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ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

A Computational and Graphical Approach to Analyze the Dynamic Wavelet Correlation among Components of a Nonlinear Dynamical System

Journal of Applied Nonlinear Dynamics 12(4) (2023) 757--766 | DOI:10.5890/JAND.2023.12.009

Josu\'e M. Polanco-Mart\'inez

GECOS, IME, University of Salamanca, 37007 Salamanca, Spain

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Abstract

An innumerable number of phenomena that take place in nature can be represented as dynamical systems, which in many cases are not linear. One of the common tasks performed in the study of these systems is to analyse through time and frequency the relationships among their components. In this work, we present, discuss, and extend for the first time in the study of nonlinear dynamical systems, a mathematical and computational tool, the wavelet local multiple correlation (WLMC) to analyse quantitatively and visually the behaviour among components of nonlinear dynamical system. The Lorenz system is used as a case study. The WLMC analysis presented shows that the WLMC is able to capture the most relevant periodic and chaotic dynamics of the Lorenz system as well as the ``dominant'' components of this dynamical system. These results confirm that the WLMC is an adequate mathematical tool to analyse nonlinear and chaotic dynamical systems with multiple components.

Acknowledgments

The author acknowledges to the SEPE (Spanish Public Service of Employment) and the Junta de Castilla y Le\'on, and the European Regional Development Fund (Grant CLU-2019-03) for partial funding support.

References

  1. [1]  Janjarasjitt, S. and Loparo, K. (2008), An approach for characterizing coupling in dynamical systems, Physica D: Nonlinear Phenomena, 237, 2482-2486.
  2. [2]  Soldatenko, S. and Chichkine, D. (2014), Correlation and spectral properties of a coupled nonlinear dynamical system in the context of numerical weather prediction and climate modeling, Discrete Dynamics in Nature and Society, 2014, 1-16.
  3. [3]  Hramov, A. and Koronovskii, A. (2004), An approach to chaotic synchronization, Chaos: An Interdisciplinary Journal of Nonlinear Science, 14, 603-610.
  4. [4]  Wang, Q., Shen, Y., and Zhang, J. (2005), A nonlinear correlation measure for multivariable data set, Physica D: Nonlinear Phenomena, 200, 287-295.
  5. [5]  M{\"u}ller, M., Baier, G., Galka, A., Stephani, U., and Muhle, H. (2005), Detection and characterization of changes of the correlation structure in multivariate time series, Physical Review E, 71, 046116.
  6. [6]  M{\"u}ller, M., Baier, G., Rummel, C., and Schindler, K. (2008), Estimating the strength of genuine and random correlations in non-stationary multivariate time series, EPL (Europhysics Letters), 84, 10009.
  7. [7]  Rummel, C., M{\"u}ller, M., Baier, G., Amor, F., and Schindler, K. (2010), Analyzing spatio-temporal patterns of genuine cross-correlations, Journal of Neuroscience Methods, 191, 94-100.
  8. [8]  Groth, A. and Ghil, M. (2011), Multivariate singular spectrum analysis and the road to phase synchronization, Physical Review E, 84, 036206.
  9. [9]  Torrence, C. and Compo, G. (1998), A practical guide to wavelet analysis, Bulletin of the American Meteorological Society, 79, 61-78.
  10. [10]  Staszewski, W. and Worden, K. (1999), Wavelet analysis of time-series: coherent structures, chaos and noise, International Journal of Bifurcation and Chaos, 9, 455-471.
  11. [11]  Torrence, C. and Webster, P. (1999), Interdecadal changes in the ENSO-monsoon system, Journal of Climate, 12, 2679-2690.
  12. [12]  Staszewski, W. (2000), Analysis of non-linear systems using wavelets, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 214, 1339-1353.
  13. [13]  Koronovskii, A. and Hramov, A. (2004), Wavelet transform analysis of the chaotic synchronization of dynamical systems, Journal of Experimental and Theoretical Physics Letters, 79, 316-319.
  14. [14]  Hramov, A. and Koronovskii, A. (2005), Time scale synchronization of chaotic oscillators, Physica D: Nonlinear Phenomena, 206, 252-264.
  15. [15]  Maraun, D. and Kurths, J. (2004), Cross wavelet analysis: significance testing and pitfalls, Nonlinear Processes in Geophysics, 11, 505-514.
  16. [16]  Grinsted, A., Moore, J., and Jevrejeva, S. (2004), Application of the cross wavelet transform and wavelet coherence to geophysical time series, Nonlinear Processes in Geophysics, 11, 561-566.
  17. [17]  Cazelles, B., Chavez, M., Berteaux, D., M{e}nard, F., Vik, J., Jenouvrier, S., and Stenseth, N. (2008), Wavelet analysis of ecological time series, Oecologia, 156, 287-304.
  18. [18]  Polanco, J., Ganzedo, U., S{a}enz, J., Caballero-Alfonso, A., and Castro-Hern{a}ndez, J. (2011), Wavelet analysis of correlation among Canary Islands octopus captures per unit effort, sea-surface temperatures and the North Atlantic Oscillation, Fisheries Research, 107, 177-183.
  19. [19]  Polanco-Mart{\i}nez, J. (2019), Dynamic relationship analysis between NAFTA stock markets using nonlinear, nonparametric, non-stationary methods, Nonlinear Dynamics, 97, 369-389.
  20. [20]  Aguiar-Conraria, L. and Soares, M. (2014), The continuous wavelet transform: moving beyond uni-and bivariate analysis, Journal of Economic Surveys, 28, 344-375.
  21. [21]  Soon, W., et~al. (2014), A review of Holocene solar-linked climatic variation on centennial to millennial timescales: Physical processes, interpretative frameworks and a new multiple cross-wavelet transform algorithm, Earth-Science Reviews, 134, 1-15.
  22. [22]  Hu, W. and Si, B. (2016), Multiple wavelet coherence for untangling scale-specific and localized multivariate relationships in geosciences, Hydrology and Earth System Sciences, 20, 3183-3191.
  23. [23]  Velasco-Herrera, V., Soon, W., Herrera, G.~V., Traversi, R., and Horiuchi, K. (2017), Generalization of the cross-wavelet function, New Astronomy, 56, 86-93.
  24. [24]  Chavez, M. and Cazelles, B. (2019), Detecting dynamic spatial correlation patterns with generalized wavelet coherence and non-stationary surrogate data, Scientific Reports, 9, 7389.
  25. [25]  Oygur, T. and Unal, G. (2021), Vector wavelet coherence for multiple time series, International Journal of Dynamics and Control, 9, 403-409.
  26. [26]  Fern{a}ndez-Macho, J. (2018), Time-localized wavelet multiple regression and correlation, Physica A: Statistical Mechanics and its Applications, 492, 1226-1238.
  27. [27]  Fern{a}ndez-Macho, J. (2012), Wavelet multiple correlation and cross-correlation: A multiscale analysis of Eurozone stock markets, Physica A: Statistical Mechanics and its Applications, 391, 1097-1104.
  28. [28]  Percival, D. and Walden, A. (2000), Wavelet methods for time series analysis, vol.~4. Cambridge University Press, reprinted 2006.
  29. [29]  Polanco-Mart{i}nez, J., Abadie, L., and Fern{a}ndez-Macho, J. (2018), A multi-resolution and multivariate analysis of the dynamic relationships between crude oil and petroleum-product prices, Applied Energy, 228, 1550-1560.
  30. [30]  Polanco-Mart{i}nez, J., Fern{a}ndez-Macho, J., and Medina-Elizalde, M. (2020), Dynamic wavelet correlation analysis for multivariate climate time series, Scientific Reports, 10, 1-11.
  31. [31]  Fern{a}ndez-Macho, J. (2018), wavemulcor: Wavelet routines for global and local multiple correlation, R package version 2.2.1.
  32. [32]  Fernandez-Macho, J. (2019), Package wavemulcor: Wavelet multiple regression and correlation in R, Computing in Science $\&$ Engineering, 21, 63-73.
  33. [33]  Daubechies, I. (1992), Ten lectures on wavelets, vol.~61. SIAM - Society for Industrial and Applied Mathematics.
  34. [34]  Gen\c{c}ay, R., Sel\c{c}uk, F., and Whitcher, B. (2002), An Introduction to Wavelets and Other Filtering Methods in Finance and Economics, Academic Press.
  35. [35]  Polanco-Mart{\i}nez, J., Fern{a}ndez-Macho, J., Neumann, M., and Faria, S. (2018), A pre-crisis vs. crisis analysis of peripheral {EU} stock markets by means of wavelet transform and a nonlinear causality test, Physica A: Statistical Mechanics and its Applications, 490, 1211-1227.
  36. [36]  Polanco-Mart{i}nez, J. (2020), RolWinMulCor: an R package for estimating rolling window multiple correlation in ecological time series, Ecological Informatics, 60, 101163.
  37. [37]  Lorenz, E. (1963), Deterministic nonperiodic flow, Journal of the Atmospheric Sciences, 20, 130-141.
  38. [38]  Saltzman, B. (1962), Finite amplitude free convection as an initial value problem\--{I}, Journal of the Atmospheric Sciences, 19, 329-341.
  39. [39]  Rayleigh, L. (1916), LIX. On convection currents in a horizontal layer of fluid, when the higher temperature is on the under side, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 32, 529-546.
  40. [40]  Nijmeijer, H. (2001), A dynamical control view on synchronization, Physica D: Nonlinear Phenomena, 154, 219-228.
  41. [41]  Festa, R., Mazzino, A., and Vincenzi, D. (2002), Lorenz-like systems and classical dynamical equations with memory forcing: An alternate point of view for singling out the origin of chaos, Physical Review E, 65, 046205.
  42. [42]  Garcia, C.A. (2022), nonlinearTseries: Nonlinear Time Series Analysis, R package version 0.2.12.
  43. [43]  Zhang, T., Ma, G., and Liu, G. (2015), Nonlinear joint dynamics between prices of crude oil and refined products, Physica A: Statistical Mechanics and its Applications, 419, 444-456.
  44. [44]  Zadourian, R. and Grassberger, P. (2017), Asymmetry of cross-correlations between intra-day and overnight volatilities, EPL (Europhysics Letters), 118, 18004.
  45. [45]  Yuan, N., Xoplaki, E., Zhu, C., and Luterbacher, J. (2016), A novel way to detect correlations on multi-time scales, with temporal evolution and for multi-variables, Scientific Reports, 6, 27707.
  46. [46]  Mihanovi{c}, H., Orli{c}, M., and Pasari{c}, Z. (2009), Diurnal thermocline oscillations driven by tidal flow around an island in the Middle Adriatic, Journal of Marine Systems, 78, S157-S168.
  47. [47]  Ng, E. and Chan, J. (2012), Geophysical applications of partial wavelet coherence and multiple wavelet coherence, Journal of Atmospheric and Oceanic Technology, 29, 1845-1853.

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