# FPGA Realization of the Parameter-Switching Method in the Chen Oscillator and Application in Image Transmission

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## Abstract

**:**

## 1. Introduction

- (i)
- FPGA realization of the parameter-switching scheme to approximate the stable cycles of the Chen oscillator, using VHDL as the implementation language with a word length of 24 bits, on the Xilinx’s Artix-7 AC701 board. The VHDL implementation on the FPGA board agreed completely with the numerical simulations done in MATLAB;
- (ii)
- FPGA realization of a secure chaos-based image transmission system on the Xilinx’s Artix-7 AC701 board, using VHDL with a 24 bit word length, whereby the parameter-switching scheme was applied as a decryption mechanism to recover chaos-encrypted RGB and grayscale images. The backbone of the secure image transmission system was a synchronized master and slave Chen system, in which the state observer was the slave system that approximated the master system. The VHDL implementation and MATLAB numerical simulations of the image transmission were in complete agreement.

## 2. Theoretical Framework

#### 2.1. Parameter-Switching Method

**R**${}^{n}$→

**R**${}^{n}$ is a Lipschitz continuous nonlinear function, p∈

**R**is the switched parameter, ${\mathit{x}}_{0}$∈

**R**${}^{n}$ represents the initial value, T > 0, and A∈L(

**R**${}^{n})$ is a constant matrix. Modeling the Chen system in Equation (1) after the IVP in Equation (2) with parameter c = p as the control parameter and giving a and b their conventional values, then:

**Notation**

**1.**

**Notation**

**2.**

**Notation**

**3.**

**Remark**

**1.**

**Remark**

**2.**

**Remark**

**3.**

**Remark**

**4.**

#### 2.2. Synchronization of Two Chen Oscillators

**R**${}^{n}$ represents the state variable and f:

**R**${}^{n}$→

**R**${}^{n}$ is the nonlinear function. The Hamiltonian forms can be described as:

**R**${}^{n}$, in which $\mathcal{M}$ is a definite matrix greater than zero, constant and symmetric. Therefore, $\frac{\partial H}{\partial x}=\mathcal{M}x$. Furthermore, $\frac{\partial H}{\partial x}$ is the gradient vector derived from $H\left(x\right)$. Matrix $\mathcal{J}\left(x\right)$ fulfills $\mathcal{J}\left(x\right)+{\mathcal{J}}^{T}\left(x\right)=0$, while $\mathcal{S}\left(x\right)$ satisfies $\mathcal{S}\left(x\right)={\mathcal{S}}^{T}\left(x\right)$ for all $x\in {\mathit{R}}^{n}$. The vector field $\mathcal{J}\left(x\right)\frac{\partial H}{\partial x}$ is the conservative part of the system. $\mathcal{S}\left(x\right)$ represents the nonconservative part. $\mathcal{F}\left(x\right)$ is the destabilizing vector.

**Definition**

**1.**

**Theorem**

**1.**

**Theorem**

**2.**

## 3. VHDL Implementation and System Co-Simulation

#### 3.1. Parameter Switching Implementation

#### 3.2. Master–Slave Synchronization

## 4. Application in Image Transmission

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

DSP | Digital signal processor |

FPGA | Field programmable gate Array |

HDL | Hardware design language |

I/O | Input/output |

IVP | Initial value problem |

LUT | Lookup table |

OGY | Ott, Grebogi, and Yorke |

PS | Parameter switching |

RAM | Random access memory |

RGB | Red, green and blue |

TRNG | True random number generator |

UPO | Unstable periodic orbit |

VHDL | Very high-speed integrated circuit Hardware Design Language |

## References

- Nicolis, G. Chapter 1—The many facets of complexity. In Complexity Science: An Introduction; Peletier, M.A., van Santen, R.A., Steur, E., Eds.; World Scientific Publishing Co.: Singapore, 2019; pp. 3–30. ISBN 978-981323960-9. [Google Scholar]
- Hu, M.; Li, F. A new method to solve numeric solution of nonlinear dynamic system. Math. Probl. Eng.
**2016**, 2016, 1485759. [Google Scholar] [CrossRef] [Green Version] - Li, X.; Liao, S. Clean numerical simulation: A new strategy to obtain reliable solutions of chaotic dynamics. Appl. Math. Mech. Engl. Ed.
**2018**, 39, 1529–1546. [Google Scholar] [CrossRef] [Green Version] - Lozi, R.; Pchelinstev, A.N. A new reliable numerical method for computing chaotic solutions of dynamical systems: The Chen attractor case. Int. J. Bifurc. Chaos
**2015**, 25, 1550187. [Google Scholar] [CrossRef] [Green Version] - Haq, B.U.; Naeem, I. First integrals and analytical solutions of some dynamical systems. Nonlinear Dyn.
**2019**, 95, 1747–1765. [Google Scholar] [CrossRef] - Mellodge, P. Chapter 4—Characteristics of Nonlinear Systems. In A Practical Approach to Dynamical Systems for Engineers; Mellodge, P., Ed.; Woodhead Publishing: Cambridge, UK, 2016; pp. 215–250. ISBN 9780081002025. [Google Scholar]
- Ling, W.K. Nonlinear Digital Filters: Analysis and Applications; Academic Press: New York, NY, USA, 2010; ISBN 9780080550015. [Google Scholar]
- Puy, A.; Daza, A.; Wagemakers, A.; Sanjuán, M.A.F. A test for fractal boundaries based on the basin entropy. Commun. Nonlinear Sci. Numer. Simul.
**2021**, 95, 105588. [Google Scholar] [CrossRef] - Goufo, E.F.D.; Khan, Y. A new auto-replication in systems of attractors with two and three merged basins of attraction via control. Commun. Nonlinear Sci. Numer. Simul.
**2021**, 96, 105709. [Google Scholar] [CrossRef] - Yan, Y.; Xu, J.; Wiercigroch, M.; Guo, Q. Statistical basin of attraction in time-delayed cutting dynamics: Modelling and computation. Phys. D Nonlinear Phenom.
**2021**, 416, 132779. [Google Scholar] [CrossRef] - Lee, M.Y.; Kim, Y.I. Development of a Family of Jarratt-Like Sixth-Order Iterative Methods for Solving Nonlinear Systems with their basins of attraction. Algorithms
**2020**, 13, 303. [Google Scholar] [CrossRef] - Rabenimananaa, T.; Walter, V.; Kacem, N.; Le Moal, P.; Bourbon, G.; Lardiès, J. Functionalization of electrostatic nonlinearities to overcome mode aliasing limitations in the sensitivity of mass microsensors based on energy localization editors-pick. Appl. Phys. Lett.
**2020**, 117, 033502. [Google Scholar] [CrossRef] - Ngo, G.Q.; George, A.; Schock, R.T.K.; Tuniz, A.; Najafidehaghani, E.; Gan, Z.; Geib, N.C.; Bucher, T.; Knopf, H.; Saravi, S.; et al. Scalable functionalization of optical fibers using atomically thin semiconductors. Adv. Mater.
**2020**, 32, e2003826. [Google Scholar] [CrossRef] - Rajagopal, K.; Akgul, A.; Jafari, S.; Karthikeyan, A.; Cavusoglu, U.; Kacar, S. An exponential jerk system, its fractional-order form with dynamical analysis and engineering application. Soft Comput.
**2020**, 24, 7469–7479. [Google Scholar] [CrossRef] - Quan, G.Z.; Ma, Y.Y.; Zhang, Y.Q.; Zhang, P.; Wang, W.Y. Separation of dynamic recrystallization parameter domains from a chaotic system for Ti–6Al–4V alloy and its application in parameter loading path design. Mater. Sci. Eng. A
**2020**, 772, 138745. [Google Scholar] [CrossRef] - Changaival, B.; Rosalie, M.; Danoy, G.; Lavangnananda, K.; Bouvry, P. Chaotic Traversal (CHAT): Very Large Graphs Traversal Using Chaotic Dynamics. Int. J. Bifurc. Chaos
**2017**, 27, 1750215. [Google Scholar] [CrossRef] [Green Version] - Li, C.T.; Lee, C.C.; Weng, C.Y.; Chen, S.J. A Secure Dynamic Identity and Chaotic Maps Based User Authentication and Key Agreement Scheme for e-Healthcare Systems. J. Med. Syst.
**2016**, 40, 233. [Google Scholar] [CrossRef] - Zang, X.; Iqbal, S.; Zhu, Y.; Liu, X.; Zhao, J. Applications of Chaotic Dynamics in Robotics. Int. J. Adv. Robot. Syst.
**2016**, 13. [Google Scholar] [CrossRef] [Green Version] - Akgul, A.; Moroz, I.; Pehlivan, I.; Vaidyanathan, S. A new four-scroll chaotic attractor and its engineering applications. Optik
**2016**, 127, 5491–5499. [Google Scholar] [CrossRef] - Ott, E. Chaos in Dynamical Systems; Cambridge University Press: Cambridge, UK, 1993. [Google Scholar]
- Din, Q. Qualitative analysis and chaos control in a density-dependent host-parasitoid system. Int. J. Dyn. Control
**2018**, 6, 778–798. [Google Scholar] [CrossRef] - Nobakhti, E.; Khaki-Sedigh, A.; Vasegh, N. Control of multichaotic systems using the extended OGY method. Int. J. Bifurc. Chaos
**2015**, 25, 1550096. [Google Scholar] [CrossRef] - Din, Q.; Elsadany, A.A.; Ibrahim, S. Bifurcation analysis and chaos control in a second order rational difference equation. Int. J. Nonlinear Sci. Numer. Simul.
**2018**, 19, 53–68. [Google Scholar] [CrossRef] - Leonov, G.A. Pyragas stabilizability via delayed feedback with periodic control gain. Syst. Control Lett.
**2014**, 69, 34–37. [Google Scholar] [CrossRef] - Xu, C.; Zhang, Q. On the chaos control of the Qi system. J. Eng. Math.
**2015**, 90, 67–81. [Google Scholar] [CrossRef] - Amster, P.; Alliera, C. Control of Pyragas applied to a coupled system with unstable periodic orbits. Bull. Math. Biol.
**2018**, 80, 2897–2916. [Google Scholar] [CrossRef] [PubMed] - Schwartz, I.B.; Triandaf, I.; Meucci, R.; Carr, T.W. Open-loop sustained chaos and control: A manifold approach. Phys. Rev. E
**2002**, 66, 026213. [Google Scholar] [CrossRef] [PubMed] - Li, Y.N.; Yang, Y.T.; Zhu, Z.M.; Zhang, C.L. Feed-forward slope compensated PFC for chaos control. J. Circuits Syst. Comput.
**2015**, 25, 1550065. [Google Scholar] [CrossRef] - Danca, M.-F.; Chattopadhyay, J. Chaos control of Hastings-Powell model by combining chaotic motions. Chaos
**2016**, 26, 043106. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Danca, M.-F.; Garrappa, R.; Tang, W.K.S.; Chen, G. Sustaining stable dynamics of a fractional-order chaotic financial system by parameter switching. Comput. Math. Appl.
**2013**, 66, 702–716. [Google Scholar] [CrossRef] - Danca, M.-F.; Lung, N. Parameter switching in a generalized Duffing system: Finding the stable attractors. Appl. Math. Comput.
**2013**, 223, 101–114. [Google Scholar] [CrossRef] [Green Version] - Núñez-Pérez, J.C.; Adeyemi, V.A.; Sandoval-Ibarra, Y.; Pérez-Pinal, F.J.; Tlelo-Cuautle, E. FPGA Realization of Spherical Chaotic System with Application in Image Transmission. Math. Probl. Eng.
**2021**, 2021, 5532106. [Google Scholar] [CrossRef] - Tuna, M.; Alçın, M.; Koyuncu, I.; Fidan, C.B.; Pehlivan, I. High speed FPGA-based chaotic oscillator design. Microprocess. Microsystem.
**2019**, 66, 72–80. [Google Scholar] [CrossRef] - Yang, C.H.; Huang, S.J. Secure color image encryption algorithm based on chaotic signals and its FPGA realization. Int. J. Circuit Theory Appl.
**2018**, 46, 2444–2461. [Google Scholar] [CrossRef] - Alçin, M.; Koyuncu, I.; Tuna, M.; Varan, M.; Pehlivan, I. A novel high speed artifical neural network-based chaotic true random number generator on field programmable gate array. Int. J. Circuit Theory Appl.
**2018**, 47, 365–378. [Google Scholar] [CrossRef] - Li, P.; Zhang, W.; Li, Z.; Liu, W.; Halang, W.A. FPGA implementation of a coupled-map-lattice-based cryptosystem. Int. J. Circuit Theory Appl.
**2010**, 38, 85–98. [Google Scholar] [CrossRef] - Tlelo-Cuautle, E.; de la Fraga, L.G.; Viet-Thanh, P.; Volos, C.; Jafari, S. Quintas-Valles, A.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] - Yu, F.; Liu, L.; He, B.; Huang, Y.; Shi, C.; Cai, S.; Song, Y.; Du, S.; Wan, Q. Analysis and FPGA realization of a novel 5D hyperchaotic four-wing memristive system, active control synchronization, and secure communication application. Complexity
**2019**, 2019, 4047957. [Google Scholar] [CrossRef] [Green Version] - Azzaz, M.S.; Tanougast, C.; Maali, A.; Benssalah, M. An efficient and lightweight multi-scroll chaos-based hardware solution for protecting fingerprint biometric templates. Int. J. Commun. Syst.
**2019**, 33, e4211. [Google Scholar] [CrossRef] - Hagras, E.A.A.; Saber, M. Low power and high-speed FPGA implementation for 4D memristor chaotic system for image encryption. Multimed. Tools Appl.
**2020**, 79, 23203–23222. [Google Scholar] [CrossRef] - Guillén-Fernández, O.; Meléndez-Cano, A.; Tlelo-Cuautle, E.; Núñez-Pérez, J.C.; Rangel-Magdaleno, J.J. On the synchronization techniques of chaotic oscillators and their FPGA-based implementation for secure image transmission. PLoS ONE
**2019**, 14, e0209618. [Google Scholar] [CrossRef] - Tlelo-Cuautle, E.; Díaz-Muñoz, J.D.; González-Zapata, A.M.; Li, R.; León-Salas, W.D.; Fernández, F.V.; Guillén-Fernández, O.; Cruz-Vega, I. Chaotic Image Encryption Using Hopfield and Hindmarsh–Rose Neurons Implemented on FPGA. Sensors
**2020**, 20, 1326. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Tlelo-Cuautle, E.; Pano-Azucena, A.D.; Guillén-Fernández, O.; Silva-Juárez, A. Synchronization and Applications of Fractional-Order Chaotic Systems. In Analog/Digital Implementation of Fractional Order Chaotic Circuits and Applications; Springer: Cham, Switzerland, 2019. [Google Scholar]
- Sivaraman, R.; Rajagopalan, S.; Amirtharajan, R. FPGA based generic RO TRNG architecture for image confusion. Multimed. Tools Appl.
**2020**, 79, 13841–13868. [Google Scholar] [CrossRef] - Zeng, J.; Wang, C. A Novel Hyperchaotic Image Encryption System Based on Particle Swarm Optimization Algorithm and Cellular Automata. Secur. Commun. Netw.
**2021**, 2021, 6675565. [Google Scholar] [CrossRef] - Wang, X.; Li, Y.; Jin, J. A new one-dimensional chaotic system with applications in image encryption. Chaos Solitons Fractals
**2020**, 139, 110102. [Google Scholar] [CrossRef] - Li, Z.; Peng, C.; Tan, W.; Li, L. An Efficient Plaintext-Related Chaotic Image Encryption Scheme Based on Compressive Sensing. Sensors
**2021**, 21, 758. [Google Scholar] [CrossRef] [PubMed] - Mohamed, A.G.; Korany, N.O.; El-Khamy, S.E. New DNA Coded Fuzzy Based (DNAFZ) S-Boxes: Application to Robust Image Encryption Using Hyper Chaotic Maps. IEEE Access
**2021**, 9, 14284–14305. [Google Scholar] [CrossRef] - Núñez Pérez, J.C.; Adeyemi, V.A.; Sandoval-Ibarra, Y.; Serrato-Andrade, R.Y.; Cárdenas, J.R.; Tlelo-Cuautle, E. Mathematical and numerical analysis of the dynamical behavior of Chen oscillator. Int. J. Dyn. Control
**2020**, 8, 386–395. [Google Scholar] [CrossRef] - Barboza, R. On the Lorenz and Chen Systems. Int. J. Bifurc. Chaos
**2018**, 28, 1850018. [Google Scholar] [CrossRef] - Chen, G.; Ueta, T. Yet another chaotic attractor. Int. J. Bifurc. Chaos
**1999**, 9, 1465–1466. [Google Scholar] [CrossRef] - Wang, X.; Chen, G. Generating Lorenz-like and Chen-like attractors from a simple algebraic structure. Sci. China Inf. Sci.
**2014**, 57, 1–7. [Google Scholar] [CrossRef] [Green Version] - Danca, M.-F.; Feckan, M. Note on a parameter-switching method for nonlinear ODEs. Math. Slovaca
**2016**, 66, 439–448. [Google Scholar] [CrossRef] [Green Version] - Tang, W.K.S.; Danca, M.-F. Emulating “Chaos + Chaos = Order” in Chen’s circuit of fractional order by parameter switching. Int. J. Bifurc. Chaos
**2016**, 26, 1650096. [Google Scholar] [CrossRef] - Danca, M.-F. Convergence of a parameter switching algorithm for a class of nonlinear continuous systems and a generalization of Parrondo’s paradox. Commun. Nonlinear Sci. Numer. Simul.
**2013**, 18, 500–510. [Google Scholar] [CrossRef] - Sira-Ramirez, H.; Cruz-Hernandez, C. Synchronization of chaotic systems: A generalized Hamiltonian systems approach. Int. J. Bifurc. Chaos
**2001**, 11, 1381–1395. [Google Scholar] [CrossRef] - Pei, L.J.; Liu, S.H. Application of generalized Hamiltonian systems to chaotic synchronization. Nonlinear Dyn. Syst. Theory
**2009**, 9, 415–432. [Google Scholar] - Danca, M.-F.; Tang, W.K.S. Parrondo’s paradox for chaos control and anticontrol of fractional-order systems. Chin. Phys. B
**2016**, 25, 010505. [Google Scholar] [CrossRef] - Harmer, G.P.; Abbott, D. A review of Parrondo’s paradox. Fluct. Noise Lett.
**2002**, 2, R71–R107. [Google Scholar] [CrossRef] [Green Version] - Kumar, V.; Aggarwal, R.; Sharma, P.; Kaur, B. Fractal basins of attraction in a binary quasar model. New Astron.
**2021**, 84, 101543. [Google Scholar] [CrossRef] - Saeed, T.; Chen, W.; Zotos, E.E. Convergence properties of equilibria in the restricted three-body problem with prolate primaries. Astron. Nachrichten
**2020**, 341, 887–898. [Google Scholar] [CrossRef] - Koppu, S.; Viswanatham, V.M. A fast enhanced secure image chaotic cryptosystem based on hybrid chaotic magic transform. Model. Simul. Eng.
**2017**, 2017, 7470204. [Google Scholar] [CrossRef] [Green Version] - Chang, D.; Li, Z.; Wang, M.; Zeng, Y. A novel digital programmable multi-scroll chaotic system and its application in FPGA-based audio secure communication. AEU Int. J. Electron. Commun.
**2018**, 88, 20–29. [Google Scholar] [CrossRef] - Tlelo-Cuautle, E.; Carbajal-Gomez, V.H.; Obeso-Rodelo, P.J.; Rangel-Magdaleno, J.J.; Núñez-Pérez, J.C. FPGA realization of a chaotic communication system applied to image processing. Nonlinear Dyn.
**2015**, 82, 1879–1892. [Google Scholar] [CrossRef] - Sadoudi, S.; Tanougast, C.; Azzaz, M.S.; Dandache, A. Design and FPGA implementation of a wireless hyperchaotic communication system for secure real-time image transmission. EURASIP J. Image Video Process.
**2013**, 43. [Google Scholar] [CrossRef] - Belo, D.; Carvalho, N.B. An OOK chirp spread spectrum backscatter communication system for wireless power transfer applications. IEEE Trans. Microw. Theory Tech.
**2021**, 69, 1838–1845. [Google Scholar] [CrossRef] - Liu, M.; Han, Y.; Chen, Y.; Song, H.; Yang, Z.; Gong, F. Modulation parameter estimation of LFM interference for direct sequence spread spectrum communication system in Alpha-Stable noise. IEEE Syst. J.
**2021**, 15, 881–892. [Google Scholar] [CrossRef] - Sheikhpour, S.; Mahani, A.; Bagheri, N. Reliable advanced encryption standard hardware implementation: 32-bit and 64-bit data-paths. Microprocess. Microsystem.
**2021**, 81, 103740. [Google Scholar] [CrossRef] - Ikhwan, A.; Rafikha Aliana, A.R.; Ehkan, P.; Yacob, Y.; Syaifuddin, M. Data Security Implementation using Data Encryption Standard Method for Student Values at the Faculty of Medicine, University of North Sumatra. J. Phys. Conf. Ser.
**2021**, 1755, 012022. [Google Scholar] [CrossRef]

**Figure 1.**Parameter-switching scheme showing the switched parameters ${p}_{k}$ and their associated sub-interval ${i}_{k}$ times.

**Figure 2.**Bifurcation diagram of the Chen oscillator (1) showing the chaotic and stable regions as parameter c is varied. The “averaged” parameter p* = 26.0858 is selected from the stable window.

**Figure 3.**Chen oscillator and parameter-switching scheme using the 4th-order Runge–Kutta method. (

**a**) Co-simulation in MATLAB/Simulink with Active-HDL. (

**b**) Active-HDL block diagram of the VHDL implementation.

**Figure 4.**Chaotic phase portraits of the Chen system (1) for the set ${\mathcal{A}}_{N}$ corresponding to ${\mathcal{P}}_{N}$.

**Figure 5.**Co-simulation results of the approximation of the Chen system’s stable cycle, illustrating the “switched” solution ${A}^{S}$ (red), “averaged” solution A* (blue), and overplots of ${A}^{S}$ and A*.

**Figure 6.**Time series of the three states x, y, and z. (

**a**) MATLAB overplots: “switched” solution ${A}^{S}$ (red) and “averaged” solution A* (blue). (

**b**) Active-HDL: “switched” solution ${A}^{S}$ (upper) and “averaged” solution A* (lower).

**Figure 7.**Hamiltonian synchronization of master and slave Chen systems. (

**a**) Co-simulation in MATLAB/Simulink with Active-HDL. (

**b**) Active-HDL block diagram of the VHDL implementation.

**Figure 8.**Basins of attraction of the Chen system (1) on $x-y$ plane at $a=35$, $b=3$, and $c=24.75$; equilibrium point $E{P}_{0}$ has the red regions as its basin; equilibrium point $E{P}_{1}$ has green regions as its basin; while equilibrium point $E{P}_{2}$ has blue regions as its basin.

**Figure 9.**Image transmission system. (

**a**) Co-simulation diagram in MATLAB/Simulink with Active-HDL. (

**b**) Active-HDL block diagram.

**Figure 10.**Waveforms of the 640 × 480 grayscale image from transmission variable x, showing the original (red), encrypted (blue), and received (magenta) recovered by the parameter-switching technique. (

**a**) Co-simulation diagram in MATLAB/Simulink with Active-HDL. (

**b**) Active-HDL.

**Figure 11.**Waveforms of the 320 × 240 RGB image data from transmission variable y, showing the original (red), encrypted (blue), and received (magenta) recovered by the parameter-switching technique. (

**a**) Co-simulation diagram in MATLAB/Simulink with Active-HDL. (

**b**) Active-HDL.

**Figure 12.**Co-simulation image data, showing the original image (left), encrypted image (middle), and received image (right) recovered by the parameter-switching technique: (

**a**) 640 × 480 grayscale image, transmitted via state variable x; (

**b**) 320 × 240 RGB image, transmitted via state variable y.

**Table 1.**Correlation coefficients’ analysis of the original, encrypted, and received RGB and grayscale images, using the parameter-switching scheme for image decryption.

Transmission Variable | Correlation | RGB Image | Grayscale Image | ||
---|---|---|---|---|---|

Red | Green | Blue | |||

x | Original and Encrypted | 0.0342 | 0.0543 | 0.0315 | 0.0352 |

Encrypted and Received | 0.0342 | 0.0543 | 0.0315 | 0.0352 | |

Original and Received | 1 | 1 | 1 | 1 | |

y | Original and Encrypted | 0.0316 | 0.0498 | 0.0296 | 0.0362 |

Encrypted and Received | 0.0316 | 0.0498 | 0.0296 | 0.0362 | |

Original and Received | 1 | 1 | 1 | 1 | |

z | Original and Encrypted | 0.0649 | 0.0747 | 0.0711 | 0.0672 |

Encrypted and Received | 0.0649 | 0.0747 | 0.0711 | 0.0672 | |

Original and Received | 1 | 1 | 1 | 1 |

**Table 2.**Utilization of FPGA resources in the Xilinx Artix-7 AC701 board to implement the: (1) Chen chaotic oscillator, (2) parameter-switching scheme applied to approximate stable cycles of the Chen oscillator, and (3) the Hamiltonian form synchronization of Chen oscillators.

Chen Oscillator | Parameter Switching | Synchronization | |||||
---|---|---|---|---|---|---|---|

Resources | Available | Used | Utilization(%) | Used | Utilization(%) | Used | Utilization(%) |

Slice LUTs | 134,600 | 5615 | 4 | 5472 | 4 | 11,935 | 9 |

Memory LUTs | 46,200 | 0 | 0 | 0 | 0 | 0 | 0 |

Registers | 269,200 | 144 | <1 | 179 | <1 | 479 | <1 |

I/O pins | 400 | 73 | 18 | 73 | 18 | 145 | 36 |

Block RAMs | 13,140,000 | 0 | 0 | 0 | 0 | 0 | 0 |

DSPs | 740 | 44 | 6 | 40 | 5 | 64 | 9 |

**Table 3.**Utilization of FPGA resources in the Xilinx Artix-7 AC701 board to implement RGB and grayscale image transmissions by Chen oscillators using the parameter-switching scheme for decryption.

Resources | Available | RGB Image | Grayscale Image | ||
---|---|---|---|---|---|

Used | Utilization (%) | Used | Utilization (%) | ||

Slice LUTs | 134,600 | 73,157 | 54 | 83,810 | 62 |

Memory LUTs | 46,200 | 0 | 0 | 0 | 0 |

Registers | 269,200 | 885 | <1 | 948 | <1 |

I/O pins | 400 | 75 | 19 | 75 | 19 |

Block RAMs | 13,140,000 | 0 | 0 | 0 | 0 |

DSPs | 740 | 104 | 14 | 104 | 14 |

Parameter | This Work (RGB) | This Work (Grayscale) | Ref. [38] | Ref. [63] | Ref. [64] | Ref. [65] |
---|---|---|---|---|---|---|

FPGA | Artix-7 | Artix-7 | ZYNQ | Cyclone IV | Stratix IV | Virtex 5 |

Slice LUTs | 54% | 62% | 43% | 33% | 27% | 24% |

Registers | <1% | <1% | 25% | 27% | <1% | 5% |

I/O pins | 19% | 19% | 24% | N/A | 22% | 32% |

Block RAMs | 0% | 0% | N/A | 96% | 40% | N/A |

DSPs | 14% | 14% | N/A | 24% | 7% | 62% |

Algorithm | RK-4 | RK-4 | RK-4 | Euler | Euler | RK-4 |

Language | VHDL | VHDL | Verilog | Verilog | VHDL | VHDL |

Number | 24 bit | 24 bit | 32 bit | 32 bit | 19 bit | 32 bit |

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**MDPI and ACS Style**

Adeyemi, V.-A.; Nuñez-Perez, J.-C.; Sandoval Ibarra, Y.; Perez-Pinal, F.-J.; Tlelo-Cuautle, E.
FPGA Realization of the Parameter-Switching Method in the Chen Oscillator and Application in Image Transmission. *Symmetry* **2021**, *13*, 923.
https://doi.org/10.3390/sym13060923

**AMA Style**

Adeyemi V-A, Nuñez-Perez J-C, Sandoval Ibarra Y, Perez-Pinal F-J, Tlelo-Cuautle E.
FPGA Realization of the Parameter-Switching Method in the Chen Oscillator and Application in Image Transmission. *Symmetry*. 2021; 13(6):923.
https://doi.org/10.3390/sym13060923

**Chicago/Turabian Style**

Adeyemi, Vincent-Ademola, Jose-Cruz Nuñez-Perez, Yuma Sandoval Ibarra, Francisco-Javier Perez-Pinal, and Esteban Tlelo-Cuautle.
2021. "FPGA Realization of the Parameter-Switching Method in the Chen Oscillator and Application in Image Transmission" *Symmetry* 13, no. 6: 923.
https://doi.org/10.3390/sym13060923