Uneingeschränkter Zugang

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
Suche nach diesem Autor auf
Sciendo|Google Scholar
Jarczewska, Kamila
  
14. Aug. 2025
Multiwavelet and multiwavelet packet analysis in qualitative assessment of the chaotic states's Cover Image
Über diesen Artikel
Vorheriger Artikel
Nächster Artikel
Zitieren
COVER HERUNTERLADEN
Artikel-Kategorie: Research Article
Online veröffentlicht: 14. Aug. 2025
Seitenbereich: 13 - 26
Eingereicht: 06. März 2025
Akzeptiert: 24. Juni 2025
DOI: https://doi.org/10.2478/sgem-2025-0017
Schlüsselwörter
© 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) .
Sprache:
Englisch
Zeitrahmen der Veröffentlichung:
4 Hefte pro Jahr
Fachgebiete der Zeitschrift:
Geowissenschaften, Geowissenschaften, andere, Materialwissenschaft, Verbundwerkstoffe, Poröse Materialien, Physik, Mechanik und Fluiddynamik
Zeitschrift RSS Feed