# A Multistable Chaotic Jerk System with Coexisting and Hidden Attractors: Dynamical and Complexity Analysis, FPGA-Based Realization, and Chaos Stabilization Using a Robust Controller

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{7}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. System Description

#### 2.1. Dynamical Analysis

#### 2.2. Coexisting Attractors

#### 2.3. Hidden Chaotic Attractor

**Definition**

**1.**

## 3. Complexity Analysis

_{0}(MC

_{0}) complexity measure. From a mathematical point of view, with the given time series $\left\{x\left(n\right),n=0,1,2,\dots ,N-1\right\},$ its current part must be removed by

_{0}complexity measure is then estimated by obtaining the ratio between the summation of the irregular part and that of the original time series, i.e.,

_{0}complexity analysis results of the proposed system as functions of the parameters $g$ and $b$. To obtain these results, the initial condition of the system is [x

_{0}, y

_{0}, z

_{0}, w

_{0}] = [−0.1, −0.2, −0.3, −0.4] and time series of 20,000 points from the variable $x$($n$) is used, where the first 10,000 points are removed. In Figure 9, the parameter $g$ varies from −4 to 0 with a step size of 0.016. In Figure 10, the parameter $b$ increases from 0.2 and 1 with an increment of 0.0032. It should be noted in Figure 9 and Figure 10 that the complexity increases with the parameters $g$ and $b$ and maintains steadiness at the end. In Figure 10, the parameter $g$ varies from −4 to 0 with a step size of 0.04, while the parameter $b$ varies from 0.2 to 1 with a step size of 0.008. The MC

_{0}and the maximum LEs-based contour plots show that the system has higher complexity in the right size of the parameter plane, thus suggesting that the system is chaotic or has higher complexity for the larger values of $g$. Meanwhile, when $g>-3$, the system has a wider complexity region with a larger parameter $b$. Thus, in real applications, the system can have a higher complexity with relatively larger parameters $g$ and $b$.

## 4. FPGA Implementation

## 5. Controller Design

#### Disturbance Observer-Based SMC

## 6. Numerical Simulations

## 7. Conclusions

_{0}complexity. The obtained results show that the system presents chaotic behavior after a period-doubling bifurcation, as well as that its complexity increases with the parameters $g$ and $b$. The existence of coexisting attractors and hidden attractors in the proposed system was also verified. Moreover, to support the possible application of the system in real-world engineering processes, a FPGA-based implementation was described and confirmed. Finally, a robust control technique was designed, and its ability to suppress the chaotic behavior of the system in a short period of time, even in the presence of unknown time-varying disturbances, was proven through numerical simulations. As a future suggestion, engineering applications such as voice encryption of the proposed system could be studied.

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Vaidyanathan, S.; Volos, C. Advances and Applications in Chaotic Systems; Springer: Berlin/Heidelberg, Germany, 2016. [Google Scholar]
- Yousefpour, A.; Jahanshahi, H.; Munoz-Pacheco, J.M.; Bekiros, S.; Wei, Z. A fractional-order hyper-chaotic economic system with transient chaos. Chaos Solitons Fractals
**2020**, 130, 109400. [Google Scholar] [CrossRef] - Pham, V.T.; Volos, C.; Kapitaniak, T.; Jafari, S.; Wang, X. Dynamics and circuit of a chaotic system with a curve of equilibrium points. Int. J. Electron.
**2017**, 105, 385–397. [Google Scholar] [CrossRef] - Rajagopal, K.; Jahanshahi, H.; Varan, M.; Bayır, I.; Pham, V.-T.; Jafari, S.; Karthikeyan, A. A hyperchaotic memristor oscillator with fuzzy based chaos control and LQR based chaos synchronization. AEU Int. J. Electron. Commun.
**2018**, 94, 55–68. [Google Scholar] [CrossRef] - Munoz-Pacheco, J.; 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] [Green Version] - Jahanshahi, H.; Yousefpour, A.; Munoz-Pacheco, J.M.; Moroz, I.; Wei, Z.; Castillo, O. A new multi-stable fractional-order four-dimensional system with self-excited and hidden chaotic attractors: Dynamic analysis and adaptive synchronization using a novel fuzzy adaptive sliding mode control method. Appl. Soft Comput.
**2020**, 87, 105943. [Google Scholar] [CrossRef] - Eisencraft, M.; Attux, R.; Suyama, R. Chaotic Signals in Digital Communications; CRC Press: Boca Raton, FL, USA, 2018. [Google Scholar]
- Jahanshahi, H.; Rajagopal, K.; Akgul, A.; Sari, N.N.; Namazi, H.; Jafari, S. Complete analysis and engineering applications of a megastable nonlinear oscillator. Int. J. Non-Linear Mech.
**2018**, 107, 126–136. [Google Scholar] [CrossRef] - Jahanshahi, H. Smooth control of HIV/AIDS infection using a robust adaptive scheme with decoupled sliding mode supervision. Eur. Phys. J. Spéc. Top.
**2018**, 227, 707–718. [Google Scholar] [CrossRef] - Yousefpour, A.; Bahrami, A.; Haeri Yazdi, M.R. Multi-frequency piezomagnetoelastic energy harvesting in the monostable mode. J. Theor. Appl. Vib. Acoust.
**2018**, 4, 1–18. [Google Scholar] - Yousefpour, A.; Vahidi-Moghaddam, A.; Rajaei, A.; Ayati, M. Stabilization of nonlinear vibrations of carbon nanotubes using observer-based terminal sliding mode control. Trans. Inst. Meas. Control
**2019**, 42, 1047–1058. [Google Scholar] [CrossRef] - Lai, Q.; Wang, L. Chaos, bifurcation, coexisting attractors and circuit design of a three-dimensional continuous autonomous system. Optik
**2016**, 127, 5400–5406. [Google Scholar] [CrossRef] - Wang, X.; Pham, V.-T.; Volos, C. Dynamics, circuit design, and synchronization of a new chaotic system with closed curve equilibrium. Complexity
**2017**, 2017, 1–9. [Google Scholar] [CrossRef] [Green Version] - Banerjee, T.; Biswas, D. Theory and experiment of a first-order chaotic delay dynamical system. Int. J. Bifurc. Chaos
**2013**, 23, 1330020. [Google Scholar] [CrossRef] - Bouali, S.; Buscarino, A.; Fortuna, L.; Frasca, M.; Gambuzza, L.V. Emulating complex business cycles by using an electronic analogue. Nonlinear Anal. Real World Appl.
**2012**, 13, 2459–2465. [Google Scholar] [CrossRef] - Jahanshahi, H.; Shahriari-Kahkeshi, M.; Alcaraz, R.; Wang, X.; Singh, V.P.; Pham, V.-T. Entropy Analysis and Neural Network-based Adaptive Control of a Non-Equilibrium Four-Dimensional Chaotic System with Hidden Attractors. Entropy
**2019**, 21, 156. [Google Scholar] [CrossRef] [Green Version] - Yousefpour, A.; Jahanshahi, H. Fast disturbance-observer-based robust integral terminal sliding mode control of a hyperchaotic memristor oscillator. Eur. Phys. J. Spéc. Top.
**2019**, 228, 2247–2268. [Google Scholar] [CrossRef] - Devaney, R. An Introduction to Chaotic Dynamical Systems; CRC Press: Boca Raton, FL, USA, 2018. [Google Scholar]
- Jahanshahi, H.; Yousefpour, A.; Wei, Z.; Alcaraz, R.; Bekiros, S. A financial hyperchaotic system with coexisting attractors: Dynamic investigation, entropy analysis, control and synchronization. Chaos Solitons Fractals
**2019**, 126, 66–77. [Google Scholar] [CrossRef] - Chen, W.; Zhuang, J.; Yu, W.; Wang, Z. Measuring complexity using FuzzyEn, ApEn, and SampEn. Med. Eng. Phys.
**2009**, 31, 61–68. [Google Scholar] [CrossRef] - Larrondo, H.; González, C.; Martin, M.; Plastino, A.; Rosso, O.A. Intensive statistical complexity measure of pseudorandom number generators. Phys. A Stat. Mech. Appl.
**2005**, 356, 133–138. [Google Scholar] [CrossRef] - Staniczenko, P.P.A.; Lee, C.F.; Jones, N.S. Rapidly detecting disorder in rhythmic biological signals: A spectral entropy measure to identify cardiac arrhythmias. Phys. Rev. E
**2009**, 79, 011915. [Google Scholar] [CrossRef] - Wei, Q.; Liu, Q.; Fan, S.-Z.; Lu, C.-W.; Lin, T.Y.; Abbod, M.; Shieh, J.-S. Analysis of EEG via Multivariate Empirical Mode Decomposition for Depth of Anesthesia Based on Sample Entropy. Entropy
**2013**, 15, 3458–3470. [Google Scholar] [CrossRef] [Green Version] - En-Hua, S.; Zhi-Jie, C.; Fan-Ji, G. Mathematical foundation of a new complexity measure. Appl. Math. Mech.
**2005**, 26, 1188–1196. [Google Scholar] [CrossRef] - Wang, S.; He, S.; Yousefpour, A.; Jahanshahi, H.; Repnik, R.; Perc, M. Chaos and complexity in a fractional-order financial system with time delays. Chaos Solitons Fractals
**2020**, 131, 109521. [Google Scholar] [CrossRef] - He, S.; Sun, K.; Zhu, C.-X. Complexity analyses of multi-wing chaotic systems. Chin. Phys. B
**2013**, 22, 050506. [Google Scholar] [CrossRef] - Sun, K.-H.; He, S.-B.; He, Y.; Yin, L.-Z. Complexity Analysis of Chaotic Pseudo-Random Sequences Based on Spectral Entropy Algorithm. Acta Phys. Sin.
**2013**, 62, 010501. [Google Scholar] - Sharma, P.R.; Shrimali, M.D.; Prasad, A.; Kuznetsov, N.V.; Leonov, G. Control of multistability in hidden attractors. Eur. Phys. J. Spéc. Top.
**2015**, 224, 1485–1491. [Google Scholar] [CrossRef] - Lai, Q.; Chen, S. Generating Multiple Chaotic Attractors from Sprott B System. Int. J. Bifurc. Chaos
**2016**, 26, 1650177. [Google Scholar] [CrossRef] - Sprott, J.C.; Jafari, S.; Khalaf, A.J.M.; Kapitaniak, T. Megastability: Coexistence of a countable infinity of nested attractors in a periodically-forced oscillator with spatially-periodic damping. Eur. Phys. J. Spéc. Top.
**2017**, 226, 1979–1985. [Google Scholar] [CrossRef] [Green Version] - Bao, B.; Chen, M.; Bao, H.; Xu, Q. Extreme multistability in a memristive circuit. Electron. Lett.
**2016**, 52, 1008–1010. [Google Scholar] [CrossRef] - Bao, B.; Jiang, T.; Xu, Q.; Chen, M.; Wu, H.; Hu, Y. Coexisting infinitely many attractors in active band-pass filter-based memristive circuit. Nonlinear Dyn.
**2016**, 86, 1711–1723. [Google Scholar] [CrossRef] - Tlelo-Cuautle, E.; Rangel-Magdaleno, J.; Pano-Azucena, A.; Obeso-Rodelo, P.; Nuñez-Perez, J.-C. FPGA realization of multi-scroll chaotic oscillators. Commun. Nonlinear Sci. Numer. Simul.
**2015**, 27, 66–80. [Google Scholar] [CrossRef] - Muñoz-Pacheco, J.M.; Tlelo-Cuautle, E.; Toxqui-Toxqui, I.; Sánchez-López, C.; Trejo-Guerra, R. Frequency limitations in generating multi-scroll chaotic attractors using CFOAs. Int. J. Electron.
**2014**, 101, 1559–1569. [Google Scholar] [CrossRef] - Pham, V.-T.; Volos, C.; Jafari, S.; Kapitaniak, T. Coexistence of hidden chaotic attractors in a novel no-equilibrium system. Nonlinear Dyn.
**2016**, 87, 2001–2010. [Google Scholar] [CrossRef] - Pham, V.-T.; Jafari, S.; Volos, C.; Gotthans, T.; Wang, X.; Vo, D.H. A chaotic system with rounded square equilibrium and with no-equilibrium. Optik
**2017**, 130, 365–371. [Google Scholar] [CrossRef] - Jafari, S.; Sprott, J.; Golpayegani, S.M.R.H. Elementary quadratic chaotic flows with no equilibria. Phys. Lett. A
**2013**, 377, 699–702. [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.; Akgul, A.; Volos, C.; Jafari, S.; Kapitaniak, T. Dynamics and circuit realization of a no-equilibrium chaotic system with a boostable variable. AEU-Int. J. Electron. Commun.
**2017**, 78, 134–140. [Google Scholar] [CrossRef] - Ren, S.; Panahi, S.; Rajagopal, K.; Akgul, A.; Pham, V.-T.; Jafari, S. A New Chaotic Flow with Hidden Attractor: The First Hyperjerk System with No Equilibrium. Z. Nat. A
**2018**, 73, 239–249. [Google Scholar] [CrossRef] - Lai, Q.; Chen, C.-Y.; Zhao, X.-W.; Kengne, J.; Volos, C. Constructing Chaotic System With Multiple Coexisting Attractors. IEEE Access
**2019**, 7, 24051–24056. [Google Scholar] [CrossRef] - Li, C.; Sprott, J.; Hu, W.; Xu, Y. Infinite Multistability in a Self-Reproducing Chaotic System. Int. J. Bifurc. Chaos
**2017**, 27, 1750160. [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] - Leonov, G.; Kuznetsov, N.V.; Vagaitsev, V. Hidden attractor in smooth Chua systems. Phys. D Nonlinear Phenom.
**2012**, 241, 1482–1486. [Google Scholar] [CrossRef] - Kuznetsov, N.V.; Leonov, G.A.; Vagaitsev, V.I. Analytical-numerical method for attractor localization of generalized Chua’s system. IFAC Proc. Vol.
**2010**, 43, 29–33. [Google Scholar] [CrossRef] - Leonov, G.; Kuznetsov, N.V.; Vagaitsev, V. Localization of hidden Chua’s attractors. Phys. Lett. A
**2011**, 375, 2230–2233. [Google Scholar] [CrossRef] - He, S.; Sun, K.; Wu, X. Fractional symbolic network entropy analysis for the fractional-order chaotic systems. Phys. Scr.
**2020**, 95, 035220. [Google Scholar] [CrossRef] - Zhang, S.; Zeng, Y. A simple Jerk-like system without equilibrium: Asymmetric coexisting hidden attractors, bursting oscillation and double full Feigenbaum remerging trees. Chaos Solitons Fractals
**2019**, 120, 25–40. [Google Scholar] [CrossRef] - Kengne, J.; Signing, V.R.F.; Chedjou, J.C.; Leutcho, G.D. Nonlinear behavior of a novel chaotic jerk system: Antimonotonicity, crises, and multiple coexisting attractors. Int. J. Dyn. Control
**2017**, 6, 468–485. [Google Scholar] [CrossRef] - Elwakil, A.S.; Kennedy, M.P. Improved implementation of Chua’s chaotic oscillator using current feedback op amp. IEEE Trans. Circuits Syst. I
**2000**, 47, 76–79. [Google Scholar] [CrossRef] [Green Version] - Chiu, R.; Mora-Gonzalez, M.; Lopez-Mancilla, D. Implementation of a chaotic oscillator into a simple microcontroller. IERI Procedia
**2013**, 4, 247–252. [Google Scholar] [CrossRef] [Green Version] - De La Hoz, M.Z.; Acho, L.; Vidal, Y. An Experimental Realization of a Chaos-Based Secure Communication Using Arduino Microcontrollers. Sci. World J.
**2015**, 2015, 1–10. [Google Scholar] [CrossRef] [Green Version] - Yau, H.-T.; Wu, C.-H.; Liang, Q.-C.; Li, S.C. Implementation of optimal PID control for chaos synchronization by FPGA chip. In Proceedings of the 2011 International Conference on Fluid Power and Mechatronics, Beijing, China, 17–20 August 2011; pp. 56–61. [Google Scholar]
- Shah, D.K.; Chaurasiya, R.B.; Vyawahare, V.A.; Pichhode, K.; Patil, M. FPGA implementation of fractional-order chaotic systems. AEU-Int. J. Electron. Commun.
**2017**, 78, 245–257. [Google Scholar] [CrossRef] - Tlelo-Cuautle, E.; De La Fraga, L.G.; Pham, V.-T.; Volos, C.; Jafari, S.; Quintas-Valles, A.D.J. Dynamics, FPGA realization and application of a chaotic system with an infinite number of equilibrium points. Nonlinear Dyn.
**2017**, 89, 1129–1139. [Google Scholar] [CrossRef] - Ya-Ming, X.; Li-Dan, W.; Shu-Kai, D. A memristor-based chaotic system and its field programmable gate array implementation. Acta Phys Sin.
**2016**, 65, 120503. [Google Scholar] - Wang, Q.; Yu, S.; Li, C.; Lu, J.; Fang, X.; Guyeux, C.; Bahi, J. Theoretical Design and FPGA-Based Implementation of Higher-Dimensional Digital Chaotic Systems. IEEE Trans. Circuits Syst. I Regul. Pap.
**2016**, 63, 401–412. [Google Scholar] [CrossRef] [Green Version] - Tuncer, T. The implementation of chaos-based PUF designs in field programmable gate array. Nonlinear Dyn.
**2016**, 86, 975–986. [Google Scholar] [CrossRef] - Dong, E.; Liang, Z.; Du, S.; Chen, Z. Topological horseshoe analysis on a four-wing chaotic attractor and its FPGA implement. Nonlinear Dyn.
**2015**, 83, 623–630. [Google Scholar] [CrossRef] - Tlelo-Cuautle, E.; Pano-Azucena, A.D.; Rangel-Magdaleno, J.; Carbajal-Gómez, V.H.; Rodriguez-Gomez, G. Generating a 50-scroll chaotic attractor at 66 MHz by using FPGAs. Nonlinear Dyn.
**2016**, 85, 2143–2157. [Google Scholar] [CrossRef] - Wang, Q.-X.; Yu, S.; Guyeux, C.; Bahi, J.; Fang, X.-L. Study on a new chaotic bitwise dynamical system and its FPGA implementation. Chin. Phys. B
**2015**, 24, 060503. [Google Scholar] [CrossRef] - Hua, Z.; Zhou, B.; Zhou, Y. Sine-Transform-Based Chaotic System With FPGA Implementation. IEEE Trans. Ind. Electron.
**2018**, 65, 2557–2566. [Google Scholar] [CrossRef] - Muñoz-Pacheco, J.M.; Tlelo-Cuautle, E.; Flores-Tiro, I.; Trejo-Guerra, R. Experimental Synchronization of two Integrated Multi-scroll Chaotic Oscillators. J. Appl. Res. Technol.
**2014**, 12, 459–470. [Google Scholar] [CrossRef] - Zambrano-Serrano, E.; Muñoz-Pacheco, J.M.; Gómez-Pavón, L.D.C.; Luis-Ramos, A.; Chen, G. Synchronization in a fractional-order model of pancreatic β-cells. Eur. Phys. J. Spéc. Top.
**2018**, 227, 907–919. [Google Scholar] [CrossRef] - Kosari, A.; Jahanshahi, H.; Razavi, S. An optimal fuzzy PID control approach for docking maneuver of two spacecraft: Orientational motion. Eng. Sci. Technol. Int. J.
**2017**, 20, 293–309. [Google Scholar] [CrossRef] [Green Version] - Kosari, A.; Jahanshahi, H.; Razavi, S.A. Optimal FPID Control Approach for a Docking Maneuver of Two Spacecraft: Translational Motion. J. Aerosp. Eng.
**2017**, 30, 04017011. [Google Scholar] [CrossRef] - Ma, Q.; Luo, W.; Jahanshahi, H.; Cavusoglu, U.; Akgul, A.; Lin, X. A novel s-box design algorithm and fuzzy-pid controller design for a 5-d burke–shaw system with hidden hyperchaos. System
**2019**, 2, 3. [Google Scholar] [CrossRef] - Jahanshahi, H.; Jafarzadeh, M.; Sari, N.N.; Pham, V.-T.; Huynh, V.; Nguyen, X.Q. Robot Motion Planning in an Unknown Environment with Danger Space. Electronics
**2019**, 8, 201. [Google Scholar] [CrossRef] [Green Version] - Sari, N.N.; Jahanshahi, H.; Fakoor, M. Adaptive Fuzzy PID Control Strategy for Spacecraft Attitude Control. Int. J. Fuzzy Syst.
**2019**, 21, 769–781. [Google Scholar] [CrossRef] - Mahmoodabadi, M.; Jahanshahi, H. Multi-objective optimized fuzzy-PID controllers for fourth order nonlinear systems. Eng. Sci. Technol. Int. J.
**2016**, 19, 1084–1098. [Google Scholar] [CrossRef] [Green Version] - Soradi-Zeid, S.; Jahanshahi, H.; Yousefpour, A.; Bekiros, S. King algorithm: A novel optimization approach based on variable-order fractional calculus with application in chaotic financial systems. Chaos Solitons Fractals
**2020**, 132, 109569. [Google Scholar] [CrossRef] - Rajagopal, K.; Jahanshahi, H.; Jafari, S.; Weldegiorgis, R.; Karthikeyan, A.; Duraisamy, P. Coexisting attractors in a fractional order hydro turbine governing system and fuzzy PID based chaos control. Asian J. Control
**2020**. [Google Scholar] [CrossRef] - Rajaei, A.; Moghaddam, A.V.; Chizfahm, A.; Sharifi, M. Control of malaria outbreak using a non-linear robust strategy with adaptive gains. IET Control Theory Appl.
**2019**, 13, 2308–2317. [Google Scholar] [CrossRef] - Sprott, J.C. Some simple chaotic jerk functions. Am. J. Phys.
**1997**, 65, 537. [Google Scholar] [CrossRef] - Harris, D.; Harris, S. Digital Design and Computer Architecture; Morgan Kaufmann: Burlington, MA, USA, 2010. [Google Scholar]
- Chen, M.; Chen, W.-H. Sliding mode control for a class of uncertain nonlinear system based on disturbance observer. Int. J. Adapt. Control Signal Process.
**2009**, 24, 51–64. [Google Scholar] [CrossRef]

**Figure 1.**Dynamics of the proposed system with the variation of the parameter $g$: (

**a**) Bifurcation diagram; (

**b**) Lyapunov exponents (LEs).

**Figure 2.**Phase diagrams of the proposed system with b = 0.7 and (

**a**) g = −3.5, (

**b**) g = −2.5, (

**c**) g = −1.5, (

**d**) g = −0.15.

**Figure 3.**Dynamics of the proposed system with the variation of the parameter b: (

**a**) Bifurcation diagram; (

**b**) Lyapunov Exponents (LEs).

**Figure 4.**Phase diagrams of the proposed system with b = 0.7 and (

**a**) g = −3.5, (

**b**) g = −2.5, (

**c**) g = −1.5, and (

**d**) g = −0.15.

**Figure 6.**Basin attraction plots of the proposed system in the initial plane y

_{0}-w

_{0}with (

**a**) $g$ = −1.5, and (

**b**) $g$= −2.

**Figure 9.**MC

_{0}complexity results as functions of (

**a**) the parameter $g$, and (

**b**) the parameter $b$.

**Figure 10.**Contour plots of the proposed system based on (

**a**) the maximum Lyapunov exponents, and (

**b**) MC

_{0}algorithm in the parameter plane $b-g$.

Resources | Used |
---|---|

Total logic elements | 1652/149,760 (1%) |

Total registers | 946 |

Total pins | 25/508 (5%) |

Total virtual pins | 0 |

Total memory bits | 192/6, 635, 520 (<1%) |

Embedded multiplier 9-bit elements | 92/720 (13%) |

Latency | 105.9 ns |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Chen, H.; He, S.; Pano Azucena, A.D.; Yousefpour, A.; Jahanshahi, H.; López, M.A.; Alcaraz, R.
A Multistable Chaotic Jerk System with Coexisting and Hidden Attractors: Dynamical and Complexity Analysis, FPGA-Based Realization, and Chaos Stabilization Using a Robust Controller. *Symmetry* **2020**, *12*, 569.
https://doi.org/10.3390/sym12040569

**AMA Style**

Chen H, He S, Pano Azucena AD, Yousefpour A, Jahanshahi H, López MA, Alcaraz R.
A Multistable Chaotic Jerk System with Coexisting and Hidden Attractors: Dynamical and Complexity Analysis, FPGA-Based Realization, and Chaos Stabilization Using a Robust Controller. *Symmetry*. 2020; 12(4):569.
https://doi.org/10.3390/sym12040569

**Chicago/Turabian Style**

Chen, Heng, Shaobo He, Ana Dalia Pano Azucena, Amin Yousefpour, Hadi Jahanshahi, Miguel A. López, and Raúl Alcaraz.
2020. "A Multistable Chaotic Jerk System with Coexisting and Hidden Attractors: Dynamical and Complexity Analysis, FPGA-Based Realization, and Chaos Stabilization Using a Robust Controller" *Symmetry* 12, no. 4: 569.
https://doi.org/10.3390/sym12040569