Figure 1
Decomposition of the multiwavelet transform for three branches of basic wavelet functions.
Source: Author’s contribution.
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).
Source: Author’s contribution.
Figure 3
Schematic of the analyzed system.
Source: Author’s contribution.
Figure 4
Bifurcation diagram (blue) and function of variation of max. Lyapunov exponent (red) for the Duffing oscillator described by equation (17) ( = 0.7, = 0.1, = 0.2, = 0.53).
Source: Author’s contribution.
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).
Source: Author’s contribution.
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 ( = 0.7, = 0.1, = 0.2, = 0.53).
Source: Author’s contribution.
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 ( = 0.7, = 0.1, = 0.2, = 0.53).
Source: Author’s contribution.
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).
Source: Author’s contribution.
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).
Source: Author’s contribution.
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).
Source: Author’s contribution.
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) ( = 0.7, = 0.1, = 0.2, = 0.53).
Source: Author’s contribution.
Figure 12
Multiwavelet expansion coefficients obtained at resolution level j = 7 of nonchaotic signal (a) and chaotic signal F = 1.329 (b) for the Duffing oscillator described by equation (17) ( = 3.3, = 0.8, = 12, = 100).
Source: Author’s contribution.
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: ( = 0.7, = 0.1, = 0.2, = 0.53) (a) and ( = 3.3, = 0.8, = 12, = 100) (b).
Source: Author’s contribution.
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.
Source: Author’s contribution.
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.
Source: Author’s contribution.
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) ( = 0.7, = 0.1, = 0.2, = 0.53).
Source: Author’s contribution.
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) ( = 0.7, = 0.1, = 0.2, = 0.53).
Source: Author’s contribution.
Figure 18
Bifurcation diagram (a) and variation function of max. Lyapunov exponent (b) for the system described by equation (18) .
Source: Author’s contribution.
Figure 19
Bifurcation diagram (a)and variation function of max. Lyapunov exponent (b) for the system described by equation (18) .
Source: Author’s contribution.
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) .
Source: Author’s contribution.
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) .
Source: Author’s contribution.
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) .
Source: Author’s contribution.