This study focuses on the qualitative and quantitative characterization of chaotic systems with the use of a symbolic description. We consider two famous systems, Lorenz and Rössler models with their iconic attractors, and demonstrate that with adequately chosen symbolic partition, three measures of complexity, such as the Shannon source entropy, the Lempel–Ziv complexity, and the Markov transition matrix, work remarkably well for characterizing the degree of chaoticity and precise detecting stability windows in the parameter space. The second message of this study is to showcase the utility of symbolic dynamics with the introduction of a fidelity test for reservoir computing for simulating the properties of the chaos in both models’ replicas. The results of these measures are validated by the comparison approach based on one-dimensional return maps and the complexity measures.
REFERENCES
1.
H. D.
Abarbanel
, R.
Brown
, J. J.
Sidorowich
, and L. S.
Tsimring
, “The analysis of observed chaotic data in physical systems
,” Rev. Mod. Phys.
65
, 1331
(1993
). 2.
H.
Kantz
and T.
Schreiber
, Nonlinear Time Series Analysis
(Cambridge University Press
, 2004
), Vol. 7.3.
E.
Bradley
and H.
Kantz
, “Nonlinear time-series analysis revisited
,” Chaos
25
, 097610
(2015
). 4.
5.
V.
Anishchenko
, T.
Vadivasova
, G.
Okrokvertskhov
, and G.
Strelkova
, “Correlation analysis of dynamical chaos
,” Physica A
325
, 199
–212
(2003
). 6.
V.
Anishchenko
, T.
Vadivasova
, G.
Okrokvertskhov
, and G.
Strelkova
, “Statistical properties of dynamical chaos
,” Phys.-Usp.
48
, 151
(2005
). 7.
V. S.
Anishchenko
, T. E.
Vadivasova
, A. S.
Kopeikin
, J.
Kurths
, and G. I.
Strelkova
, “Effect of noise on the relaxation to an invariant probability measure of nonhyperbolic chaotic attractors
,” Phys. Rev. Lett.
87
, 054101
(2001
). 8.
V. S.
Anishchenko
, T. E.
Vadivasova
, J.
Kurths
, G. A.
Okrokvertskhov
, and G. I.
Strelkova
, “Autocorrelation function and spectral linewidth of spiral chaos in a physical experiment
,” Phys. Rev. E
69
, 036215
(2004
). 9.
V. S.
Afraimovich
, V. V.
Bykov
, and L. P.
Shilnikov
, “The origin and structure of the Lorenz attractor
,” Sov. Phys. Dokl.
22
, 253
–255
(1977
).10.
V. S.
Afraimovich
and L. P.
Shilnikov
, “Strange attractors and quasiattractors,” in Nonlinear Dynamics and Turbulence (Pitman, Boston, MA, 1983).11.
V. S.
Afraimovich
, V. V.
Bykov
, and L. P.
Shilnikov
, “On structurally unstable attracting limit sets of Lorenz attractor type
,” Trans. Moscow Math. Soc.
44
, 153
–216
(1983
).12.
L. P.
Shilnikov
, “Bifurcation theory and turbulence,” in Methods of Qualitative Theory of Differential Equations (Gorky State University, 1986), pp. 150–163.13.
V.
Bykov
and A.
Shilnikov
, “On the boundaries of the domain of existence of the Lorenz attractor,” in Methods of Qualitative Theory and Theory of Bifurcations (Gorky State University, Gorky, 1989), pp. 151–159.14.
S. V.
Gonchenko
, L. P.
Shil’nikov
, and D. V.
Turaev
, “On models with non-rough Poincaré homoclinic curves
,” Physica D
62
(1–4
), 1
–14
(1993
). 15.
O. E.
Rössler
, “An equation for continuous chaos
,” Phys. Lett. A
57
, 397
–398
(1976
). 16.
O. E.
Rössler
, “Continuous chaos four prototype equations
,” Ann. N. Y. Acad. Sci.
316
, 376
–392
(1979
). 17.
L. P.
Shilnikov
, “A case of the existence of a denumerable set of periodic motions
,” Dokl. Akad. Nauk
160
, 558
–561
(1965
).18.
V. S.
Afraimovich
, S. V.
Gonchenko
, L. M.
Lerman
, A. L.
Shilnikov
, and D. V.
Turaev
, “Scientific heritage of L. P. Shilnikov
,” Regul. Chaotic Dyn.
19
, 435
–460
(2014
). 19.
S.
Malykh
, Y.
Bakhanova
, A.
Kazakov
, K.
Pusuluri
, and A. L.
Shilnikov
, “Homoclinic chaos in the Rössler model
,” Chaos
30
, 113126
(2020
). 20.
D. V.
Turaev
and L. P.
Shil’nikov
, “An example of a wild strange attractor
,” Sb. Math.
189
, 291
(1998
). 21.
D.
Turaev
and L. P.
Shilnikov
, “Pseudohyperbolicity and the problem on periodic perturbations of Lorenz-type attractors,” in Doklady Mathematics (SP MAIK Nauka/Interperiodica, 2008), Vol. 77, pp. 17–21.22.
S.
Gonchenko
, L. P.
Shil’nikov
, and D.
Turaev
, “Quasiattractors and homoclinic tangencies
,” Comput. Math. Appl.
34
, 195
–227
(1997
). 23.
S.
Gonchenko
, A.
Kazakov
, and D.
Turaev
, “Wild pseudohyperbolic attractor in a four-dimensional Lorenz system
,” Nonlinearity
34
, 2018
(2021
). 24.
C. E.
Shannon
, “A mathematical theory of communication
,” Bell Syst. Tech. J.
27
, 379
–423
(1948
). 25.
W.
Ebeling
and G.
Nicolis
, “Word frequency and entropy of symbolic sequences: A dynamical perspective
,” Chaos, Solitons Fractals
2
, 635
–650
(1992
). 26.
W.
Ebeling
, “Prediction and entropy of nonlinear dynamical systems and symbolic sequences with LRO
,” Physica D
109
, 42
–52
(1997
). 27.
J.-P.
Eckmann
and D.
Ruelle
, “Ergodic theory of chaos and strange attractors,” in The Theory of Chaotic Attractors (Springer, 1985), pp. 273–312.28.
Y.
Sinai
, “Kolmogorov-Sinai entropy
,” Scholarpedia
4
, 2034
(2009
), revision #91407. 29.
V.
Basios
and D.
Mac Kernan
, “Symbolic dynamics, coarse graining and the monitoring of complex systems
,” Int. J. Bifurcation Chaos
21
, 3465
–3475
(2011
). 30.
H.
Herzel
, A.
Schmitt
, and W.
Ebeling
, “Finite sample effects in sequence analysis
,” Chaos, Solitons Fractals
4
, 97
–113
(1994
). 31.
P.
Grassberger
, “Finite sample corrections to entropy and dimension estimates
,” Phys. Lett. A
128
, 369
–373
(1988
). 32.
J.
Ziv
and A.
Lempel
, “Compression of individual sequences via variable-rate coding
,” IEEE Trans. Inf. Theor.
24
, 530
–536
(1978
). 33.
D.
Ruelle
, “An inequality for the entropy of differentiable maps
,” Bull. Braz. Math. Soc.
9
, 83
–87
(1978
). 34.
T.
Xing
, R.
Barrio
, and A.
Shilnikov
, “Symbolic quest into homoclinic chaos
,” Int. J. Bifurcation Chaos
24
, 1440004
(2014
). 35.
K.
Pusuluri
and A.
Shilnikov
, “Homoclinic chaos and its organization in a nonlinear optics model
,” Phys. Rev. E
98
, 040202
(2018
). 36.
K.
Pusuluri
, H. G. E.
Meijer
, and A. L.
Shilnikov
, “Homoclinic puzzles and chaos in a nonlinear laser model
,” J. Commun. Nonlinear Sci. Numer. Simul.
93
, 105503
(2021
).37.
T.
Xing
, K.
Pusuluri
, and A. L.
Shilnikov
, “Ordered intricacy of Shilnikov saddle-focus homoclinics in symmetric systems
,” J. Chaos
31
, 073143
(2021
).38.
H.
Jaeger
, “The ‘echo state’ approach to analysing and training recurrent neural networks—With an erratum note,” German National Research Center for Information Technology (GMD) Technical Report No. 148, Bonn, 2001.39.
B.
Schrauwen
, D.
Verstraeten
, and J.
Campenhout
, “An overview of reservoir computing: Theory, applications and implementations,” in 15th European Symposium on Artificial Neural Networks (ESANN 2007), Bruges, Belgium, 25–27 April 2007 (dblp, 2007), pp. 471–482.40.
J.
Pathak
, B.
Hunt
, M.
Girvan
, Z.
Lu
, and E.
Ott
, “Model-free prediction of large spatiotemporally chaotic systems from data: A reservoir computing approach
,” Phys. Rev. Lett.
120
, 024102
(2018
). 41.
J.
Pathak
, Z.
Lu
, B. R.
Hunt
, M.
Girvan
, and E.
Ott
, “Using machine learning to replicate chaotic attractors and calculate Lyapunov exponents from data
,” Chaos
27
, 121102
(2017
). 42.
M.
Lukoševičius
, “A practical guide to applying echo state networks,” in Neural Networks: Tricks of the Trade (Springer, 2012), pp. 659–686.43.
J.
Scully
, Fidelity test toolkit (2021), GitHub. https://github.com/jamesjscully/rossleresn44.
C.
Rackauckas
and Q.
Nie
, “Differentialequations.jl—A performant and feature-rich ecosystem for solving differential equations in Julia
,” J. Open Res. Softw.
5
, 15
(2017
). 45.
J.
Bezanson
, A.
Edelman
, S.
Karpinski
, and V. B.
Shah
, “Julia: A fresh approach to numerical computing
,” SIAM Rev.
59
, 65
–98
(2017
). 46.
F.
Martinuzzi
, “Reservoircomputing.jl”; see https://github.com/SciML/ReservoirComputing.jl.© 2021 Author(s). Published under an exclusive license by AIP Publishing
2021
Author(s)
You do not currently have access to this content.
