# A New Variable-Boostable 3D Chaotic System with Hidden and Coexisting Attractors: Dynamical Analysis, Periodic Orbit Coding, Circuit Simulation, and Synchronization

^{*}

## Abstract

**:**

## 1. Introduction

## 2. New Hidden Chaotic System with Two Stable Equilibria

#### 2.1. Basic Properties of New Chaotic System

#### 2.2. Observation of Chaotic and Complex Dynamics

#### 2.2.1. Fix $b=100,\phantom{\rule{3.33333pt}{0ex}}c=10$, and $k=4.6$ and Vary a

#### 2.2.2. Fix $a=12$, $c=10$, and $k=4.6$ and Vary b

#### 2.2.3. Fix $a=12$, $b=100$, and $k=4.6$ and Vary c

#### 2.2.4. Fix $a=12$, $b=100$, and $c=10$ and Vary k

#### 2.2.5. Division of Different Parameters

## 3. Variational Method

## 4. Symbolic Encoding of Unstable Periodic Orbits in the Hidden Chaotic Attractor with Six Letters

## 5. Circuitry of Proposed System

## 6. Offset Boosting Control and Adaptive Synchronization of New System

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Altan, A.; Karasu, S.; Bekiros, S. Digital currency forecasting with chaotic meta-heuristic bio-inspired signal processing techniques. Chaos Solitons Fractals
**2019**, 126, 325–336. [Google Scholar] [CrossRef] - 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]
- Lorenz, E.N. Deterministic nonperiodic flow. J. Atmos. Sci.
**1963**, 20, 130–141. [Google Scholar] [CrossRef] - Šhil’nikov, L.P. A case of the existence of a denumerable set of periodic motions. Sov. Math. Dokl.
**1965**, 6, 163–166. [Google Scholar] - Chen, G.; Ueta, T. Yet another chaotic attractor. Int. J. Bifurc. Chaos
**1999**, 9, 1465–1466. [Google Scholar] [CrossRef] - Qi, G.; Chen, G.; Du, S.; Chen, Z.; Yuan, Z. Analysis of a new chaotic system. Phys. A
**2005**, 352, 295–308. [Google Scholar] [CrossRef] - Lü, J.; Chen, G. A new chaotic attractor coined. Int. J. Bifurc. Chaos
**2002**, 12, 659–661. [Google Scholar] [CrossRef] [Green Version] - Rössler, O.E. An equation for continuous chaos. Phys. Lett. A
**1976**, 57, 397–398. [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. Bifurc. Chaos
**2013**, 23, 1330002. [Google Scholar] [CrossRef] [Green Version] - Wang, S.; Wang, C.; Xu, C. An image encryption algorithm based on a hidden attractor chaos system and the Knuth-Durstenfeld algorithm. Opt. Lasers Eng.
**2020**, 128, 105995. [Google Scholar] [CrossRef] - Khalaf, A.J.M.; Abdolmohammadi, H.R.; Ahmadi, A.; Moysis, L.; Volos, C.; Hussain, I. Extreme multi-stability analysis of a novel 5D chaotic system with hidden attractors, line equilibrium, permutation entropy and its secure communication scheme. Eur. Phys. J. Spec. Top.
**2020**, 229, 1175–1188. [Google Scholar] [CrossRef] - Lai, Q.; Wang, Z.; Kuate, P.D.K. Dynamical analysis, FPGA implementation and synchronization for secure communication of new chaotic system with hidden and coexisting attractors. Mod. Phys. Lett. B
**2022**, 36, 2150538. [Google Scholar] [CrossRef] - Leonov, G.A.; Kuznetsov, N.V.; Kuznetsova, O.A.; Seledzhi, S.M.; Vagaitsev, V.I. Hidden oscillations in dynamical systems. Trans. Syst. Contr.
**2011**, 6, 54–67. [Google Scholar] - Jafari, S.; Pham, V.T.; Kapitaniak, T. Multiscroll chaotic sea obtained from a simple 3D System without equilibrium. Int. J. Bifurc. Chaos
**2016**, 26, 1650031. [Google Scholar] [CrossRef] - Pham, V.T.; Jafari, S.; Volos, C.; Wang, X.; Golpayegani, S. Is that really hidden? The presence of complex fixed-points in chaotic flows with no equilibria. Int. J. Bifurc. Chaos
**2014**, 24, 1450146. [Google Scholar] [CrossRef] - Jafari, S.; Sprott, J.C.; Golpayegani, S. Elementary quadratic chaotic flows with no equilibria. Phys. Lett. A
**2013**, 377, 699–702. [Google Scholar] [CrossRef] - Dong, C.; Wang, J. Hidden and coexisting attractors in a novel 4D hyperchaotic system with no equilibrium point. Fractal Fract.
**2022**, 6, 306. [Google Scholar] [CrossRef] - Wang, X.; Chen, G. A chaotic system with only one stable equilibrium.Commun. Nonlinear Sci. Numer. Simul.
**2012**, 17, 1264–1272. [Google Scholar] [CrossRef] [Green Version] - Bao, B.; Li, Q.; Wang, N.; Xu, Q. Multistability in Chua’s circuit with two stable node-foci. Chaos
**2016**, 26, 043111. [Google Scholar] [CrossRef] - Molaie, M.; Jafari, S.; Sprott, J.C.; Golpayegani, S. Simple chaotic flows with one stable equilibrium. Int. J. Bifurc. Chaos
**2013**, 23, 1350188. [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] - Gotthans, T.; Petržela, J. New class of chaotic systems with circular equilibrium. Nonlinear Dyn.
**2015**, 81, 1143–1149. [Google Scholar] [CrossRef] [Green Version] - Feng, Y.; Rajagopal, K.; Khalaf, A.J.M.; Alsaadi, F.E.; Alsaadi, F.E.; Pham, V.T. A new hidden attractor hyperchaotic memristor oscillator with a line of equilibria. Eur. Phys. J. Spec. Top.
**2020**, 229, 1279–1288. [Google Scholar] [CrossRef] - Zhang, X.; Wang, C. Multiscroll hyperchaotic system with hidden attractors and its circuit implementation. Int. J. Bifurc. Chaos
**2019**, 29, 1950117. [Google Scholar] [CrossRef] - Bao, J.; Chen, D. Coexisting hidden attractors in a 4D segmented disc dynamo with one stable equilibrium or a line equilibrium. Chin. Phys. B
**2017**, 26, 080201. [Google Scholar] [CrossRef] - Jafari, S.; Sprott, J.C. Simple chaotic flows with a line equilibrium. Chaos Solitons Fractals
**2013**, 57, 79–84. [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] - Cang, S.; Li, Y.; Zhang, R.; Wang, Z. Hidden and self-excited coexisting attractors in a Lorenz-like system with two equilibrium points. Nonlinear Dyn.
**2019**, 95, 381–390. [Google Scholar] [CrossRef] - Wei, Z. Dynamical behaviors of a chaotic system with no equilibria. Phys. Lett. A
**2011**, 376, 102–108. [Google Scholar] [CrossRef] - Pham, V.T.; Volos, C.; Jafari, S.; Wei, Z.; Wang, X. Constructing a novel no-equilibrium chaotic system. Int. J. Bifurc. Chaos
**2014**, 24, 1450073. [Google Scholar] [CrossRef] - Huang, L.; Wang, Y.; Jiang, Y.; Lei, T. A novel memristor chaotic system with a hidden attractor and multistability and its implementation in a circuit. Math. Probl. Eng.
**2021**, 2021, 7457220. [Google Scholar] [CrossRef] - Jafari, S.; Ahmadi, A.; Khalaf, A.J.M.; Abdolmohammadi, H.R.; Pham, V.T.; Alsaadi, F.E. A new hidden chaotic attractor with extreme multi-stability. AEU-Int. J. Electron. Commun.
**2018**, 89, 131–135. [Google Scholar] [CrossRef] - Goufo, E.F.D. On chaotic models with hidden attractors in fractional calculus above power law. Chaos Solitons Fractals
**2019**, 127, 24–30. [Google Scholar] [CrossRef] - Cui, L.; Lu, M.; Ou, Q.; Duan, H.; Luo, W. Analysis and circuit implementation of fractional order multi-wing hidden attractors. Chaos Solitons Fractals
**2020**, 138, 109894. [Google Scholar] [CrossRef] - Clemente-López, D.; Tlelo-Cuautle, E.; de la Fraga, L.G.; de Jesús Rangel-Magdaleno, J.; Munoz-Pacheco, J.M. Poincaré maps for detecting chaos in fractional-order systems with hidden attractors for its Kaplan-Yorke dimension optimization. AIMS Math.
**2022**, 7, 5871–5894. [Google Scholar] [CrossRef] - Almatroud, A.O.; Matouk, A.E.; Mohammed, W.W.; Iqbal, N.; Alshammari, S. Self-excited and hidden chaotic attractors in Matouk’s hyperchaotic systems. Discret. Dyn. Nat. Soc.
**2022**, 2022, 1–14. [Google Scholar] [CrossRef] - Yuan, F.; Wang, G.; Wang, X. Extreme multistability in a memristor-based multi-scroll hyper-chaotic system. Chaos
**2016**, 26, 073107. [Google Scholar] [CrossRef] - Bao, B.; Bao, H.; Wang, N.; Chen, M.; Xu, Q. Hidden extreme multistability in memristive hyperchaotic system. Chaos Solitons Fractals
**2017**, 94, 102–111. [Google Scholar] [CrossRef] - Wang, L.; Zhang, S.; Zeng, Y.; Li, Z. Generating hidden extreme multistability in memristive chaotic oscillator via micro-perturbation. Electron. Lett.
**2018**, 54, 808–810. [Google Scholar] [CrossRef] - Mezatio, B.A.; Motchongom, M.T.; Tekam, B.R.W.; Kengne, R.; Tchitnga, R.; Fomethe, A. A novel memristive 6D hyperchaotic autonomous system with hidden extreme multistability. Chaos Solitons Fractals
**2019**, 120, 100–115. [Google Scholar] [CrossRef] - Wang, Z.; Sun, W.; Wei, Z.; Zhang, S. Dynamics and delayed feedback control for a 3D jerk system with hidden attractor. Nonlinear Dyn.
**2015**, 82, 577–588. [Google Scholar] [CrossRef] - Li, P.; Zheng, T.; Li, C.; Wang, X.; Hu, W. A unique jerk system with hidden chaotic oscillation. Nonlinear Dyn.
**2016**, 86, 197–203. [Google Scholar] [CrossRef] - Bao, B.; Jiang, T.; Wang, G.; Jin, P.; Bao, H.; Chen, M. Two-memristor-based Chua’s hyperchaotic circuit with plane equilibrium and its extreme multistability. Nonlinear Dyn.
**2017**, 89, 1157–1171. [Google Scholar] [CrossRef] - Singh, J.P.; Roy, B.K.; Jafari, S. New family of 4-D hyperchaotic and chaotic systems with quadric surfaces of equilibria. Chaos Solitons Fractals
**2018**, 106, 243–257. [Google Scholar] [CrossRef] - Lin, Y.; Wang, C.; Xu, H. Grid multi-scroll chaotic attractors in hybrid image encryption algorithm based on current conveyor. Acta Phys. Sin.
**2012**, 61, 240503. [Google Scholar] [CrossRef] - Sprott, J.C.; Li, C. Asymmetric bistability in the Rössler system. Acta Phys. Pol. B
**2017**, 48, 97. [Google Scholar] [CrossRef] - Li, C.; Sprott, J.C.; Zhang, X.; Chai, L.; Liu, Z. Constructing conditional symmetry in symmetric chaotic systems. Chaos Solitons Fractals
**2022**, 155, 111723. [Google Scholar] [CrossRef] - Li, C.; Sprott, J.C.; Hu, W.; Xu, Y. Infinite multistability in a self-reproducing chaotic system. Int. J. Bifurc. Chaos
**2017**, 27, 1750160. [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. [Google Scholar]
- Wei, Z.; Yang, Q. Dynamical analysis of the generalized Sprott C system with only two stable equilibria. Nonlinear Dyn.
**2012**, 68, 543–554. [Google Scholar] [CrossRef] - Munoz-Pacheco, J.M.; Volos, C.; Serrano, F.E.; Jafari, S.; Kengne, J.; Rajagopal, K. Stabilization and synchronization of a complex hidden attractor chaotic system by backstepping technique. Entropy
**2021**, 23, 921. [Google Scholar] [CrossRef] - Strogatz, S.H. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering; Perseus Books: Reading, MA, USA, 1994. [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] - Press, W.H.; Teukolsky, S.A.; Veterling, W.T.; Flannery, B.P. Numerical Recipes in C; Cambridge University Press: Cambridge, UK, 1992. [Google Scholar]
- Dong, C.; Liu, H.; Jie, Q.; Li, H. Topological classification of periodic orbits in the generalized Lorenz-type system with diverse symbolic dynamics. Chaos Solitons Fractals
**2022**, 154, 111686. [Google Scholar] [CrossRef] - Lan, Y.; Cvitanović, P. Unstable recurrent patterns in Kuramoto-Sivashinsky dynamics. Phys. Rev. E
**2008**, 78, 026208. [Google Scholar] [CrossRef] - Dong, C.; Liu, H.; Li, H. Unstable periodic orbits analysis in the generalized Lorenz-type system. J. Stat. Mech. Theory Exp.
**2020**, 2020, 073211. [Google Scholar] [CrossRef] - Artuso, R.; Aurell, E.; Cvitanović, P. Recycling of strange sets: I. Cycle expansions. Nonlinearity
**1990**, 3, 325–359. [Google Scholar] [CrossRef] - Lan, Y. Cycle expansions: From maps to turbulence. Commun. Nonlinear Sci. Numer. Simul.
**2010**, 15, 502–526. [Google Scholar] [CrossRef] - Hao, B.L.; Zheng, W.M. Applied Symbolic Dynamics and Chaos; World Scientic: Singapore, 1998; pp. 11–13. [Google Scholar]
- Munoz-Pacheco, J.M.; Tlelo-Cuautle, E.; Toxqui-Toxqui, I.; Sanchez-Lopez, C.; Trejo-Guerra, R. Frequency limitations in generating multi-scroll chaotic attractors using CFOAs. Int. J. Electron.
**2014**, 101, 1559–1569. [Google Scholar] [CrossRef] - Sayed, W.S.; Roshdy, M.; Said, L.A.; Herencsar, N.; Radwan, A.G. CORDIC-based FPGA realization of a spatially rotating translational fractional-order multi-scroll grid chaotic system. Fractal Fract.
**2022**, 6, 432. [Google Scholar] [CrossRef] - Li, C.; Sprott, J.C. Variable-boostable chaotic flows. Optik
**2016**, 127, 10389–10398. [Google Scholar] [CrossRef] - Zhang, X.; Li, C.; Dong, E.; Zhao, Y.; Liu, Z. A conservative memristive system with amplitude control and offset boosting. Int. J. Bifurc. Chaos
**2022**, 32, 2250057. [Google Scholar] [CrossRef] - 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. [Google Scholar] [CrossRef] - Munoz-Pacheco, J.M.; Zambrano-Serrano, E.; Volos, C.; Jafari, S.; Kengne, J.; Rajagopal, K. A new fractional-order chaotic system with different families of hidden and self-excited attractors. Entropy
**2018**, 20, 564. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Munoz-Pacheco, J.M. Infinitely many hidden attractors in a new fractional-order chaotic system based on a fracmemristor. Eur. Phys. J. Spec. Top.
**2019**, 228, 2185–2196. [Google Scholar] [CrossRef]

**Figure 2.**Continuous broadband power spectrum of the new chaotic system for $(a,b,c,k)=(12,100,10,4.6)$.

**Figure 3.**Projection in different two-dimensional (2D) phase spaces of system (2) for $(a,b,c,k)=(12,100,10,4.6)$: (

**a**) x-y plane, (

**b**) x-z plane, and (

**c**) y-z plane.

**Figure 4.**(

**a**) Three colored basins of attraction at $z=-10$ on $(x,y)$ plane. (

**b**) Three-dimensional (3D) views of the chaotic attractor and two fixed point attractors. (

**c**) Coexisting time-series diagram of $x\left(t\right)$.

**Figure 5.**(

**a**) Bifurcation diagram and (

**b**) largest Lyapunov exponent spectrum versus a, where $b=100$, $c=10$, and $k=4.6$.

**Figure 6.**Two-dimensional views of coexisting chaotic attractors and periodic attractors in system (2) with parameters $b=100$, $c=10$, and $k=4.6:$ (

**a**) $a=27.35,({x}_{0},{y}_{0},{z}_{0})=(1,1,1)$, (

**b**) $a=27.35,$$({x}_{0},{y}_{0},{z}_{0})=(-1,-1,1)$, (

**c**) $a=29,({x}_{0},{y}_{0},{z}_{0})=(1,1,1)$, and (

**d**) $a=29,({x}_{0},{y}_{0},{z}_{0})=(-1,-1,1)$.

**Figure 7.**(

**a**) Bifurcation diagram of $\left|y\right|$ with b as the varied parameter and (

**b**) the largest Lyapunov exponent spectrum, where $a=12,\phantom{\rule{3.33333pt}{0ex}}c=10,$ and $k=4.6$.

**Figure 8.**Three-dimensional views of various limit cycles with parameters $a=12$, $c=10$, and $k=4.6$: (

**a**) $b=45,({x}_{0},{y}_{0},{z}_{0})=(1,1,1)$, (

**b**) $b=47,({x}_{0},{y}_{0},{z}_{0})=(1,1,1)$, (

**c**) $b=47,({x}_{0},{y}_{0},{z}_{0})=(-1,-1,1)$, and (

**d**) $b=65,({x}_{0},{y}_{0},{z}_{0})=(1,1,1)$.

**Figure 9.**(

**a**) Bifurcation diagram of $\left|y\right|$ with c as the varied parameter and (

**b**) the largest Lyapunov exponent spectrum, where $a=12$, $b=100$, and $k=4.6$.

**Figure 10.**Three-dimensional views of rich dynamical behaviors with parameters $a=12$, $b=100$, and $k=4.6$: (

**a**) $c=-10$, (

**b**) $c=-2$, (

**c**) $c=8$, and (

**d**) $c=15$. The initial values of $({x}_{0},{y}_{0},{z}_{0})=(1,1,1)$ were selected.

**Figure 11.**(

**a**) Bifurcation diagram of $\left|y\right|$ with k as the varied parameter and (

**b**) the largest Lyapunov exponent spectrum, where $a=12$, $b=100$, and $c=10$.

**Figure 12.**Diagram of largest Lyapunov exponents with different parameters: (

**a**) division for parameters c and k ($a=12$ and $b=100$) and (

**b**) division for parameters a and b ($c=10$ and $k=4.6$).

**Figure 14.**Six building block cycles for system (2), $(a,b,c,k)=(12,100,10,4.6)$: (

**a**) cycle 0, (

**b**) cycle 1, (

**c**) cycle 2, (

**d**) cycle 3, (

**e**) cycle 4, and (

**f**) cycle 5.

**Figure 15.**Unstable periodic orbits with topological length of 2 in the new system: (

**a**) cycle 01, (

**b**) 23, (

**c**) 45, (

**d**) 03, (

**e**) 24, and (

**f**) 25.

**Figure 16.**Unstable periodic orbits with a topological length of 3 in the new system, where the two stable equilibria are marked with “+”: (

**a**) cycle 001, (

**b**) 013, (

**c**) 023, (

**d**) 033, (

**e**) 003, (

**f**) 243, (

**g**) 245, (

**h**) 253, and (

**i**) 345.

**Figure 19.**(

**a**) Chaotic attractors with different offsets w in the y-z plane; (

**b**) State z with different values of the offset boosting controller w.

**Figure 20.**Time evolution diagrams of the master and slave systems showing results of the complete synchronization of the respective states: (

**a**) x variable, (

**b**) y variable, and (

**c**) z variable.

**Figure 21.**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 ${e}_{k}$.

**Table 1.**Lyapunov exponents and Kaplan–Yorke dimension of system (2): $(a,b,c)=(12,100,10)$ and $({x}_{0},{y}_{0},{z}_{0})=(1,1,1)$.

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

1 | −0.794149 | −0.795286 | −20.4107 | 0 | Equilibrium |

3 | 0.961912 | 0 | −22.9613 | 2.0419 | Chaos |

4.225 | 0 | −0.0340439 | −21.9686 | 1.0 | Period |

5 | 0.920123 | 0 | −22.9146 | 2.0399 | Chaos |

5.54 | 0 | −0.0956672 | −21.9091 | 1.0 | Period |

13 | 0.742037 | 0 | −22.7375 | 2.0324 | Chaos |

Length | Itineraries | Periods | Length | Itineraries | Periods | Length | Itineraries | Periods |
---|---|---|---|---|---|---|---|---|

1 | 0 | 0.645509 | 3 | 223 | 3.552630 | 3 | 001 | 2.324411 |

1 | 0.645509 | 233 | 3.552630 | 011 | 2.324411 | |||

2 | 1.186404 | 033 | 2.981070 | 123 | — | |||

3 | 1.186404 | 122 | 2.981070 | 032 | — | |||

2 | 01 | 1.559290 | 021 | 2.653639 | 003 | 2.401931 | ||

23 | 2.366105 | 013 | 2.653639 | 112 | 2.401931 | |||

12 | 1.792160 | 031 | — | 113 | — | |||

03 | 1.792160 | 012 | — | 002 | — | |||

02 | — | 132 | 2.962501 | 022 | — | |||

13 | — | 023 | 2.962501 | 133 | — | |||

1 | 4 | 1.515729 | 3 | 445 | 4.815813 | 3 | 354 | 4.221816 |

5 | 1.515729 | 455 | 4.815813 | 234 | 3.878345 | |||

2 | 24 | 2.695403 | 344 | 4.221765 | 325 | 3.878345 | ||

35 | 2.695403 | 255 | 4.221765 | 225 | 3.892262 | |||

25 | 2.706161 | 335 | 3.881997 | 334 | 3.892262 | |||

34 | 2.706161 | 224 | 3.881997 | 254 | 4.211233 | |||

45 | 3.031718 | 244 | 4.211320 | 345 | 4.211233 | |||

3 | 235 | 3.889210 | 355 | 4.211320 | ||||

324 | 3.889210 | 245 | 4.221816 |

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**MDPI and ACS Style**

Wang, J.; Dong, C.; Li, H.
A New Variable-Boostable 3D Chaotic System with Hidden and Coexisting Attractors: Dynamical Analysis, Periodic Orbit Coding, Circuit Simulation, and Synchronization. *Fractal Fract.* **2022**, *6*, 740.
https://doi.org/10.3390/fractalfract6120740

**AMA Style**

Wang J, Dong C, Li H.
A New Variable-Boostable 3D Chaotic System with Hidden and Coexisting Attractors: Dynamical Analysis, Periodic Orbit Coding, Circuit Simulation, and Synchronization. *Fractal and Fractional*. 2022; 6(12):740.
https://doi.org/10.3390/fractalfract6120740

**Chicago/Turabian Style**

Wang, Jiahui, Chengwei Dong, and Hantao Li.
2022. "A New Variable-Boostable 3D Chaotic System with Hidden and Coexisting Attractors: Dynamical Analysis, Periodic Orbit Coding, Circuit Simulation, and Synchronization" *Fractal and Fractional* 6, no. 12: 740.
https://doi.org/10.3390/fractalfract6120740