# FPGA-Based Implementation of a New 3-D Multistable Chaotic Jerk System with Two Unstable Balance Points

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- We propose a new dissipative chaotic jerk system having three quadratic nonlinear terms.
- (2)
- We establish that the new chaotic jerk system has two unstable equilibrium points, which implies that the new system exhibits a self-excited chaotic attractor.
- (3)
- We carry out a detailed bifurcation analysis of the new jerk system which shows the changes in the dynamic behavior of the jerk system with respect to changes in the system parameters.
- (4)
- We establish that the new jerk system has multistability with coexisting chaotic attractors.
- (5)
- We provide a control application of the new jerk system, viz. complete synchronization of the new jerk systems via backstepping control.
- (6)
- We design an FPGA implementation of the new chaotic jerk system.

## 2. A New 3-D Jerk System

## 3. Bifurcation Analysis for the New 3-D Jerk System

#### 3.1. When the Parameter a Varies

#### 3.2. When the Parameter b Varies

#### 3.3. When the Parameter c Varies

## 4. Multistability of the Jerk System

## 5. Complete Synchronization of the New Jerk Systems Using Backstepping Control

**Theorem**

**1.**

**Proof.**

## 6. FPGA Implementation of the New Jerk System

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Reis, E.V.; Savi, M.A. Spatiotemporal chaos in a conservative Duffing-type system. Chaos Solitons Fractals
**2022**, 165, 112776. [Google Scholar] [CrossRef] - Cai, C.; Shen, Y.; Wen, S. Primary and super-harmonic simultaneous resonance of van der Pol oscillator. Int. J. Non-Linear Mech.
**2022**, 147, 104237. [Google Scholar] [CrossRef] - Madiot, G.; Correia, F.; Barbay, S.; Braive, R. Random number generation with a chaotic electromechanical resonator. Nanotechnology
**2022**, 33, 475204. [Google Scholar] [CrossRef] - Balamurali, R.; Telem, A.N.K.; Kengne, J.; Rajagopal, K.; Hermann-Dior, M.E. On the mechanism of multiscroll chaos generation in coupled non-oscillatory Rayleigh-Duffing oscillators. Phys. Scr.
**2022**, 97, 105207. [Google Scholar] [CrossRef] - Liu, H. Audio block encryption using 3D chaotic system with adaptive parameter perturbation. Multimed. Tools Appl.
**2023**, 82, 27973–27987. [Google Scholar] [CrossRef] - Dongmo, E.D.; Ramadoss, J.; Tchamda, A.R.; Sone, M.E.; Rajagopal, K. FPGA implementation, controls and synchronization of autonomous Josephson junction jerk oscillator. Phys. Scr.
**2023**, 98, 035224. [Google Scholar] [CrossRef] - Mohamed, S.M.; Sayed, W.S.; Madian, A.H.; Radwan, A.G.; Said, L.A. An Encryption Application and FPGA Realization of a Fractional Memristive Chaotic System. Electronics
**2023**, 12, 1219. [Google Scholar] [CrossRef] - Dridi, F.; El Assad, S.; Youssef, W.E.H.; Machhout, M. Design, Hardware Implementation on FPGA and Performance Analysis of Three Chaos-Based Stream Ciphers. Fractal Fract.
**2023**, 7, 197. [Google Scholar] [CrossRef] - Sun, J.Y.; Cai, H.; Wang, G.; Gao, Z.B.; Zhang, H. FPGA image encryption-steganography using a novel chaotic system with line equilibria. Digit. Signal Process.
**2023**, 134, 103889. [Google Scholar] - Sprott, J.C. Some simple chaotic jerk functions. Am. J. Phys.
**1997**, 65, 537–543. [Google Scholar] [CrossRef] - Sun, K.H.; Sprott, J.C. A simple jerk system with piecewise exponential nonlinearity. Int. J. Nonlinear Sci. Numer. Simul.
**2000**, 10, 1443–1450. [Google Scholar] [CrossRef] - Liu, M.; Sang, B.; Wang, N.; Ahmad, I. Chaotic dynamics by some quadratic jerk systems. Axioms
**2021**, 10, 227. [Google Scholar] [CrossRef] - Vaidyanathan, S.; Volos, C.K.; Kyprianidis, I.M.; Stouboulos, I.N.; Pham, V.T. Analysis, adaptive control and anti-synchronization of a six-term novel jerk chaotic system with two exponential nonlinearities and its circuit simulation. J. Eng. Sci. Technol. Rev.
**2021**, 8, 24–36. [Google Scholar] [CrossRef] - Rajagopal, K.; Pham, V.T.; Tahir, F.R.; Akgul, A.; Abdolmohammadi, H.R.; Jafari, S. A chaotic jerk system with non-hyperbolic equilibrium: Dynamics, effect of time delay and circuit realisation. Pramana J. Phys.
**2018**, 90, 52. [Google Scholar] [CrossRef] - Ramakrishnan, B.; Tamba, V.K.; Natiq, H.; Tsafack, A.S.K.; Karthikeyan, A. Dynamical analysis of autonomous Josephson junction jerk oscillator with cosine interference term embedded in FPGA and investigation of its collective behavior in a network. Eur. Phys. J. B
**2022**, 95, 145. [Google Scholar] [CrossRef] - Tagne, S.; Bodo, B.; Eyebe, G.F.V.A.; Fouda, J.S.A.E. PIC micro-controller based synchronization of two fractional order jerk systems. Sci. Rep.
**2022**, 12, 14281. [Google Scholar] [CrossRef] - Wang, Q.; Tian, Z.; Wu, X.; Tan, W. Coexistence of multiple attractors in a novel simple jerk chaotic circuit with CFOAs implementation. Front. Phys.
**2022**, 10, 835188. [Google Scholar] [CrossRef] - Wei, M.; Han, X.; Ma, X.; Zou, Y.; Bi, Q. Bursting patterns with complex structures in a parametrically and externally excited Jerk circuit system. Eur. Phys. J. Spec. Top.
**2022**, 231, 2265–2275. [Google Scholar] [CrossRef] - Kamdem Tchiedjo, S.; Kamdjeu Kengne, L.; Kengne, J.; Djuidje Kenmoe, G. Dynamical behaviors of a chaotic jerk circuit based on a novel memristive diode emulator with a smooth symmetry control. Eur. Phys. J. Plus
**2022**, 137, 940. [Google Scholar] [CrossRef] - Kengne, L.K.; Muni, S.S.; Chedjou, J.C.; Kyandoghere, K. Various coexisting attractors, asymmetry analysis and multistability control in a 3D memristive jerk system. Eur. Phys. J. Plus
**2022**, 137, 848. [Google Scholar] [CrossRef] - Njitacke, Z.T.; Feudjio, C.; Signing, V.F.; Koumetio, B.N.; Tsafack, N.; Awrejcewicz, J. Circuit and microcontroller validation of the extreme multistable dynamics of a memristive Jerk system: Application to image encryption. Eur. Phys. J. Plus
**2022**, 137, 619. [Google Scholar] [CrossRef] - Méndez-Ramírez, R.; Arellano-Delgado, A.; Cruz-Hernández, C.; Martínez-Clark, R. A new simple chaotic LLorenz-type system and its digital realization using a TFT touch-screen display embedded system. Complexity
**2017**, 2017, 6820492. [Google Scholar] [CrossRef] [Green Version] - Gupta, A.; Dubey, B. Bifurcation and chaos in a delayed eco-epidemic model induced by prey configuration. Chaos Solitons Fractals
**2022**, 165, 112785. [Google Scholar] [CrossRef] - Xing, J.; Yang, Z.; Ren, Y. Analysis of bifurcation and chaotic behavior for the flexspline of an electromagnetic harmonic drive system with movable teeth transmission. Appl. Math. Model.
**2022**, 112, 467–485. [Google Scholar] [CrossRef] - Yan, H.; Qiao, Y.; Ren, Z.; Duan, L.; Miao, J. Master–slave synchronization of fractional-order memristive MAM neural networks with parameter disturbances and mixed delays. Commun. Nonlinear Sci. Numer. Simul.
**2023**, 120, 107152. [Google Scholar] [CrossRef] - Kumar, S.; Prasad, R.P.; Nishad, C.; Tiwary, A.K.; Khan, F. Analysis and chaos synchronization of Genesio–Tesi system applying sliding mode control techniques. Int. J. Dyn. Control
**2023**, 11, 656–665. [Google Scholar] [CrossRef] - Dousseh, Y.P.; Monwanou, A.V.; Koukpémèdji, A.A.; Miwadinou, C.H.; Chabi Orou, J.B. Dynamics analysis, adaptive control, synchronization and anti-synchronization of a novel modified chaotic financial system. Int. J. Dyn. Control
**2023**, 11, 862–876. [Google Scholar] [CrossRef] - Benkouider, K.; Vaidyanathan, S.; Sambas, A.; Tlelo-Cuautle, E.; Abd El-Latif, A.A.; Abd-El-Atty, B.; Bermudez-Marquez, C.F.; Sulaiman, I.M.; Awwal, A.M.; Kumam, P. A New 5-D Multistable Hyperchaotic System With Three Positive Lyapunov Exponents: Bifurcation Analysis, Circuit Design, FPGA Realization and Image Encryption. IEEE Access
**2022**, 10, 90111–90132. [Google Scholar] [CrossRef] - Valencia-Ponce, M.A.; Castañeda-Aviña, P.R.; Tlelo-Cuautle, E.; Carbajal-Gómez, V.H.; González-Díaz, V.R.; Sandoval-Ibarra, Y.; Nuñez-Perez, J.C. CMOS OTA-based filters for designing fractional-order chaotic oscillators. Fractal Fract.
**2021**, 5, 122. [Google Scholar] [CrossRef] - Vaidyanathan, S.; Azar, A.T. Backstepping Control of Nonlinear Dynamical Systems; Academic Press: New York, NY, USA, 2021. [Google Scholar]
- Dong, H.; Cao, J.; Liu, H. Observers-based event-triggered adaptive fuzzy backstepping synchronization of uncertain fractional order chaotic systems. Chaos
**2023**, 33, A366. [Google Scholar] [CrossRef] - Yan, S.; Wang, J.; Wang, E.; Wang, Q.; Sun, X.; Li, L. A four-dimensional chaotic system with coexisting attractors and its backstepping control and synchronization. Integration
**2023**, 91, 67–78. [Google Scholar] [CrossRef] - He, J.; Qiu, W.; Cai, J. Synchronization of hyperchaotic systems based on intermittent control and its application in secure communication. J. Adv. Comput. Intell. Intell. Inform.
**2023**, 27, 292–303. [Google Scholar] [CrossRef] - Gong, X.; Wang, H.; Ji, Y.; Zhang, Y. Optical chaos generation and synchronization in secure communication with electro-optic coupling mutual injection. Opt. Commun.
**2022**, 521, 128565. [Google Scholar] [CrossRef]

**Figure 1.**MATLAB simulation plots in 2-D planes and 3-D space for the new 3-D jerk system (4) corresponding to the parameter data $(a,b,c)=(1,1,0.1)$ and initial data $Z\left(0\right)=(0.3,0.1,0.3)$.

**Figure 2.**(

**a**) Bifurcation diagram, (

**b**) LE spectrum and (

**c**) period-doubling route for the jerk system (4) when $a\in [0.65,1]$, $b=1$ and $c=0.1$.

**Figure 4.**(

**a**) Bifurcation diagram, (

**b**) LE spectrum, (

**c**) the first cascade of reverse-period doubling, (

**d**) the second cascade of reverse-period doubling for the jerk system (4) when $a=1$, $b\in [0.8,2.1]$ and $c=0.1$.

**Figure 7.**(

**a**) Bifurcation diagram, (

**b**) LEs, (

**c**) antimonotonocity phenomena, and (

**d**) reverse-period doubling route for the jerk system (4) when $a=1$, $b=1$ and $c\in [0,0.5]$.

**Figure 8.**Phase plots of the new jerk system (4) for c equal to: (

**a**) 0.29, (

**b**) 0.32, (

**c**) 0.353, (

**d**) 0.36, (

**e**) 0.39, and (

**f**) 0.48.

**Figure 9.**Coexisting attractors of the jerk system (4): (

**a**) Two coexisting periodic attractors, (

**b**) coexistence of one periodic attractor and one chaotic attractor, and (

**c**) two coexisting chaotic attractors.

**Figure 11.**Block diagram for the hardware design of the proposed new 3-D chaotic jerk system from (40).

**Figure 13.**Experimental views for the attractors $x-y$, $x-z$, and $y-z$, generated by setting $a=1$, $b=1$, and $c=0.1$, with initial conditions $(0.3,$$0.2,$$0.3)$, and $h=0.001$.

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

Slice | 134 | 1.01% |

LUTs | 304 | 0.57% |

FFs | 292 | 0.21% |

DSPs | 10 | 3.18% |

Frequency Max | 111 MHz | – |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Vaidyanathan, S.; Tlelo-Cuautle, E.; Benkouider, K.; Sambas, A.; Ovilla-Martínez, B.
FPGA-Based Implementation of a New 3-D Multistable Chaotic Jerk System with Two Unstable Balance Points. *Technologies* **2023**, *11*, 92.
https://doi.org/10.3390/technologies11040092

**AMA Style**

Vaidyanathan S, Tlelo-Cuautle E, Benkouider K, Sambas A, Ovilla-Martínez B.
FPGA-Based Implementation of a New 3-D Multistable Chaotic Jerk System with Two Unstable Balance Points. *Technologies*. 2023; 11(4):92.
https://doi.org/10.3390/technologies11040092

**Chicago/Turabian Style**

Vaidyanathan, Sundarapandian, Esteban Tlelo-Cuautle, Khaled Benkouider, Aceng Sambas, and Brisbane Ovilla-Martínez.
2023. "FPGA-Based Implementation of a New 3-D Multistable Chaotic Jerk System with Two Unstable Balance Points" *Technologies* 11, no. 4: 92.
https://doi.org/10.3390/technologies11040092