# Synchronization, Control and Data Assimilation of the Lorenz System

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Master–Slave Synchronization

#### 2.1. Pecora–Carrol Synchronization

#### 2.2. Conditional Coupling

#### 2.3. Partial Conditional Coupling

#### 2.4. Intermittent Synchronization

#### 2.5. Generalized Synchronization

## 3. Parameter Estimation

## 4. Pruned-Enriching Approach

Algorithm 1 Pruned-enriching algorithm |

Require: $\left\{{\mathit{w}}_{n}\right\},\delta ,{\mathit{Q}}^{\left(i\right)},dt,\tau ,M,{T}_{0}$$m\leftarrow 0$ for
$m<M$
do$t\leftarrow 0$ ${\mathit{R}}_{0}^{\left(i\right)}\leftarrow \mathrm{random}\phantom{\rule{4.pt}{0ex}}\mathrm{in}\phantom{\rule{4.pt}{0ex}}[a,b]$ $\mathit{W}\leftarrow \mathrm{Ensemble}\phantom{\rule{4.pt}{0ex}}\mathrm{evolution}\phantom{\rule{4.pt}{0ex}}\mathrm{up}\phantom{\rule{4.pt}{0ex}}\mathrm{to}\phantom{\rule{4.pt}{0ex}}t={T}_{0}\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}\mathrm{store}\phantom{\rule{4.pt}{0ex}}\mathrm{measure}\phantom{\rule{4.pt}{0ex}}\mathrm{every}\phantom{\rule{4.pt}{0ex}}\tau $ for $t<T$ doif $t\propto \tau $ then$\mathit{W}\leftarrow \mathrm{Update}\phantom{\rule{4.pt}{0ex}}\mathrm{with}\phantom{\rule{4.pt}{0ex}}f\left(\mathit{R}\right)$ $d\leftarrow \mathrm{Euclidean}\phantom{\rule{4.pt}{0ex}}\mathrm{distance}\phantom{\rule{4.pt}{0ex}}d({w}_{n},{\mathit{W}}_{n})$ $\mathit{Q}\leftarrow \mathrm{Parameter}\phantom{\rule{4.pt}{0ex}}\mathrm{updating}\phantom{\rule{4.pt}{0ex}}\mathrm{step}$ ▷Algorithm 2 $\tilde{\mathit{Q}}\leftarrow \mathrm{Mean}\phantom{\rule{4.pt}{0ex}}\mathrm{of}\phantom{\rule{4.pt}{0ex}}\mathrm{first}\phantom{\rule{4.pt}{0ex}}R/2\phantom{\rule{4.pt}{0ex}}\mathrm{ensemble}\phantom{\rule{4.pt}{0ex}}\mathrm{elements}$ end if$\mathit{R}\leftarrow \mathrm{Evolution}\phantom{\rule{4.pt}{0ex}}\mathrm{step}\phantom{\rule{4.pt}{0ex}}\mathrm{with}\phantom{\rule{4.pt}{0ex}}\mathrm{time}\phantom{\rule{4.pt}{0ex}}\mathrm{step}\phantom{\rule{4.pt}{0ex}}dt\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}\mathit{Q}\phantom{\rule{4.pt}{0ex}}\mathrm{as}\phantom{\rule{4.pt}{0ex}}\mathrm{parameters}$ $t\leftarrow t+dt$ end forend forreturn
$\tilde{\mathit{Q}}$ |

Algorithm 2 Parameter updating step |

Require: $R,\delta ,d,\mathit{W},\mathit{w}\left(t\right)$$\mathrm{index}\leftarrow \mathrm{argsort}\left(\mathrm{d}\right)$ ▷ return index of d sorted in ascending order for
$i\phantom{\rule{4.pt}{0ex}}\mathrm{in}\phantom{\rule{4.pt}{0ex}}\mathrm{range}(R/2,R)$
do${\mathit{Q}}_{i}\leftarrow {\mathit{Q}}_{j}\phantom{\rule{4.pt}{0ex}}\mathrm{with}\phantom{\rule{4.pt}{0ex}}j\phantom{\rule{4.pt}{0ex}}\mathrm{random}\phantom{\rule{4.pt}{0ex}}\mathrm{integer}\phantom{\rule{4.pt}{0ex}}\mathrm{in}\phantom{\rule{4.pt}{0ex}}(0,R/2)$ if $\mathrm{random}(0,1)<0.5$ then$k\leftarrow \mathrm{random}\phantom{\rule{4.pt}{0ex}}\mathrm{integer}\phantom{\rule{4.pt}{0ex}}\mathrm{in}\phantom{\rule{4.pt}{0ex}}(1,2,3)$ ${\mathit{Q}}_{ik}\leftarrow {\mathit{Q}}_{jk}+\mathrm{random}(-\delta ,\delta )$ while ${\mathit{Q}}_{ik}<0$ do${\mathit{Q}}_{ik}\leftarrow {\mathit{Q}}_{jk}+\mathrm{random}(0,\delta )$ end whileelse${\mathit{R}}_{ik}\leftarrow {\mathit{R}}_{jk}+\mathrm{random}(-\delta ,\delta )$ end ifend forreturn
$\mathit{Q}$ |

## 5. Conclusions and Future Prospects

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Lorenz, E. The Essence of Chaos (Jessie and John Danz Lectures); University of Washington Press: Washington, DC, USA, 1995; p. 240. [Google Scholar]
- Evensen, G.; Vossepoel, F.C.; van Leeuwen, P.J. Data Assimilation Fundamentals: A Unified Formulation of the State and Parameter Estimation Problem (Springer Textbooks in Earth Sciences, Geography and Environment); Springer: Berlin/Heidelberg, Germany, 2022; p. 415. [Google Scholar]
- Geer, A.J. Learning earth system models from observations: Machine learning or data assimilation? Philos. Trans. R. Soc. Math. Phys. Eng. Sci.
**2021**, 379, 20200089. [Google Scholar] [CrossRef] [PubMed] - Bonavita, M.; Alan Geer, P.L.; Massart, S.; Chrust, M. Data Assimilation or Machine Learning? ECMWF Newsletter Number 167; Springer: Berlin/Heidelberg, Germany, 2021; Available online: https://www.ecmwf.int/en/newsletter/167/meteorology/data-assimilation-or-machine-learning (accessed on 13 February 2023).
- Strogatz, S.H. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering (Studies in Nonlinearity); CRC Press: Boca Raton, FL, USA, 1994; p. 512. [Google Scholar]
- Pikovsky, A.; Rosenblum, M.; Kurths, J. Cambridge Nonlinear Science Series: Synchronization: A Universal Concept in Nonlinear Sciences Series Number 12; Cambridge nonlinear science series; Cambridge University Press: Cambridge, UK, 2003. [Google Scholar]
- Bagnoli, F.; Rechtman, R. Synchronization and maximum Lyapunov exponents of cellular automata. Phys. Rev. E
**1999**, 59, R1307–R1310. [Google Scholar] [CrossRef] - Lorenz, E.N. Deterministic Nonperiodic Flow. J. Atmos. Sci.
**1963**, 20, 130–141. [Google Scholar] [CrossRef] - Chua, L.O. The Genesis of Chua’s Circuit. Archiv Eelektrischen Übertragung
**1992**, 46, 250–257. [Google Scholar] - Chua, L.O. A zoo of strange attractors from the canonical Chua’s circuits. In Proceedings of the 35th Midwest Symposium on Circuits and Systems, Washington, DC, USA, 9–12 August 1992; Volume 2, p. 916. [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] - Kiseleva, M.A.; Kudryashova, E.V.; Kuznetsov, N.V.; Kuznetsova, O.A.; Leonov, G.A.; Yuldashev, M.V.; Yuldashev, R.V. Hidden and self-excited attractors in Chua circuit: Synchronization and SPICE simulation. Int. J. Parallel Emergent Distrib. Syst.
**2017**, 33, 513–523. [Google Scholar] [CrossRef] - Zaqueros-Martinez, J.; Rodriguez-Gomez, G.; Tlelo-Cuautle, E.; Orihuela-Espina, F. Fuzzy Synchronization of Chaotic Systems with Hidden Attractors. Entropy
**2023**, 25, 495. [Google Scholar] [CrossRef] [PubMed] - Pecora, L.M.; Carroll, T.L. Synchronization in chaotic systems. Phys. Rev. Lett.
**1990**, 64, 821–824. [Google Scholar] [CrossRef] [PubMed] - Pecora, L.M.; Carroll, T.L. Synchronization of chaotic systems. Chaos: Interdiscip. J. Nonlinear Sci.
**2015**, 25, 097611. [Google Scholar] [CrossRef] [PubMed] - Rössler, O. An equation for continuous chaos. Phys. Lett.
**1976**, 57, 397–398. [Google Scholar] [CrossRef] - Newcomb, R.; Sathyan, S. An RC op amp chaos generator. IEEE Trans. Circuits Syst.
**1983**, 30, 54–56. [Google Scholar] [CrossRef] - May, R.M. Simple mathematical models with very complicated dynamics. Nature
**1976**, 261, 459–467. [Google Scholar] [CrossRef] [PubMed] - Rulkov, N.F.; Sushchik, M.M.; Tsimring, L.S.; Abarbanel, H.D.I. Generalized synchronization of chaos in directionally coupled chaotic systems. Phys. Rev. E
**1995**, 51, 980–994. [Google Scholar] [CrossRef] [PubMed] - Kirkpatrick, S.; Gelatt, C.D.; Vecchi, M.P. Optimization by Simulated Annealing. Science
**1983**, 220, 671–680. [Google Scholar] [CrossRef] [PubMed] - Takens, F. Detecting strange attractors in turbulence. In Lecture Notes in Mathematics; Springer: Berlin/Heidelberg, Germany, 1981; pp. 366–381. [Google Scholar] [CrossRef]
- Packard, N.H.; Crutchfield, J.P.; Farmer, J.D.; Shaw, R.S. Geometry from a Time Series. Phys. Rev. Lett.
**1980**, 45, 712–716. [Google Scholar] [CrossRef] - Rosenbluth, M.N.; Rosenbluth, A.W. Monte Carlo Calculation of the Average Extension of Molecular Chains. J. Chem. Phys.
**1955**, 23, 356–359. [Google Scholar] [CrossRef] - Grassberger, P. Pruned-enriched Rosenbluth method: Simulations of θ polymers of chain length up to 1,000,000. Phys. Rev. E
**1997**, 56, 3682–3693. [Google Scholar] [CrossRef] - Mitchell, M. An Introduction to Genetic Algorithms; MIT Press: London, UK, 1996. [Google Scholar]

**Figure 1.**Conditional Lyapunov maximal exponent for different coupling directions: ${C}_{x}=[1,0,0]$, ${C}_{y}=[0,1,0]$, ${C}_{z}=[0,0,1]$. The black dotted lined marks the zero value while the blue continuous line marks the value of the Lyapunov exponent. In the last one, in orange, we also show the distance d (normalized at its maximum value obtained in the simulation) between the master and slave state variables for different coupling strengths (see also Figure 2).

**Figure 2.**Asymptotic distance d as a function of the coupling strength p for different coupling directions. From left to right: $\mathit{C}=[1,1,1]$, ${\mathit{C}}_{x}=[1,0,0]$, ${\mathit{C}}_{y}=[0,1,0]$.

**Figure 3.**The dependence of the synchronization threshold on the intermittent parameter k such that $\tau =k\Delta t$ for different coupling directions and its linear fit obtained using the first 20 time steps. The other parameters of the simulation are: $dt={10}^{-3}$ and ${k}_{max}=50$ for $\mathit{C}=[1,0,0]$, ${k}_{max}=100$ for the others.

**Figure 4.**Heat map of the state–variable distance d for different values of parameter coupling $D=\pi {D}_{0}$ ($0\le \pi \le 1$) and state–variable coupling p for some state–variable coupling directions C and parameter coupling direction $\chi $. (

**a**) $\mathit{C}=[1,1,1]$, $\chi =[1,1,1]$; (

**b**) ${\mathit{C}}_{x}=[1,0,0]$, $\chi =[1,1,1]$; (

**c**) ${\mathit{C}}_{z}=[0,0,1]$, $\chi =[1,1,1]$;. The line $\pi =0$ corresponds to Figure 2.

**Figure 5.**The state–variable distance d (dashed line) and parameter distance (color) as a function of temperature $\theta $ for $p=0.6\gg {p}_{c}$ and different coupling directions C. (

**a**) $C=[1,1,1]$; (

**b**) $C=[1,0,0]$; (

**c**) $C=[0,1,0]$. We set $\theta =1$, $\u03f5={10}^{-4}$, $T=100$.

**Figure 6.**The schematic of the pruned-enriching method. Lines denotes schematically the trajectories of replicas in the space of state variables and parameters. The dashed lines marks the trajectory of master system. Disks marks the elimination of replicas which are farther from the master one. Black dotted lines marks the pruning and enriching times, and the duplication of replicas are marked by the dashed colored lines with arrows. The variation of the duplication of the nearer replicas is either on one of the state variables or one of parameters.

**Figure 7.**Parameter distance D after M repetitions for different amplitudes $\delta $ with (

**a**) ${\mathit{Q}}^{\left(i\right)}$ randomly initialized in the interval $(0.5,30)$ and (

**b**) ${\mathit{Q}}^{\left(i\right)}$ initialized near the “true” values $\mathit{Q}$ by adding a random noise of amplitude $\u03f5=0.5$. (

**c**) Variance of the distance D for different amplitudes $\delta $ for the last interval. The vertical lines indicate when the ensemble was restored to $t=0$. The ${\mathit{Q}}^{\left(i\right)}$ are initialized as in (

**a**).

**Figure 8.**The distance between variables (black points, right axis) and parameters (color points, left axis) as a function of iterations (repetitions times number of samples of a trajectory) with the pruned-enriching method for 50 randomly selected replicas of the $h=\mathrm{10,000}$ used to estimate the parameters. We used $\mathrm{d}t={10}^{-2}$, $\tau =0.2$ and $T=100$, so the number of samples (time series) of a trajectory is 500, and we show $M=8$ iterations. We consider the situation where only measurements in the coupling direction $C=[1,0,0]$ are available at every $k=20$ temporal steps $\mathrm{d}t$, and we used an embedding space of dimension ${d}_{e}=5$.

**Figure 9.**Estimation (x) and standard deviation (blue area) of the parameters computed using the first half of the ensemble (

**a**) with and (

**b**) without cross-over. The true values are also shown (dotted orange lines).

**Table 1.**Maximum conditional Lyapunov exponent estimated using the asymptotic distance $\left|d\right|$ between master and coupled slave systems (${\lambda}_{\mathrm{sim}}$) and derived from linear fit (${\lambda}_{\mathrm{fit}}$).

$\mathit{C}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}[1,1,1]$ | ${\mathit{C}}_{\mathit{x}}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}[1,0,0]$ | ${\mathit{C}}_{\mathit{y}}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}[0,1,0]$ | |
---|---|---|---|

${\lambda}_{\mathrm{lin}}$ | $0.995$ | $8.840$ | $2.670$ |

${\lambda}_{\mathrm{fit}}$ | $0.961$ | $8.619$ | $2.555$ |

**Table 2.**Estimated parameters obtained using the pruned-enriching algorithm with $\delta $ variable. In the last column, we also show the standard deviation on the ensemble members. Simulation data as in Figure 8.

Real | ens. Mean | $\mathit{\sigma}$ ens | |
---|---|---|---|

$\sigma $ | $10.000000$ | $9.997584$ | $0.062233$ |

$\beta $ | $2.666667$ | $2.664959$ | $0.031623$ |

$\rho $ | $28.000000$ | $28.017719$ | $0.067698$ |

**Table 3.**Estimated parameters obtained using the pruned-enriching algorithm with $\delta $ variable, averaged over $\mathsf{\Omega}=50$ realizations.

Real | Mean | $\mathit{\sigma}$ | |
---|---|---|---|

$\sigma $ | $10.000000$ | $9.999346$ | $0.013472$ |

$\beta $ | $2.666667$ | $2.666094$ | $0.005040$ |

$\rho $ | $28.000000$ | $28.003983$ | $0.042175$ |

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**

Bagnoli, F.; Baia, M.
Synchronization, Control and Data Assimilation of the Lorenz System. *Algorithms* **2023**, *16*, 213.
https://doi.org/10.3390/a16040213

**AMA Style**

Bagnoli F, Baia M.
Synchronization, Control and Data Assimilation of the Lorenz System. *Algorithms*. 2023; 16(4):213.
https://doi.org/10.3390/a16040213

**Chicago/Turabian Style**

Bagnoli, Franco, and Michele Baia.
2023. "Synchronization, Control and Data Assimilation of the Lorenz System" *Algorithms* 16, no. 4: 213.
https://doi.org/10.3390/a16040213