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December  2019, 13(6): 1189-1212. doi: 10.3934/ipi.2019053

Robust parameter estimation of chaotic systems

1. 

Lappeenranta University of Technology, Department of Computational Engineering, Lappeenranta, 53850, Finland

2. 

Finnish Meteorological Institute, Atmospheric Remote Sensing Group, Helsinki, 00560, Finland

3. 

University of Montana, Department of Mathematical Sciences, 59812, Missoula, Department of Mathematical Sciences, Missoula, 59812, Montana

* Corresponding author: Sebastian Springer

Received  August 2018 Revised  May 2019 Published  October 2019

Fund Project: The first author is supported by the CoE, Academy of Finland, decision number 312 122.

Reliable estimation of parameters of chaotic dynamical systems is a long standing problem important in numerous applications. We present a robust method for parameter estimation and uncertainty quantification that requires neither the knowledge of initial values for the system nor good guesses for the unknown model parameters. The method uses a new distance concept recently introduced to characterize the variability of chaotic dynamical systems. We apply it to cases where more traditional methods, such as those based on state space filtering, are no more applicable. Indeed, the approach combines concepts from chaos theory, optimization and statistics in a way that enables solving problems considered as 'intractable and unsolved' in prior literature. We illustrate the results with a large number of chaotic test cases, and extend the method in ways that increase the accuracy of the estimation results.

Citation: Sebastian Springer, Heikki Haario, Vladimir Shemyakin, Leonid Kalachev, Denis Shchepakin. Robust parameter estimation of chaotic systems. Inverse Problems & Imaging, 2019, 13 (6) : 1189-1212. doi: 10.3934/ipi.2019053
References:
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M. A. BeaumontW. Y. Zhang and D. J. Balding, Approximate bayesian computation in population genetics, Genetics, 162 (2002), 2025-2035.   Google Scholar

[2]

S. BorovkovaR. Burton and H. Dehling, Limit theorems for functionals of mixing processes with applications to $U$-statistics and dimension estimation, Transactions of the American Mathematical Society, 353 (2001), 4261-4318.  doi: 10.1090/S0002-9947-01-02819-7.  Google Scholar

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[5]

P. R. ConradY. M. MarzoukN. S. Pillai and A. Smith, Accelerating asymptotically exact MCMC for computationally intensive models via local approximations, Journal of the American Statistical Association, 111 (2016), 1591-1607.  doi: 10.1080/01621459.2015.1096787.  Google Scholar

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G. Coper, Aizawa strange attractor, (2018), http://www.algosome.com/articles/aizawa-attractor-chaos.html. Google Scholar

[7] J. Durbin and S. J. Koopman, Time Series Analysis by State Space Methods, Oxford Statistical Science Series, 38. Oxford University Press, Oxford, 2012.  doi: 10.1093/acprof:oso/9780199641178.001.0001.  Google Scholar
[8]

V. Feoktistov, Differntial Evolution: In Search of Solutions, Springer Optimization and Its Applications, 5. Springer, New York, 2006.  Google Scholar

[9]

R. Genesio and A. Tesi, Harmonic balance methods for the analysis of chaotic dynamics in nonlinear systems, Automatica, 28 (1992), 531-548.  doi: 10.1016/0005-1098(92)90177-H.  Google Scholar

[10]

H. HaarioE. Saksman and J. Tamminen, An adaptive Metropolis algorithm, Bernoulli Society for Mathematical Statistics and Probability, 7 (2001), 223-242.  doi: 10.2307/3318737.  Google Scholar

[11]

H. HaarioM. LaineA. Mira and E. Saksman, DRAM: Efficient adaptive MCMC, Statistics and Computing, 16 (2006), 339-354.  doi: 10.1007/s11222-006-9438-0.  Google Scholar

[12]

H. Haario, L. Kalachev and J. Hakkarainen, Generalized correlation integral vectors: A distance concept for chaotic dynamical systems, Chaos: An Interdisciplinary Journal of Nonlinear Science, 25 (2015), 063102, 10 pp. doi: 10.1063/1.4921939.  Google Scholar

[13]

J. HakkarainenA. IlinA. SolonenM. LaineH. HaarioJ. TamminenE. Oja and H. Järvinen, On closure parameter estimation in chaotic systems, Nonlinear Processes in Geophysics, 19 (2012), 127-143.  doi: 10.5194/npg-19-127-2012.  Google Scholar

[14]

J. Hakkarainen, A. Solonen, A. Ilin, J. Susiluoto, M. Laine, H. Haario and H. Järvinen, A dilemma of the uniqueness of weather and climate model closure parameters, Tellus A: Dynamic Meteorology and Oceanography, 65 (2013), 20147. doi: 10.3402/tellusa.v65i0.20147.  Google Scholar

[15]

H. JärvinenP. RäisänenM. LaineJ. TamminenA. IlinE. OjaA. Solonen and H. Haario, Estimation of ECHAM5 climate model closure parameters with adaptive MCMC, Atmospheric Chemistry and Physics, 10 (2010), 9993-10002.   Google Scholar

[16]

J. H. LüG. R. Chen and S. C. Zhang, Dynamical analysis of a new chaotic attractor, International Journal of Bifurcation and Chaos, 12 (2002), 1001-1015.  doi: 10.1142/S0218127402004851.  Google Scholar

[17] E. Kalnay, Atmospheric Modeling, Data Assimilation and Predictability, Cambridge University Press, 2002.  doi: 10.1017/CBO9780511802270.  Google Scholar
[18]

E. N. Lorenz, Deterministic nonperiodic flow, Journal of the Atmospheric Sciences, 20 (1963), 130-141.   Google Scholar

[19]

E. N. Lorenz, An equation for continuous chaos, Physics Letters A, 57 (1976), 397-398.   Google Scholar

[20]

T. Matsumoto, A chaotic attractor from Chua's circuit, IEEE Transactions on Circuits and Systems, 31 (1984), 1055-1058.  doi: 10.1109/TCS.1984.1085459.  Google Scholar

[21]

T. Matsumoto, Generation of a new three dimension autonomous chaotic attractor and its four wing type, ETASR Engineering, Technology and Applied Science Research, 3 (2013), 352-358.   Google Scholar

[22]

D. McFadden, A method of simulated moments for estimation of discrete response models without numerical integration, Econometrica, 57 (1989), 995-1026.  doi: 10.2307/1913621.  Google Scholar

[23]

J. Meier, Lorenz Mod 2 strange attractor, (2018), http://3d-meier.de/tut19/Seite80.html. Google Scholar

[24]

S. Handrock-MeyerL. V. Kalachev and K. R. Schneider, A method to determine the dimension of long-time dynamics in multi-scale systems, Journal of Mathematical Chemistry, 30 (2001), 133-160.  doi: 10.1023/A:1017960802671.  Google Scholar

[25]

L. PanL. Zhou and D. Q. Li, Synchronization of three-scroll unified chaotic system (TSUCS) and its hyper-chaotic system using active pinning control, Nonlinear Dynamics, 73 (2013), 2059-2071.  doi: 10.1007/s11071-013-0922-8.  Google Scholar

[26]

B. PengB. LiuF. Y. Zhang and L. Wang, Differential evolution algorithm-based parameter estimation for chaotic systems, Chaos, Solitons and Fractals, 39 (2009), 2110-2118.   Google Scholar

[27]

K. V. Price, R. M. Storn and J. A. Lampinen, Differential Evolution: A Practical Approach to Global Optimization, With 1 CD-ROM (Windows, Macintosh and UNIX), Natural Computing Series, Springer-Verlag, Berlin, 2005.  Google Scholar

[28]

S. Rahnamayan, H. R. Tizhoosh and M. M. A. Salama, Opposition-based differential evolution, Advances in Differential Evolution, (2008), 155–171. doi: 10.1007/978-3-540-68830-3_6.  Google Scholar

[29]

J. Rougier, 'Intractable and unsolved': Some thoughts on statistical data assimilation with uncertain static parameters, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 371 (2013), 20120297, 12 pp. doi: 10.1098/rsta.2012.0297.  Google Scholar

[30] S. Särkkä, Bayesian Filtering and Smoothing, Cambridge University Press, 2013.   Google Scholar
[31]

V. Shemyakin and H. Haario, Online identification of large scale chaotic system, Nonlinear Dynamics, 93 (2018), 961-975.  doi: 10.1007/s11071-018-4239-5.  Google Scholar

[32]

A. SolonenP. OllinahoM. LaineH. HaarioJ. Tamminen and H. Järvinen, Asystematic approach to generating n-scroll attractorst, International Society for Bayesian Analysis, 12 (2012), 715-736.   Google Scholar

[33]

R. Storn and K. Price, Differential evolution: A simple and efficient heuristic for global optimization over continuous spaces, Journal of Global Optimization, 11 (1997), 341-359.  doi: 10.1023/A:1008202821328.  Google Scholar

[34]

Z. H. WangY. X. SunB. J. van WykG. Y. Qi and M. A. van Wyk, A 3-D four-wing attractor and its analysis, Brazilian Journal of Physics, 39 (2009), 547-553.   Google Scholar

[35]

W.-H. HoJ.-H. Chou and C.-Y. Guo, Parameter identification of chaotic systems using improved differential evolution algorithm, Nonlinear Dynamics, 61 (2010), 29-41.  doi: 10.1007/s11071-009-9629-2.  Google Scholar

[36]

S. N. Wood, Statistical inference for noisy nonlinear ecological dynamic systems, Nature, 466 (2010), 1102-1104.  doi: 10.1038/nature09319.  Google Scholar

[37]

X. T. Li and M. H. Yin, Parameter estimation for chaotic systems by hybrid differential evolution algorithm and artificial bee colony algorithm, Nonlinear Dynamics, 77 (2014), 61-71.  doi: 10.1007/s11071-014-1273-9.  Google Scholar

[38]

M. E. YalcinJ. A. K. Suykens and J. Vandewalle, Experimental confirmation of 3- and 5-scroll attractors from a generalized Chua's circuit, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 47 (2000), 425-429.  doi: 10.1109/81.841929.  Google Scholar

[39]

G.-Q. ZhongK.-F. Man and G. R. Chen, Asystematic approach to generating n-scroll attractorst, International Journal of Bifurcation and Chaos, 12 (2002), 2907-2915.   Google Scholar

show all references

References:
[1]

M. A. BeaumontW. Y. Zhang and D. J. Balding, Approximate bayesian computation in population genetics, Genetics, 162 (2002), 2025-2035.   Google Scholar

[2]

S. BorovkovaR. Burton and H. Dehling, Limit theorems for functionals of mixing processes with applications to $U$-statistics and dimension estimation, Transactions of the American Mathematical Society, 353 (2001), 4261-4318.  doi: 10.1090/S0002-9947-01-02819-7.  Google Scholar

[3]

U. K. Chakraborty, Computational intelligence, Springer: Verlag, 143 (2008), 338. Google Scholar

[4]

H.-K. Chen and C.-I. Lee, Anti-control of chaos in rigid body motion, Chaos, Solitons and Fractals, 21 (2004), 957-965.  doi: 10.1016/j.chaos.2003.12.034.  Google Scholar

[5]

P. R. ConradY. M. MarzoukN. S. Pillai and A. Smith, Accelerating asymptotically exact MCMC for computationally intensive models via local approximations, Journal of the American Statistical Association, 111 (2016), 1591-1607.  doi: 10.1080/01621459.2015.1096787.  Google Scholar

[6]

G. Coper, Aizawa strange attractor, (2018), http://www.algosome.com/articles/aizawa-attractor-chaos.html. Google Scholar

[7] J. Durbin and S. J. Koopman, Time Series Analysis by State Space Methods, Oxford Statistical Science Series, 38. Oxford University Press, Oxford, 2012.  doi: 10.1093/acprof:oso/9780199641178.001.0001.  Google Scholar
[8]

V. Feoktistov, Differntial Evolution: In Search of Solutions, Springer Optimization and Its Applications, 5. Springer, New York, 2006.  Google Scholar

[9]

R. Genesio and A. Tesi, Harmonic balance methods for the analysis of chaotic dynamics in nonlinear systems, Automatica, 28 (1992), 531-548.  doi: 10.1016/0005-1098(92)90177-H.  Google Scholar

[10]

H. HaarioE. Saksman and J. Tamminen, An adaptive Metropolis algorithm, Bernoulli Society for Mathematical Statistics and Probability, 7 (2001), 223-242.  doi: 10.2307/3318737.  Google Scholar

[11]

H. HaarioM. LaineA. Mira and E. Saksman, DRAM: Efficient adaptive MCMC, Statistics and Computing, 16 (2006), 339-354.  doi: 10.1007/s11222-006-9438-0.  Google Scholar

[12]

H. Haario, L. Kalachev and J. Hakkarainen, Generalized correlation integral vectors: A distance concept for chaotic dynamical systems, Chaos: An Interdisciplinary Journal of Nonlinear Science, 25 (2015), 063102, 10 pp. doi: 10.1063/1.4921939.  Google Scholar

[13]

J. HakkarainenA. IlinA. SolonenM. LaineH. HaarioJ. TamminenE. Oja and H. Järvinen, On closure parameter estimation in chaotic systems, Nonlinear Processes in Geophysics, 19 (2012), 127-143.  doi: 10.5194/npg-19-127-2012.  Google Scholar

[14]

J. Hakkarainen, A. Solonen, A. Ilin, J. Susiluoto, M. Laine, H. Haario and H. Järvinen, A dilemma of the uniqueness of weather and climate model closure parameters, Tellus A: Dynamic Meteorology and Oceanography, 65 (2013), 20147. doi: 10.3402/tellusa.v65i0.20147.  Google Scholar

[15]

H. JärvinenP. RäisänenM. LaineJ. TamminenA. IlinE. OjaA. Solonen and H. Haario, Estimation of ECHAM5 climate model closure parameters with adaptive MCMC, Atmospheric Chemistry and Physics, 10 (2010), 9993-10002.   Google Scholar

[16]

J. H. LüG. R. Chen and S. C. Zhang, Dynamical analysis of a new chaotic attractor, International Journal of Bifurcation and Chaos, 12 (2002), 1001-1015.  doi: 10.1142/S0218127402004851.  Google Scholar

[17] E. Kalnay, Atmospheric Modeling, Data Assimilation and Predictability, Cambridge University Press, 2002.  doi: 10.1017/CBO9780511802270.  Google Scholar
[18]

E. N. Lorenz, Deterministic nonperiodic flow, Journal of the Atmospheric Sciences, 20 (1963), 130-141.   Google Scholar

[19]

E. N. Lorenz, An equation for continuous chaos, Physics Letters A, 57 (1976), 397-398.   Google Scholar

[20]

T. Matsumoto, A chaotic attractor from Chua's circuit, IEEE Transactions on Circuits and Systems, 31 (1984), 1055-1058.  doi: 10.1109/TCS.1984.1085459.  Google Scholar

[21]

T. Matsumoto, Generation of a new three dimension autonomous chaotic attractor and its four wing type, ETASR Engineering, Technology and Applied Science Research, 3 (2013), 352-358.   Google Scholar

[22]

D. McFadden, A method of simulated moments for estimation of discrete response models without numerical integration, Econometrica, 57 (1989), 995-1026.  doi: 10.2307/1913621.  Google Scholar

[23]

J. Meier, Lorenz Mod 2 strange attractor, (2018), http://3d-meier.de/tut19/Seite80.html. Google Scholar

[24]

S. Handrock-MeyerL. V. Kalachev and K. R. Schneider, A method to determine the dimension of long-time dynamics in multi-scale systems, Journal of Mathematical Chemistry, 30 (2001), 133-160.  doi: 10.1023/A:1017960802671.  Google Scholar

[25]

L. PanL. Zhou and D. Q. Li, Synchronization of three-scroll unified chaotic system (TSUCS) and its hyper-chaotic system using active pinning control, Nonlinear Dynamics, 73 (2013), 2059-2071.  doi: 10.1007/s11071-013-0922-8.  Google Scholar

[26]

B. PengB. LiuF. Y. Zhang and L. Wang, Differential evolution algorithm-based parameter estimation for chaotic systems, Chaos, Solitons and Fractals, 39 (2009), 2110-2118.   Google Scholar

[27]

K. V. Price, R. M. Storn and J. A. Lampinen, Differential Evolution: A Practical Approach to Global Optimization, With 1 CD-ROM (Windows, Macintosh and UNIX), Natural Computing Series, Springer-Verlag, Berlin, 2005.  Google Scholar

[28]

S. Rahnamayan, H. R. Tizhoosh and M. M. A. Salama, Opposition-based differential evolution, Advances in Differential Evolution, (2008), 155–171. doi: 10.1007/978-3-540-68830-3_6.  Google Scholar

[29]

J. Rougier, 'Intractable and unsolved': Some thoughts on statistical data assimilation with uncertain static parameters, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 371 (2013), 20120297, 12 pp. doi: 10.1098/rsta.2012.0297.  Google Scholar

[30] S. Särkkä, Bayesian Filtering and Smoothing, Cambridge University Press, 2013.   Google Scholar
[31]

V. Shemyakin and H. Haario, Online identification of large scale chaotic system, Nonlinear Dynamics, 93 (2018), 961-975.  doi: 10.1007/s11071-018-4239-5.  Google Scholar

[32]

A. SolonenP. OllinahoM. LaineH. HaarioJ. Tamminen and H. Järvinen, Asystematic approach to generating n-scroll attractorst, International Society for Bayesian Analysis, 12 (2012), 715-736.   Google Scholar

[33]

R. Storn and K. Price, Differential evolution: A simple and efficient heuristic for global optimization over continuous spaces, Journal of Global Optimization, 11 (1997), 341-359.  doi: 10.1023/A:1008202821328.  Google Scholar

[34]

Z. H. WangY. X. SunB. J. van WykG. Y. Qi and M. A. van Wyk, A 3-D four-wing attractor and its analysis, Brazilian Journal of Physics, 39 (2009), 547-553.   Google Scholar

[35]

W.-H. HoJ.-H. Chou and C.-Y. Guo, Parameter identification of chaotic systems using improved differential evolution algorithm, Nonlinear Dynamics, 61 (2010), 29-41.  doi: 10.1007/s11071-009-9629-2.  Google Scholar

[36]

S. N. Wood, Statistical inference for noisy nonlinear ecological dynamic systems, Nature, 466 (2010), 1102-1104.  doi: 10.1038/nature09319.  Google Scholar

[37]

X. T. Li and M. H. Yin, Parameter estimation for chaotic systems by hybrid differential evolution algorithm and artificial bee colony algorithm, Nonlinear Dynamics, 77 (2014), 61-71.  doi: 10.1007/s11071-014-1273-9.  Google Scholar

[38]

M. E. YalcinJ. A. K. Suykens and J. Vandewalle, Experimental confirmation of 3- and 5-scroll attractors from a generalized Chua's circuit, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 47 (2000), 425-429.  doi: 10.1109/81.841929.  Google Scholar

[39]

G.-Q. ZhongK.-F. Man and G. R. Chen, Asystematic approach to generating n-scroll attractorst, International Journal of Bifurcation and Chaos, 12 (2002), 2907-2915.   Google Scholar

Figure 1.  Example of possible measurements of the states $ (X, Y) $ done beyond the deterministic intervals for the Lorenz 63 chaotic attractor for 64 simulations with slightly randomized initial conditions
Figure 2.  The uncertainty quantification study of the first numerical experiment
Figure 3.  Lorenz 63: Analysis performed after the addition of the second feature vector $ (S, \dot S) $
Figure 4.  The solid line represents to the projection of the solution trajectory onto an $ (X, V) $ plane and the dashed line is the corresponding attractor ($ V = 0 $). The solution is already on the attractor for $ Y $ and $ Z $ variables (the initial values were taken to be $ X(0) = 2.51 $, $ Y(0) = 2.51 $, $ Z(0) = 19.92 $, $ V(0) = -20 $). The selected points on the trajectory projection corresponding to different instants of time are marked with the stars ($ t_1 = 1.0005 $, $ t_2 = 1.0015 $, $ t_3 = 1.0025 $, and $ t_4 = 1.0035 $). To make the results more visual, different scales were used for the $ X $ and $ V $ axes (otherwise the trajectory would have looked like a vertical line drawn to $ V $ axis representing the attractor); as a result, for the same reason of making the illustration more comprehensible, we have drawn the ellipses with main axes $ 1.3\times10^{-4} $ and $ 2 $ instead of circles of radius $ \tilde{R} = 2 $. The corresponding values of $ \frac{1}{K}|\hat{g}_4^0| $ for the chosen time points $ t_1 $, $ t_2 $, $ t_3 $, and $ t_4 $ are $ 13.67 $, $ 5.03 $, $ 1.85 $, and $ 0.68 $, respectively
Figure 5.  Parameter uncertainty quantification for the Wang system
Figure 6.  Chua 7 model: parameters uncertainty quantification
Table 1.  DE settings used for all parameter estimations cases, see Appendix 3
Name Notation Value
Crossover probability $ Cr $ 0.9
Lower bound of scale factor $ F_l $ 0.55
Upper bound of scale factor $ F_h $ 1.1
Randomization factor $ \delta $ 0.001
Generation jumping probability $ Jp $ 0.3
Population size $ SoP $ $ 20 \times $number-of-parameters
Name Notation Value
Crossover probability $ Cr $ 0.9
Lower bound of scale factor $ F_l $ 0.55
Upper bound of scale factor $ F_h $ 1.1
Randomization factor $ \delta $ 0.001
Generation jumping probability $ Jp $ 0.3
Population size $ SoP $ $ 20 \times $number-of-parameters
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