# Dynamics and Complexity of a New 4D Chaotic Laser System

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## Abstract

**:**

## 1. Inroduction

- (i)
- We derive a new 4D chaotic laser system with three equilibria from Lorenz-Haken equations;
- (ii)
- We investigate the stability of the symmetric equilibria, and the existence of coexisting multiple Hopf bifurcations on these equilibria;
- (iii)
- We analyze the presence of complex coexisting behaviors in the laser system;
- (iv)
- We use the complexity of the laser system time series to locate the regions of coexisting attractors when the parameters and initial values vary;
- (v)
- Based on the complexity of the system time series, we study the randomness of multistability regions.

## 2. A New 4D Chaotic Laser System From Lorenz-Haken Model

#### 2.1. Chaotic Behavior Regions

#### 2.2. Dissipation and Symmetry

#### 2.3. Equilibria and Stability

## 3. Local Bifurcation Analysis and Numerical Simulations

#### 3.1. Hopf Bifurcation

- (A)
- nondegeneracy condition: the Jacobian matrix ${J}_{({x}_{0},{\zeta}_{0})}$ has one pair of purely imaginary roots, and other roots have nonzero real parts;
- (B)
- transversality condition: the real part of differentiation characteristic equation with respect to the parameter $\zeta $ satisfy$$\begin{array}{c}\hfill \begin{array}{c}\hfill Re\left(\frac{d\lambda}{d\zeta}\right){|}_{\zeta ={\zeta}_{0}}\ne 0;\end{array}\end{array}$$
- (C)
- the first Lyapunov coefficient ${l}_{1}$ is nonzero.

#### 3.2. Numerical Simulations

## 4. Multistability Behavior

## 5. Complexity and Randomness of Multistability Regions

#### 5.1. Sample Entropy

- (A)
- Reconstructing phase-space: for a given embedding dimension m and time delay $\tau $, the reconstruction sequences are given by$$\begin{array}{c}\hfill \begin{array}{c}\hfill {Y}_{i}=\{{y}_{i},{y}_{i+\tau},...,{y}_{i+(m-1)\tau}\},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{y}_{i}\in {R}^{m}\end{array}\end{array}$$
- (B)
- Counting the vector pairs: let ${B}_{i}$ be the number of vector ${Y}_{j}$ such that$$\begin{array}{c}\hfill \begin{array}{c}\hfill d[{Y}_{i},{Y}_{j}]\le r,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}i\ne j\end{array}\end{array}$$$$\begin{array}{c}\hfill \begin{array}{c}\hfill d[{Y}_{i},{Y}_{j}]=max\left\{\right|y(i+k)-y(j+k)|:0\le k\le m-1\}.\end{array}\end{array}$$
- (C)
- Calculating probability: according to the obtained number of vector pairs, we can obtain$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {C}_{i}^{m}\left(r\right)& ={\displaystyle \frac{{B}_{i}}{M-(m-1)\tau}},\hfill \end{array}\end{array}$$$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\varphi}^{m}\left(r\right)& ={\displaystyle \frac{{\sum}_{i=1}^{M-(m-1)\tau}ln{C}_{i}^{m}\left(r\right)}{[M-(m-1\left)\tau \right]}}\hfill \end{array}\end{array}$$
- (D)
- Calculating SamEn: repeating the above steps we can obtain ${\varphi}^{m+1}\left(r\right)$, then SamEn is given by$$\begin{array}{c}\hfill \begin{array}{c}\hfill SamEn(m,r,M)={\varphi}^{m}\left(r\right)-{\varphi}^{m+1}\left(r\right).\end{array}\end{array}$$

#### 5.2. Chaos-Based PRNG

Algorithm 1 The generation of chaos-based PRNG |

Input: The initial values of system (2).
- 1:
**for**$i=1$ to 4**do**- 2:
**for**$r=27$ to 29**do**- 3:
- Truncate a chaotic sequence ${C}_{i}$ from the trajectory of ${x}_{i}$;
- 4:
- Convert the floating number ${C}_{i}$ of ${x}_{i}$ into a 32-bit binary using the IEEE-754-Standard;
- 5:
- Fetch the last 16th digital number of the obtained binary string;
- 6:
**end for**- 7:
**end****for**
Output: Four PRNG are generated from of ${x}_{1}$, ${x}_{2}$, ${x}_{3}$ and ${x}_{4}$ |

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Banerjee, S.; Rondoni, L.; Mukhopadhyay, S.; Misra, A.P. Synchronization of spatiotemporal semiconductor lasers and its application in color image encryption. Opt. Commun.
**2011**, 284, 2278–2291. [Google Scholar] [CrossRef] [Green Version] - Valli, D.; Banerjee, S.; Ganesan, K.; Muthuswamy, B.; Subramaniam, C.K. Chaotic time delay systems and field programmable gate array realization. In Chaos, Complexity and Leadership 2012; Springer: Dordrecht, The Netherlands, 2014; pp. 9–16. [Google Scholar]
- Banerjee, S.; Saha, P.; Chowdhury, A.R. Chaotic scenario in the Stenflo equations. Phys. Scr.
**2001**, 63, 177. [Google Scholar] [CrossRef] - Natiq, H.; Banerjee, S.; He, S.; Said, M.R.M.; Kilicman, A. Designing an M-dimensional nonlinear model for producing hyperchaos. Chaos Solitons Fractals
**2018**, 114, 506–515. [Google Scholar] [CrossRef] - Ghosh, D.; Banerjee, S.; Chowdhury, A.R. Synchronization between variable time-delayed systems and cryptography. Europhys. Lett.
**2007**, 80, 30006. [Google Scholar] [CrossRef] - Banerjee, S. (Ed.) Chaos Synchronization and Cryptography for Secure Communications: Applications for Encryption; IGI Global: Hershey, PA, USA, 2010. [Google Scholar]
- Saha, P.; Banerjee, S.; Chowdhury, A.R. Chaos, signal communication and parameter estimation. Phys. Lett. A
**2004**, 326, 133–139. [Google Scholar] [CrossRef] - Fataf, N.A.A.; Palit, S.K.; Mukherjee, S.; Said, M.R.M.; Son, D.H.; Banerjee, S. Communication scheme using a hyperchaotic semiconductor laser model: Chaos shift key revisited. Eur. Phys. J. Plus
**2017**, 132, 492. [Google Scholar] [CrossRef] - Banerjee, S.; Pizzi, M.; Rondoni, L. Modulation of output power in the spatio-temporal analysis of a semi conductor laser. Opt. Commun.
**2012**, 285, 1341–1346. [Google Scholar] [CrossRef] - Rondoni, L.; Ariffin, M.R.K.; Varatharajoo, R.; Mukherjee, S.; Palit, S.K.; Banerjee, S. Optical complexity in external cavity semiconductor laser. Opt. Commun.
**2017**, 387, 257–266. [Google Scholar] [CrossRef] - Mukherjee, S.; Palit, S.K.; Banerjee, S.; Ariffin, M.R.K.; Rondoni, L.; Bhattacharya, D.K. Can complexity decrease in congestive heart failure? Phys. A Stat. Mech. Appl.
**2015**, 439, 93–102. [Google Scholar] [CrossRef] [Green Version] - Banerjee, S.; Palit, S.K.; Mukherjee, S.; Ariffin, M.R.K.; Rondoni, L. Complexity in congestive heart failure: A time-frequency approach. Chaos Interdiscip. J. Nonlinear Sci.
**2016**, 26, 033105. [Google Scholar] [CrossRef] - Pham, V.-T.; Vaidyanathan, S.; Volos, C.K.; Jafari, S. Hidden attractors in a chaotic system with an exponential nonlinear term. Eur. Phys. J. Spec. Top.
**2015**, 224, 1507–1517. [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] - Pham, V.T.; Volos, C.; Jafari, S.; Wang, X.; Vaidyanathan, S. Hidden hyperchaotic attractor in a novel simple memristive neural network. Optoelectron. Adv. Mater. Rapid Commun.
**2014**, 8, 1157–1163. [Google Scholar] - Dudkowski, D.; Jafari, S.; Kapitaniak, T.; Kuznetsov, N.V.; Leonov, G.A.; Prasad, A. Hidden attractors in dynamical systems. Phys. Rep.
**2016**, 637, 1–50. [Google Scholar] [CrossRef] - Tlelo-Cuautle, E.; de la Fraga, L.G.; Pham, V.T.; Volos, C.; Jafari, S.; de Jesus Quintas-Valles, A. 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] - Pham, V.-T.; Volos, C.; Gambuzza, L.V. A memristive hyperchaotic system without equilibrium. Sci. World J.
**2014**, 2014, 368986. [Google Scholar] [CrossRef] [PubMed] - Natiq, H.; Said, M.R.M.; Ariffin, M.R.K.; He, S.; Rondoni, L.; Banerjee, S. Self-excited and hidden attractors in a novel chaotic system with complicated multistability. Eur. Phys. J. Plus
**2018**, 133, 557. [Google Scholar] [CrossRef] - Wang, X.; Pham, V.T.; Jafari, S.; Volos, C.; Munoz-Pacheco, J.M.; Tlelo-Cuautle, E. A new chaotic system with stable equilibrium: From theoretical model to circuit implementation. IEEE Access
**2017**, 5, 8851–8858. [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] - Pham, V.T.; Jafari, S.; Volos, C.; Giakoumis, A.; Vaidyanathan, S.; Kapitaniak, T. A chaotic system with equilibria located on the rounded square loop and its circuit implementation. IEEE Trans. Circuits Syst. II Express Briefs
**2016**, 63, 878–882. [Google Scholar] [CrossRef] - Arecchi, F.; Meucci, R.; Puccioni, G.; Tredicce, J. Experimental evidence of subharmonic bifurcations, multistability, and turbulence in a q-switched gas laser. Phys. Rev. Lett.
**1982**, 49, 1217. [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] - Wang, C.; Ding, Q. A New Two-Dimensional Map with Hidden Attractors. Entropy
**2018**, 20, 322. [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] - Sparrow, C. The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors; Springer Science & Business Media: New York, NY, USA, 2012; Volume 41. [Google Scholar]
- Pereira, U.; Coullet, P.; Tirapegui, E. The Bogdanov—Takens normal form: A minimal model for single neuron dynamics. Entropy
**2015**, 17, 7859–7874. [Google Scholar] [CrossRef] - Zhan, X.; Ma, J.; Ren, W. Research entropy complexity about the nonlinear dynamic delay game model. Entropy
**2017**, 19, 22. [Google Scholar] [CrossRef] - Han, Z.; Ma, J.; Si, F.; Ren, W. Entropy complexity and stability of a nonlinear dynamic game model with two delays. Entropy
**2016**, 18, 317. [Google Scholar] [CrossRef] - Dang, T.S.; Palit, S.K.; Mukherjee, S.; Hoang, T.M.; Banerjee, S. Complexity and synchronization in stochastic chaotic systems. Eur. Phys. J. Spec. Top.
**2016**, 225, 159–170. [Google Scholar] [CrossRef] - He, S.; Sun, K.; Wang, H. Complexity analysis and DSP implementation of the fractional-order Lorenz hyperchaotic system. Entropy
**2015**, 17, 8299–8311. [Google Scholar] [CrossRef] - Ma, J.; Ma, X.; Lou, W. Analysis of the Complexity Entropy and Chaos Control of the Bullwhip Effect Considering Price of Evolutionary Game between Two Retailers. Entropy
**2016**, 18, 416. [Google Scholar] [CrossRef] - He, S.; Li, C.; Sun, K.; Jafari, S. Multivariate Multiscale Complexity Analysis of Self-Reproducing Chaotic Systems. Entropy
**2018**, 20, 556. [Google Scholar] [CrossRef] - Haken, H. Analogy between higher instabilities in fluids and lasers. Phys. Lett. A
**1975**, 53, 77–78. [Google Scholar] [CrossRef] - Banerjee, S.; Saha, P.; Chowdhury, A.R. Chaotic aspects of lasers with host-induced nonlinearity and its control. Phys. Lett. A
**2001**, 291, 103–114. [Google Scholar] [CrossRef] - Van Tartwijk, G.H.M.; Agrawal, G.P. Nonlinear dynamics in the generalized Lorenz-Haken model. Opt. Commun.
**1997**, 133, 565–577. [Google Scholar] [CrossRef] - Kuznetsov, Y.A. Numerical Analysis of Bifurcations. In Elements of Applied Bifurcation Theory; Springer: New York, NY, USA, 2004; pp. 505–585. [Google Scholar]
- Kaffashi, F.; Foglyano, R.; Wilson, C.G.; Loparo, K.A. The effect of time delay on approximate & sample entropy calculations. Phys. D Nonlinear Phenom.
**2008**, 237, 3069–3074. [Google Scholar] - Richman, J.S.; Moorman, J.R. Physiological time-series analysis using approximate entropy and sample entropy. Am. J. Physiol. Heart Circ. Physiol.
**2000**, 278, H2039–H2049. [Google Scholar] [CrossRef] [PubMed] - Volos, C.K.; Kyprianidis, I.M.; Stouboulos, I.N. Fingerprint images encryption process based on a chaotic true random bits generator. Int. J. Multimedia Intell. Secur.
**2010**, 1, 320–335. [Google Scholar] [CrossRef] - Rukhin, A.; Soto, J.; Nechvatal, J.; Smid, M.; Barker, E. A Statistical Test Suite for Random and Pseudorandom Number Generators for Cryptographic Applications; Booz-Allen and Hamilton Inc.: Mclean, VA, USA, 2001. [Google Scholar]
- Natiq, H.; Al-Saidi, N.M.G.; Said, M.R.M.; Kilicman, A. A new hyperchaotic map and its application for image encryption. Eur. Phys. J. Plus
**2018**, 133, 6. [Google Scholar] [CrossRef] - Rodríguez-Orozco, E.; García-Guerrero, E.; Inzunza-Gonzalez, E.; López-Bonilla, O.; Flores-Vergara, A.; Cárdenas-Valdez, J.; Tlelo-Cuautle, E. FPGA-based Chaotic Cryptosystem by Using Voice Recognition as Access Key. Electronics
**2018**, 7, 414. [Google Scholar] [CrossRef]

**Figure 1.**Dynamics of the system (2) versus the parameter b for the initial values $(2,1,1,2)$ and with $\sigma =4$, $\delta =0.5$, $r=27$: (

**a**) bifurcation diagram; (

**b**) Lyapunov exponents.

**Figure 2.**Different orientations on a two-scroll chaotic attractor of the system (2) for the initial values $(2,1,1,2)$ and with the parameters $\sigma =4$, $\delta =0.5$, $r=27$, $b=2$. (

**a**) $({x}_{2},{x}_{3},{x}_{1})$ space; (

**b**) $({x}_{4},{x}_{1},{x}_{2})$ space; (

**c**) $({x}_{4},{x}_{3},{x}_{2})$ space; (

**d**) $({x}_{4},{x}_{3},{x}_{1})$ space.

**Figure 3.**Hopf bifurcation of the system (2): (

**a**) $r=5.5<{r}_{0}$, the orbit of the system is attracted to the stable symmetric equilibria ${E}_{2}$ and ${E}_{3}$; (

**b**) $r=6.5>{r}_{0}$, the orbit of the system is attracted to a stable limit cycle emerging from the symmetric equilibria ${E}_{2}$ and ${E}_{3}$.

**Figure 4.**Bifurcation diagrams versus parameter r for illustrating the two and three coexisting attractors of the system (2): (

**a**) $\sigma =2$, $\delta =1.5$, $b=0.7$ for the initial values $(1,1,1,1)$ (red) and $(-2,1,1,1)$ (blue); (

**b**) $\sigma =4$, $\delta =0.5$, $b=2$ for the initial values $(2,1,1,2)$ (blue), $(-2,1,1,-2)$ (red) and $(2,1,1,-2)$ (green).

**Figure 5.**Multiple coexisting chaotic attractors of the system (2) when $\sigma =2$, $\delta =1.5$, $b=0.7$, $r=9.41$ for the initial values $(1,1,1,1)$ (red) and $(-2,1,1,1)$ (blue). (

**a**) ${x}_{1}$–${x}_{2}$ plane; (

**b**) ${x}_{2}$–${x}_{3}$ plane; (

**c**) ${x}_{1}$–${x}_{4}$ plane; (

**d**) ${x}_{2}$–${x}_{4}$ plane.

**Figure 6.**Three coexisting attractors with $\sigma =4$, $\delta =0.5$, $b=2$, $r=27$: (

**a**,

**c**,

**e**) different perspectives on the coexistence of the chaotic and two stable fixed-point attractors for the initial values $(2,1,1,2)$ (blue), $(-2,1,1,-2)$ (red) and $(2,1,1,-2)$ (green); (

**b**,

**d**,

**f**) the corresponding time series of the state variables ${x}_{1}$, ${x}_{2}$ and ${x}_{4}$, respectively.

**Figure 7.**SamEn in the parameter r-initial value plane for $\sigma =4$, $\delta =0.5$, $b=2$: (

**a**) $r-{x}_{10}$ plane; (

**b**) $r-{x}_{20}$ plane; (

**c**) $r-{x}_{30}$ plane; (

**d**) $r-{x}_{40}$ plane.

**Figure 8.**SamEn versus varying two of the initial values for $\sigma =4$, $\delta =0.5$, $b=2$, $r=27$: (

**a**) $({x}_{10},{x}_{20},1,2)$; (

**b**) $(2,{x}_{20},{x}_{30},2)$; (

**c**) $({x}_{10},1,1,{x}_{40})$.

**Figure 9.**The statistical tests NIST SP800-22 of the pseudorandom number generator (PRNG) that generated by ${x}_{1}$, ${x}_{2}$, ${x}_{3}$, ${x}_{4}$ of the system (2) with $\sigma =4$, $\delta =0.5$, $b=2$, $r\in [27,29]$ and for the initial values $(2,1,1,-2)$. (

**a**) Block-Frequency, Discrete Fourier Transform, Frequency (Monobit), Random Excursions, Random Excursions Variant, Serial-1, Serial-2, Linear Complexity, and Longest Run of Ones, respectively; (

**b**) Approximate Entropy, Cumulative Sums (Forward), Cumulative Sums (Reverse), Lempel-ziv Compression, Non-overlapping Template, Overlapping Template, Binary Matrix Rank, Runs, and Universal Statistical.

**Table 1.**NIST-800-22 tests results of binary sequences generated by PRNG of ${x}_{1}$, ${x}_{2}$, ${x}_{3}$ and ${x}_{4}$ outputs.

Each Sequence to be Tested Consists of 1,000,000 Bits | ||||||
---|---|---|---|---|---|---|

NIST-800-22 Tests | $\mathit{p}$-Value (${\mathit{x}}_{\mathbf{1}}$) | $\mathit{p}$-Value (${\mathit{x}}_{\mathbf{2}}$) | $\mathit{p}$-Value (${\mathit{x}}_{\mathbf{3}}$) | $\mathit{p}$-Value (${\mathit{x}}_{\mathbf{4}}$) | Result | |

1. | Block-Frequency (m = 128) | 0.2116 | 0.8460 | 0.8313 | 0.0210 | Random |

2. | Frequency (Monobit) | 0.7611 | 0.0380 | 0.6570 | 0.3503 | Random |

3. | Discrete Fourier Transform | 0.3602 | 0.1792 | 0.1478 | 0.1225 | Random |

4. | Approximate Entropy (m = 10) | 0.9592 | 0.6512 | 0.6343 | 0.3659 | Random |

5. | Cumulative Sums (Forward) | 0.7617 | 0.0721 | 0.7280 | 0.5832 | Random |

Cumulative Sums (Reverse) | 0.5578 | 0.0320 | 0.5106 | 0.1816 | Random | |

6. | Serial-1 (m = 16) | 0.7937 | 0.2948 | 0.1635 | 0.9706 | Random |

Serial-2 (m = 16) | 0.8885 | 0.7628 | 0.5357 | 0.9530 | Random | |

7. | Runs | 0.9649 | 0.6196 | 0.4751 | 0.1530 | Random |

8. | Longest Run of Ones | 0.2568 | 0.0965 | 0.8242 | 0.2420 | Random |

9. | Overlapping Template (m = 9) | 0.7032 | 0.6461 | 0.5603 | 0.7085 | Random |

10. | Non-overlapping Template (m = 9) | 0.4960 | 0.5403 | 0.5150 | 0.5117 | Random |

11. | Linear Complexity (m = 500) | 0.4091 | 0.7263 | 0.1607 | 0.8582 | Random |

12. | Binary Matrix Rank | 0.2618 | 0.1029 | 0.2843 | 0.2376 | Random |

13. | Lempel-ziv Compression | 0.0769 | 0.2343 | 0.1411 | 0.9581 | Random |

14. | Random Excursions | 0.4628 | 0.2379 | 0.4787 | 0.3931 | Random |

15. | Random Excursions Variant | 0.6141 | 0.1814 | 0.3977 | 0.2865 | Random |

16. | Universal Statistical | 0.4931 | 0.7326 | 0.6056 | 0.1038 | Random |

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## Share and Cite

**MDPI and ACS Style**

Natiq, H.; Said, M.R.M.; Al-Saidi, N.M.G.; Kilicman, A.
Dynamics and Complexity of a New 4D Chaotic Laser System. *Entropy* **2019**, *21*, 34.
https://doi.org/10.3390/e21010034

**AMA Style**

Natiq H, Said MRM, Al-Saidi NMG, Kilicman A.
Dynamics and Complexity of a New 4D Chaotic Laser System. *Entropy*. 2019; 21(1):34.
https://doi.org/10.3390/e21010034

**Chicago/Turabian Style**

Natiq, Hayder, Mohamad Rushdan Md Said, Nadia M. G. Al-Saidi, and Adem Kilicman.
2019. "Dynamics and Complexity of a New 4D Chaotic Laser System" *Entropy* 21, no. 1: 34.
https://doi.org/10.3390/e21010034