional*. 2022; 6(6):306.
https://doi.org/10.3390/fractalfract6060306

**Chicago/Turabian Style**

Dong, Chengwei, and Jiahui Wang.
2022. "Hidden and Coexisting Attractors in a Novel 4D Hyperchaotic System with No Equilibrium Point" *Fractal and Fractional* 6, no. 6: 306.
https://doi.org/10.3390/fractalfract6060306