Multiwavelet and multiwavelet packet analysis in qualitative assessment of the chaotic states

[1] Baker, G.L., Gollub, J.P. (1998). Introduction to the Dynamics of Chaotic Systems. Wydawnictwo Naukowe PWN, Warszawa, (in Polish). Baker G.L. Gollub J.P. 1998 Introduction to the Dynamics of Chaotic Systems Wydawnictwo Naukowe PWN Warszawa (in Polish). Search in Google Scholar

[2] Ott, E. (1997). Chaos in Dynamical Systems. Wydawnictwo Naukowo Techniczne, Warszawa, (in Polish). Ott E. 1997 Chaos in Dynamical Systems Wydawnictwo Naukowo Techniczne Warszawa (in Polish). Search in Google Scholar

[3] Łuczko, J. (2008). Regular and Chaotic Vibrations in Nonlinear Mechanical Systems. Kraków University of Technology Publishing House, Kraków, (in Polish). Łuczko J. 2008 Regular and Chaotic Vibrations in Nonlinear Mechanical Systems Kraków University of Technology Publishing House Kraków (in Polish). Search in Google Scholar

[4] Glabisz, W. (2003). Mathematica in Structure Mechanics Problems. Wrocław University of Technology Publishing House, Wrocław, (in Polish). Glabisz W. 2003 Mathematica in Structure Mechanics Problems Wrocław University of Technology Publishing House Wrocław (in Polish). Search in Google Scholar

[5] Awrejcewicz, J. (1997). Secrets of Nonlinear Dynamics. Lódź University of Technology Publishing House, Lódź, (in Polish). Awrejcewicz J. 1997 Secrets of Nonlinear Dynamics Lódź University of Technology Publishing House Lódź (in Polish). Search in Google Scholar

[6] Argyris, J., Faust, G., Hase, M. (1994). An Exploration of Chaos. North-Holland Elsevier, Amsterdam. Argyris J. Faust G. Hase M. 1994 An Exploration of Chaos North-Holland Elsevier Amsterdam Search in Google Scholar

[7] Parker T.S., Chua L.O. (1989). Practical Numerical Algorithms for Chaotic Systems. Springer-Verlag, New York. Parker T.S. Chua L.O. 1989 Practical Numerical Algorithms for Chaotic Systems Springer-Verlag New York Search in Google Scholar

[8] Medio, A., Lines, M. (2001). Nonlinear Dynamics. Cambridge University Press, United Kingdom. Medio A. Lines M. 2001 Nonlinear Dynamics Cambridge University Press United Kingdom Search in Google Scholar

[9] Permann, D., Hamilton, I. (1992). Wavelet Analysis of the time series for the Duffing oscillator: The detection of order within chaos. Physical Review Letters, 69, 2607-2610. Permann D. Hamilton I. 1992 Wavelet Analysis of the time series for the Duffing oscillator: The detection of order within chaos Physical Review Letters 69 2607 2610 Search in Google Scholar

[10] Glabisz, W. (2004). Packet Wavelet Analysis in Mechanics Problems. Wrocław University of Technology Publishing House, Wrocław, (in Polish). Glabisz W. 2004 Packet Wavelet Analysis in Mechanics Problems Wrocław University of Technology Publishing House Wrocław (in Polish). Search in Google Scholar

[11] Staszewski, W.J., Worden, K. (1996). The analysis of chaotic behaviour using fractal and wavelet theory. Proceedings of the International on Nonlinearity, Bifurcation and Chaos, Lódź, Poland, pp. 222-226. Staszewski W.J. Worden K. 1996 The analysis of chaotic behaviour using fractal and wavelet theory Proceedings of the International on Nonlinearity, Bifurcation and Chaos Lódź, Poland pp. 222 226 Search in Google Scholar

[12] Jibing, Z., Hangshan, G., Yinchao, G. (1998). Application of wavelet transform to bifurcation and chaos study. Applied Mathematics and Mechanics, 19, 593-599. Jibing Z. Hangshan G. Yinchao G. 1998 Application of wavelet transform to bifurcation and chaos study Applied Mathematics and Mechanics 19 593 599 Search in Google Scholar

[13] Billings, S.A., Coca, D. (1999). Discrete Wavelet models for identification and Qualitative analysis of chaotic systems. International Journal of Bifurcation and Chaos, 9(7), 1263-1284. Billings S.A. Coca D. 1999 Discrete Wavelet models for identification and Qualitative analysis of chaotic systems International Journal of Bifurcation and Chaos 9 7 1263 1284 Search in Google Scholar

[14] Nakao, H., Mishiro, T., Yamada, M. (2001). Visualization of correlation cascade in spatiotemporal chaos using wavelets. International Journal of Bifurcation and Chaos, 11(5), 1483-1493. Nakao H. Mishiro T. Yamada M. 2001 Visualization of correlation cascade in spatiotemporal chaos using wavelets International Journal of Bifurcation and Chaos 11 5 1483 1493 Search in Google Scholar

[15] Constantine, W.L.B., Reinhall, P.G. (2001). Wavelet based in band denoising technique for chaotic sequences. International Journal of Bifurcation and Chaos, 11(2), 483-495. Constantine W.L.B. Reinhall P.G. 2001 Wavelet based in band denoising technique for chaotic sequences International Journal of Bifurcation and Chaos 11 2 483 495 Search in Google Scholar

[16] Mastroddi, F., Bettoli, A. (1999). Wavelet analysis for Hopf bifurcations whit aeroelastic applications. Journal of Sound and Vibration, 228, 199-210. Mastroddi F. Bettoli A. 1999 Wavelet analysis for Hopf bifurcations whit aeroelastic applications Journal of Sound and Vibration 228 199 210 Search in Google Scholar

[17] Shi, Z., Yang, X., Li, Y., Yu, G. (2021). Wavelet-based Synchroextracting Transform: An effective TFA tool for machinery fault diagnosis. Control Engineering Practice, 144, 104884. Shi Z. Yang X. Li Y. Yu G. 2021 Wavelet-based Synchroextracting Transform: An effective TFA tool for machinery fault diagnosis Control Engineering Practice 144 104884 Search in Google Scholar

[18] Varanis, M., Balthazar, J.M., Tusset, A., Ribeiro, M.A., De Oliveira, C. (2024). Signal analysis in chaotic systems: A comprehensive assessment through time-frequency analysis. New Insights on Oscillators and Their Applications to Engineering and Science, IntechOpen, London, UK. doi: 10.5772/intechopen.111087. Varanis M. Balthazar J.M. Tusset A. Ribeiro M.A. De Oliveira C. 2024 Signal analysis in chaotic systems: A comprehensive assessment through time-frequency analysis New Insights on Oscillators and Their Applications to Engineering and Science IntechOpen London, UK 10.5772/intechopen.111087 Open DOISearch in Google Scholar

[19] Alpert, B.K, Beylkin, G., Gines, D., Vozovoi, L. (2002). Adaptive solution of partial differential equations in multiwavelet bases. Journal of Computational Physics, 182, 149-190. Alpert B.K Beylkin G. Gines D. Vozovoi L. 2002 Adaptive solution of partial differential equations in multiwavelet bases Journal of Computational Physics 182 149 190 Search in Google Scholar

[20] Alpert, B.K. (1993). A class of bases in L2 for the sparse representation of integral operators. SIAM Journal on Mathematical Analysis, 24, 246-262. Alpert B.K. 1993 A class of bases in L2 for the sparse representation of integral operators SIAM Journal on Mathematical Analysis 24 246 262 Search in Google Scholar

[21] Strela, V. (1996). Multiwavelts: Theory and Applications. (PhD thesis). Massachusetts Institute of Technology, Cambridge, Massachusetts, US. Strela V. 1996 Multiwavelts: Theory and Applications PhD thesis Massachusetts Institute of Technology Cambridge, Massachusetts, US Search in Google Scholar

[22] Chui, C.K., Lian J. (1996). A study on orthonormal multiwavelets. Applied Numerical Mathematics, 20, 273-298. Chui C.K. Lian J. 1996 A study on orthonormal multiwavelets Applied Numerical Mathematics 20 273 298 Search in Google Scholar

[23] Fann, G., Beylkin, G., Harrison, R.J., Jordan K.E. (2004). Singular operators in multiwavelets bases. IBM Journal of Research and Development, 48(2), 161-171. Fann G. Beylkin G. Harrison R.J. Jordan K.E. 2004 Singular operators in multiwavelets bases IBM Journal of Research and Development 48 2 161 171 Search in Google Scholar

[24] Keinert, F. (2004). Wavelets and Multiwavelets. Chapman & Hall/CRC Press Company, Boca Raton, Florida. Keinert F. 2004 Wavelets and Multiwavelets Chapman & Hall/CRC Press Company Boca Raton, Florida Search in Google Scholar

[25] Averbuch, A., Israeli, M., Vozovoi, L. (1999). Solution of time-dependent diffusion equations with variable coefficients using multiwavelets. Journal of Computational Physics, 150, 394-424. Averbuch A. Israeli M. Vozovoi L. 1999 Solution of time-dependent diffusion equations with variable coefficients using multiwavelets Journal of Computational Physics 150 394 424 Search in Google Scholar

[26] Holmes, P. (1979). Nonlinear oscillator with a strange attractore. Philosophical Transactions of the Royal Society of London, 294, 419-448. Holmes P. 1979 Nonlinear oscillator with a strange attractore Philosophical Transactions of the Royal Society of London 294 419 448 Search in Google Scholar

[27] Holmes, P., Moon, F.C. (1983). Strange attractors and chaos in nonlinear mechanics. Journal of Applide Mechanics, 50, 1021-1032. Holmes P. Moon F.C. 1983 Strange attractors and chaos in nonlinear mechanics Journal of Applide Mechanics 50 1021 1032 Search in Google Scholar

[28] Koruba, Z., Osiecki, J. (2007). Elements of Advanced Mechanics. Świętokrzyski University of Technology Publishing House, Kielce, (in Polish). Koruba Z. Osiecki J. 2007 Elements of Advanced Mechanics Świętokrzyski University of Technology Publishing House Kielce (in Polish). Search in Google Scholar

[29] Napiórkowska-Ałykow, M., Glabisz, W. (2005). Parametric identification procedure based on the Walsh wavelet packet approach for estimation of signal function derivatives. Archives of Civil and Mechanical Engineering, 5(4), 5-26. Napiórkowska-Ałykow M. Glabisz W. 2005 Parametric identification procedure based on the Walsh wavelet packet approach for estimation of signal function derivatives Archives of Civil and Mechanical Engineering 5 4 5 26 Search in Google Scholar