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

Kamila Jarczewska

Kamila Jarczewska

Faculty of Civil Engineering, Wrocław University of Science and Technology, Wyb. Wyspiańskiego 27Wrocław, Poland
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Jarczewska, Kamila
  
Aug 14, 2025
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Article Category: Research Article
Published Online: Aug 14, 2025
Page range: 13 - 26
Received: Mar 06, 2025
Accepted: Jun 24, 2025
DOI: https://doi.org/10.2478/sgem-2025-0017
Keywords
© 2025 Kamila Jarczewska, published by Sciendo
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Figure 1

Decomposition of the multiwavelet transform for three branches of basic wavelet functions.
Decomposition of the multiwavelet transform for three branches of basic wavelet functions.

Figure 2

Basic multiscaling functions of Legendre multiwavelet order k = 3 (a) and basic multiwavelet functions of Legendre multiwavelet order k = 3 (b), and second term multiwavelet packet functions of Legendre multiwavelet order k = 3 for j = 3, from final decomposition stage (c).
Basic multiscaling functions of Legendre multiwavelet order k = 3 (a) and basic multiwavelet functions of Legendre multiwavelet order k = 3 (b), and second term multiwavelet packet functions of Legendre multiwavelet order k = 3 for j = 3, from final decomposition stage (c).

Figure 3

Schematic of the analyzed system.
Schematic of the analyzed system.

Figure 4

Bifurcation diagram (blue) and function of variation of max. Lyapunov exponent (red) for the Duffing oscillator described by equation (17) ( ω \omega = 0.7, c c = 0.1, α \alpha = 0.2, β \beta = 0.53).
Bifurcation diagram (blue) and function of variation of max. Lyapunov exponent (red) for the Duffing oscillator described by equation (17) ( ω \omega = 0.7, c c = 0.1, α \alpha = 0.2, β \beta = 0.53).

Figure 5

Convergence of the highest Lyapunov exponent over time at F = 6.5 (a) and F = 16 (b) for the Duffing oscillator described by equation (17).
Convergence of the highest Lyapunov exponent over time at F = 6.5 (a) and F = 16 (b) for the Duffing oscillator described by equation (17).

Figure 6

Phase trajectory (a) Poincaré cross sections, (b) power spectra (Fourier analysis), (c) for nonchaotic signal (F = 6.5) described by equation (17) with parameters ( ω \omega = 0.7, c c = 0.1, α \alpha = 0.2, β \beta = 0.53).
Phase trajectory (a) Poincaré cross sections, (b) power spectra (Fourier analysis), (c) for nonchaotic signal (F = 6.5) described by equation (17) with parameters ( ω \omega = 0.7, c c = 0.1, α \alpha = 0.2, β \beta = 0.53).

Figure 7

Phase trajectory (a) Poincaré cross section, (b) power spectra (Fourier analysis), (c) for chaotic signal (F = 16) described by equation (17) with parameters ( ω \omega = 0.7, c c = 0.1, α \alpha = 0.2, β \beta = 0.53).
Phase trajectory (a) Poincaré cross section, (b) power spectra (Fourier analysis), (c) for chaotic signal (F = 16) described by equation (17) with parameters ( ω \omega = 0.7, c c = 0.1, α \alpha = 0.2, β \beta = 0.53).

Figure 8

Phase trajectory (a), Poincaré cross sections (b), and power spectra (Fourier analysis) (c) obtained from wavelet packet transform of nonchaotic signal (F = 6.5) described by equation (17).
Phase trajectory (a), Poincaré cross sections (b), and power spectra (Fourier analysis) (c) obtained from wavelet packet transform of nonchaotic signal (F = 6.5) described by equation (17).

Figure 9

Phase trajectory (a), Poincaré cross section (b), and power spectra (Fourier analysis) (c) obtained from wavelet packet transform of chaotic signal (F = 16) described by equation (17).
Phase trajectory (a), Poincaré cross section (b), and power spectra (Fourier analysis) (c) obtained from wavelet packet transform of chaotic signal (F = 16) described by equation (17).

Figure 10

Selected resolution levels j of multiwavelet analysis coefficients of the Duffing oscillator described by equation (17) for nonchaotic signal F = 6.5 (left part of figure) and chaotic signal F = 16 (right part of figure).
Selected resolution levels j of multiwavelet analysis coefficients of the Duffing oscillator described by equation (17) for nonchaotic signal F = 6.5 (left part of figure) and chaotic signal F = 16 (right part of figure).

Figure 11

Multiwavelet expansion coefficients obtained at resolution j = 7 of nonchaotic signal F = 6.5 (a) and chaotic signal F = 16 (b) for the Duffing system described by equation (17) ( ω \omega = 0.7, c c = 0.1, α \alpha = 0.2, β \beta = 0.53).
Multiwavelet expansion coefficients obtained at resolution j = 7 of nonchaotic signal F = 6.5 (a) and chaotic signal F = 16 (b) for the Duffing system described by equation (17) ( ω \omega = 0.7, c c = 0.1, α \alpha = 0.2, β \beta = 0.53).

Figure 12

Multiwavelet expansion coefficients obtained at resolution level j = 7 of nonchaotic signal F = 2.5 F=2.5\hspace{.25em} (a) and chaotic signal F = 1.329 (b) for the Duffing oscillator described by equation (17) ( ω \omega = 3.3, c c = 0.8, α \alpha = 12, β \beta = 100).
Multiwavelet expansion coefficients obtained at resolution level j = 7 of nonchaotic signal F = 2.5 F=2.5\hspace{.25em} (a) and chaotic signal F = 1.329 (b) for the Duffing oscillator described by equation (17) ( ω \omega = 3.3, c c = 0.8, α \alpha = 12, β \beta = 100).

Figure 13

Nonchaotic signal energy (F = 6.5, F = 2.5 – dashed line) and chaotic signal energy (F = 16, F = 1.329 – solid line) versus the number of multiwavelet expansion coefficients for the Duffing oscillator: ( ω \omega = 0.7, c c = 0.1, α \alpha = 0.2, β \beta = 0.53) (a) and ( ω \omega = 3.3, c c = 0.8, α \alpha = 12, β \beta = 100) (b).
Nonchaotic signal energy (F = 6.5, F = 2.5 – dashed line) and chaotic signal energy (F = 16, F = 1.329 – solid line) versus the number of multiwavelet expansion coefficients for the Duffing oscillator: ( ω \omega = 0.7, c c = 0.1, α \alpha = 0.2, β \beta = 0.53) (a) and ( ω \omega = 3.3, c c = 0.8, α \alpha = 12, β \beta = 100) (b).

Figure 14

Selected resolution levels j of multiwavelet packet analysis coefficients for nonchaotic signal F = 6.5 of the Duffing oscillator described by equation (17) using packets of Legender’s multiwavelets k3_2.
Selected resolution levels j of multiwavelet packet analysis coefficients for nonchaotic signal F = 6.5 of the Duffing oscillator described by equation (17) using packets of Legender’s multiwavelets k3_2.

Figure 15

Selected resolution levels j of multiwavelet packet analysis coefficients for chaotic signal F = 16 of the Duffing oscillator described by equation (17) using packets of Legender’s multiwavelets k3_2.
Selected resolution levels j of multiwavelet packet analysis coefficients for chaotic signal F = 16 of the Duffing oscillator described by equation (17) using packets of Legender’s multiwavelets k3_2.

Figure 16

Selected resolution levels j = {4, 5, 6, 7} of wavelet analysis coefficients for nonchaotic signal F = 6.5 (a) and chaotic signal F = 16 (b) of Duffing oscillator described by equation (17) ( ω \omega = 0.7, c c = 0.1, α \alpha = 0.2, β \beta = 0.53).
Selected resolution levels j = {4, 5, 6, 7} of wavelet analysis coefficients for nonchaotic signal F = 6.5 (a) and chaotic signal F = 16 (b) of Duffing oscillator described by equation (17) ( ω \omega = 0.7, c c = 0.1, α \alpha = 0.2, β \beta = 0.53).

Figure 17

Expansion coefficients of wavelet analysis (a) and (b) and packet wavelet analysis (c) and (d), obtained for nonchaotic signal F = 6.5 (a), (c) and chaotic signal F = 16 (b), (d) of Duffing oscillator described by equation (17) ( ω \omega = 0.7, c c = 0.1, α \alpha = 0.2, β \beta = 0.53).
Expansion coefficients of wavelet analysis (a) and (b) and packet wavelet analysis (c) and (d), obtained for nonchaotic signal F = 6.5 (a), (c) and chaotic signal F = 16 (b), (d) of Duffing oscillator described by equation (17) ( ω \omega = 0.7, c c = 0.1, α \alpha = 0.2, β \beta = 0.53).

Figure 18

Bifurcation diagram (a) and variation function of max. Lyapunov exponent (b) for the system described by equation (18) L = 0.5 m , m = 1 kg , k = 100 N m , k 1 = 1,000 N m 3 , ω = 10 rad/s \left(\phantom{\rule[-0.75em]{}{0ex}},L=0.5\hspace{.5em}\text{m},\hspace{.5em}m=1\hspace{.5em}\text{kg},\hspace{.5em}k=\frac{100\hspace{.5em}\text{N}}{m},\hspace{.5em}{k}_{1}=\frac{\mathrm{1,000}\hspace{.5em}\text{N}}{{m}^{3}},\hspace{.5em}\omega =10\hspace{.5em}\text{rad/s}\right) .
Bifurcation diagram (a) and variation function of max. Lyapunov exponent (b) for the system described by equation (18) L = 0.5 m , m = 1 kg , k = 100 N m , k 1 = 1,000 N m 3 , ω = 10 rad/s \left(\phantom{\rule[-0.75em]{}{0ex}},L=0.5\hspace{.5em}\text{m},\hspace{.5em}m=1\hspace{.5em}\text{kg},\hspace{.5em}k=\frac{100\hspace{.5em}\text{N}}{m},\hspace{.5em}{k}_{1}=\frac{\mathrm{1,000}\hspace{.5em}\text{N}}{{m}^{3}},\hspace{.5em}\omega =10\hspace{.5em}\text{rad/s}\right) .

Figure 19

Bifurcation diagram (a)and variation function of max. Lyapunov exponent (b) for the system described by equation (18) L = 0.5 m , m = 1 kg , k = 100 N m , k 1 = 1,000 N m 3 , ω = 15 rad/s \left(\phantom{\rule[-0.75em]{}{0ex}},L=0.5\hspace{.5em}\text{m},\hspace{.5em}m=1\hspace{.5em}\text{kg},\hspace{.5em}k=\frac{100\hspace{.5em}\text{N}}{m},\hspace{.5em}{k}_{1}=\frac{\mathrm{1,000}\hspace{.5em}\text{N}}{{m}^{3}},\hspace{.5em}\omega =15\hspace{.5em}\text{rad/s}\right) .
Bifurcation diagram (a)and variation function of max. Lyapunov exponent (b) for the system described by equation (18) L = 0.5 m , m = 1 kg , k = 100 N m , k 1 = 1,000 N m 3 , ω = 15 rad/s \left(\phantom{\rule[-0.75em]{}{0ex}},L=0.5\hspace{.5em}\text{m},\hspace{.5em}m=1\hspace{.5em}\text{kg},\hspace{.5em}k=\frac{100\hspace{.5em}\text{N}}{m},\hspace{.5em}{k}_{1}=\frac{\mathrm{1,000}\hspace{.5em}\text{N}}{{m}^{3}},\hspace{.5em}\omega =15\hspace{.5em}\text{rad/s}\right) .

Figure 20

Selected resolution levels j of multiwavelet signal analysis coefficients in pre-critical state P = 25 N (a) and in post-critical state P = 30 N (b) for the system described by equation (18) L = 0.5 m , m = 1 kg , k = 100 N m , k 1 = 1,000 N m 3 , ω = 10 rad/s \left(\phantom{\rule[-0.75em]{}{0ex}},L=0.5\hspace{.5em}\text{m},\hspace{.5em}m=1\hspace{.5em}\text{kg},\hspace{.5em}k=\frac{100\hspace{.5em}\text{N}}{m},\hspace{.5em}{k}_{1}=\frac{\mathrm{1,000}\hspace{.5em}\text{N}}{{m}^{3}},\hspace{.25em}\omega =10\hspace{.5em}\text{rad/s}\right) .
Selected resolution levels j of multiwavelet signal analysis coefficients in pre-critical state P = 25 N (a) and in post-critical state P = 30 N (b) for the system described by equation (18) L = 0.5 m , m = 1 kg , k = 100 N m , k 1 = 1,000 N m 3 , ω = 10 rad/s \left(\phantom{\rule[-0.75em]{}{0ex}},L=0.5\hspace{.5em}\text{m},\hspace{.5em}m=1\hspace{.5em}\text{kg},\hspace{.5em}k=\frac{100\hspace{.5em}\text{N}}{m},\hspace{.5em}{k}_{1}=\frac{\mathrm{1,000}\hspace{.5em}\text{N}}{{m}^{3}},\hspace{.25em}\omega =10\hspace{.5em}\text{rad/s}\right) .

Figure 21

Multiwavelet expansion coefficients obtained at resolution level j = 6 in pre-critical state P = 25 N (a) and in post-critical state P = 30 N (b) for the system described by equation (18) L = 0.5 m , m = 1 kg , k = 100 N m , k 1 = 1,000 N m 3 , ω = 10 rad/s \left(\phantom{\rule[-0.75em]{}{0ex}},L=0.5\hspace{.5em}\text{m},\hspace{.5em}m=1\hspace{.5em}\text{kg},\hspace{.5em}k=\frac{100\hspace{.5em}\text{N}}{m},\hspace{.5em}{k}_{1}=\frac{\mathrm{1,000}\hspace{.5em}\text{N}}{{m}^{3}},\hspace{.5em}\omega =10\hspace{.5em}\text{rad/s}\right) .
Multiwavelet expansion coefficients obtained at resolution level j = 6 in pre-critical state P = 25 N (a) and in post-critical state P = 30 N (b) for the system described by equation (18) L = 0.5 m , m = 1 kg , k = 100 N m , k 1 = 1,000 N m 3 , ω = 10 rad/s \left(\phantom{\rule[-0.75em]{}{0ex}},L=0.5\hspace{.5em}\text{m},\hspace{.5em}m=1\hspace{.5em}\text{kg},\hspace{.5em}k=\frac{100\hspace{.5em}\text{N}}{m},\hspace{.5em}{k}_{1}=\frac{\mathrm{1,000}\hspace{.5em}\text{N}}{{m}^{3}},\hspace{.5em}\omega =10\hspace{.5em}\text{rad/s}\right) .

Figure 22

Multiwavelet expansion coefficients obtained at resolution level j = 6 in pre-critical state P = 40 N (a) and in post-critical state P = 55 N (b) for the system described by equation (18) L = 0.5 m , m = 1 kg , k = 100 N m , k 1 = 1,000 N m 3 , ω = 15 rad/s \left(\phantom{\rule[-0.75em]{}{0ex}},L=0.5\hspace{.5em}\text{m},\hspace{.5em}m=1\hspace{.5em}\text{kg},\hspace{.5em}k=\frac{100\hspace{.5em}\text{N}}{m},\hspace{.5em}{k}_{1}=\frac{\mathrm{1,000}\hspace{.5em}\text{N}}{{m}^{3}},\hspace{.5em}\omega =15\hspace{.5em}\text{rad/s}\right) .
Multiwavelet expansion coefficients obtained at resolution level j = 6 in pre-critical state P = 40 N (a) and in post-critical state P = 55 N (b) for the system described by equation (18) L = 0.5 m , m = 1 kg , k = 100 N m , k 1 = 1,000 N m 3 , ω = 15 rad/s \left(\phantom{\rule[-0.75em]{}{0ex}},L=0.5\hspace{.5em}\text{m},\hspace{.5em}m=1\hspace{.5em}\text{kg},\hspace{.5em}k=\frac{100\hspace{.5em}\text{N}}{m},\hspace{.5em}{k}_{1}=\frac{\mathrm{1,000}\hspace{.5em}\text{N}}{{m}^{3}},\hspace{.5em}\omega =15\hspace{.5em}\text{rad/s}\right) .
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