Signal Analysis in Chaotic Systems: A Comprehensive Assessment through Time-Frequency Analysis

Abstract

Non-stationary and nonlinear signals, which can bring important applications in chaotic dynamics, and are found in several scientific and engineering fields. Several processing techniques have been used to understand and extract information from these signals, and the literature shows that time-frequency analysis techniques are suitable tools for this characterization. They allow to examine the time-varying characteristics of the signals. In this chapter, we will explore time-frequency methods applied especially to nonlinear signals. First, we discuss the diverse range of dynamical systems. Then, we introduce the classical time-frequency methods, including the Short-Time Fourier Transform, the Wavelet Transform, the Hilbert Transform, and the Wigner-Ville distribution. These methods have been widely used in the literature in the study of non-stationary operations. Thus, we present emerging methods of time-frequency analysis, taking advantage of post-processing and synchrosqueezing techniques to improve the accuracy and resolution of the time-frequency representation. We present a comprehensive analysis of these emerging methods, comparing them with classical approaches to show their contributions. Our main goal is to highlight the capabilities of these emerging time-frequency analysis methods in capturing and understanding chaotic patterns in signals.

1. Introduction

The analysis of non-stationary and non-linear signals is fundamental in understanding various phenomena in scientific and engineering domains. Chaotic behavior, observed in nonlinear systems such as vibrations in mechanical systems, electronic circuits, biological processes and weather patterns, has been widely studied using signal processing and time-frequency analysis (TFA) techniques. However, the analysis of this class of signals presents challenges due to their non-stationary and non-linear nature.

Traditional signal processing methods, based on the Fourier Transform (FT), have been used to analyze stationary and deterministic signals. However, these methods are not suitable for non-stationary and non-linear signals [1, 2]. One of the main limitations of TF-based TFA methods, such as the Short Time Fourier Transform (STFT), is the fixed window size constraint. STFT represents the signal in the time-frequency domain by computing the Fourier transform of fixed-size windows with overlap. This choice of window size directly affects the trade-off between time and frequency resolution. Larger windows provide better frequency resolution but sacrifice time resolution, while smaller windows improve time resolution but offer less accurate frequency localization. In addition, fixed window sizes may not capture the rapid spectral changes often observed in chaotic signals. In this context, the wavelet transform emerges as a more suitable tool to capture time-varying behavior, transients, and abrupt variations in the frequency spectrum.

The Continuous Wavelet Transform (CWT) is a TFA method that overcomes the window size limitation of the STFT. CWT uses wavelet functions, known as mother wavelets, to analyze signals at multiple scales and resolutions simultaneously [1, 3]. It is an adaptive approach that allows high time resolution analysis for transient events and fine frequency resolution for temporal variations. Simultaneous localization in the time and frequency domains makes CWT suitable for non-stationary analysis. In addition, CWT offers flexibility in choosing the appropriate wavelet function, allowing the selection of a parent wavelet that best captures the signal characteristics. CWT in diverse fields including mechanical systems [4, 5, 6, 7, 8], biomedical signal analysis [9, 10, 11], geophysics [12, 13] and financial data analysis [14, 15, 16].

Another important TFA is the Wigner-Ville decomposition. In [17] it is described that one of the main challenges is the interference between terms, which occurs when different frequency components overlap in time and become difficult to distinguish in the time-frequency representation. In addition, WVD is sensitive to noise, which can affect the accuracy of energy estimation at different frequencies. Another consideration is that the Wigner-Ville transform is computationally intensive [2, 17]. It requires performing convolutions for all possible combinations of frequencies and time instants, which results in a high computational cost. This can be a problem when analyzing long duration signals or in real-time applications where computational efficiency is a concern. WVD has advantages in terms of time-frequency resolution and preservation of phase information, it is necessary to consider the disadvantages related to cross-term interference, sensitivity to noise and the computational cost associated with its implementation.

Emerging time-frequency analysis methods, such as the Synchrosqueezing Transform (SST), MultiSynchrosqueezing Transform (MSST), and Synchroextracting Transform (SET), have gained attention in recent years. These methods are known as post-processing techniques and they can be improve the TFR obtained from classical methods, such as CWT and the STFT [18, 19]. These classical methods are the foundation for modern time-frequency analysis but may have limitations in terms of resolution, adaptability, or noise sensitivity. The emerging methods, including the synchrosqueezing-based techniques, offer improved accuracy, enhanced time-frequency localization, and better noise robustness [20, 21]. They complement and enhance the results obtained from classical methods, providing more detailed and informative time-frequency representations of complex signals.

SST aims to improve the time-frequency localization of signal components with variable IFs [18]. By applying reassignment techniques, it concentrates the energy around the true instantaneous frequencies, providing a clearer time-varying behavior in the TFR [21]. Building on this approach, MSST extends the feature by considering multiple reference frequencies, allowing simultaneous extraction of multiple modes or components of a signal [22]. In addition, SET focuses on isolating specific modes or components of interest from a signal by separating them from background and noise [23, 24]. These methods have demonstrated a success in the analysis of mechanical systems, structural health monitoring, fault detection and a wide variety of biological signals, including electrocardiograms (ECG), electroencephalograms (EEG) and blood pressure signals.

It is important to emphasize that these methods are known as post-processing techniques as they refine and enhance the results obtained from classical methods such as STFT and CWT [24, 25]. They extract more accurate information from the original TFR, improving the understanding and enabling the analysis of signals with rich dynamics [20]. Post-processing-based TFA methods have shown to be promising tools for the analysis of non-stationary and nonlinear signals. The concentration of energy in the time-frequency representation allows for a better understanding of strong variations of IF’s in the frequency spectrum. The post-processing operations of TFA methods are being increasingly used in various science and engineering applications, improving the results obtained through classical methods.

In this chapter, we present a review of the application of TFA in the analysis of non-stationary systems. We will present a strongly nonlinear system, with cubic nonlinearity and fractional damping, and from which we obtain signals for analysis. This system was studied in depth in [26]. In this way, we present the classic TFA methods, including the STFT, CWT and WVD.

Despite the wide range of applications of classical TFA methods, they have some limitations in terms of resolution and accuracy when analyzing non-stationary and nonlinear signals. Therefore, our study focuses in application of emerging time-frequency techniques based on post-processing operations, which have shown superior performance in some analyses. We will compare these emerging techniques with classical methods and illustrate their applications in characterizing nonlinear and chaotic patterns in signals. The main objective of this work is to highlight the potential of post-processing operations in characterizing of nonlinear signals. We will discuss the advantages, limitations and applications of each method.

2. Mathematical background: time-frequency methods

Frequently used technique for analyzing non-stationary signals is the Short Time Fourier Transform (STFT). Gabor introduced this method by adapting the Fourier Transform to focus on specific segments of the signal in the time domain. With STFT, we can observe the signal in two dimensions: time and frequency. The expression for STFT, as presented in Eq. (1), involves windowing the signal x(t) with a function w(t − t₀) centered around a specific time t [2, 7].

$$X\left(\omega ,t_0\right)={\int }_{-\infty }^{\infty }x\left(t\right)w\left(t-t_0\right)e^{-\mathit{j\omega t}}\mathit{dt}$$E1

The signal in the frequency domain is denoted as X(ω, t₀), ω represents the angular frequency and t₀ indicates the window center time. In the time domain, x(t) refers to the original signal, while w(t − t₀) represents the window function centered at t₀. The window function ω(t) determines the properties of the analysis, including time and frequency resolution. Commonly used window functions include the Hamming window, the rectangular window, and the Gaussian window, among others [27]. The choice of window function is strongly associated with the nature of the signal. Thus, STFT provides a time-frequency representation that shows how the frequency content of the signal changes over time. However, as already mentioned, STFT has limitations in terms of time-frequency resolution due to its fixed window size. a shorter window gives better time resolution but worse frequency resolution and vice versa. A long study and robust formulation of the STFT can be found in [27]. Even with all its known limitations, the STFT remains an important tool for time-frequency analysis, massively used in several applications and serves as the basis for many other techniques, such as the Continuous Wavelet Transform and methods based on synchrosqueezing [7, 19].

The CWT is a TFA technique that provides a representation of the frequency content of a signal over time, very similar to the process presented in STFT. However, unlike STFT, CWT uses a set of wavelet functions at different scales to analyze the signal [1].

In [7], the CWT of a signal x(t) on a specific scale a and time t is described as follows:

$$X\left(a,t\right)={\int }_{-\infty }^{\infty }x\left(t\right){\mathrm{\Psi }}^{\ast }\left(\frac{t-t^{\prime }}{a}\right)d{t}^{\prime }$$E2

where X(a, t) is the CWT of the signal or the signal represented in the wavelet domain, ψ.

represents the mother wavelet function, a denotes the scale parameter and t represents the time parameter. The wavelet function ψ is a scaled and translated version of the parent wavelet and serves as a time-varying analysis function. In ψ we have the main element that denotes the variation window in the formulation [7]. CWT decomposes the signal into different frequency components at different scales, providing a localized time-frequency representation. This allows for better time resolution at high frequencies and better frequency resolution at low frequencies. Commonly used wavelet functions include the Morlet wavelet, Mexican hat wavelet, and Daubechies wavelet, among others. The choice of wavelet function depends on the specific characteristics and requirements of the signal being analyzed. CWT provides a flexible and adaptive time-frequency representation that can capture localized variations in the frequency content of the signal, such as transients, spikes and variations in the spectrum.

The symbol ψ represents the parent wavelet function in the CWT formulation. The specific form of the mother wavelet function depends on the chosen wavelet family. Below are some examples of commonly used wavelet functions. The Morlet wavelet is frequently used in many applications and frequently found in the literature [21]. It is defined as a complex value Gaussian window modulated by a complex sinusoid. The expression for the Morlet wavelet is given by:

$$\mathrm{\Psi }\left(t\right)={\pi }^{-\frac{1}{4}}{e}^{j{\omega }_{0}t}{e}^{-\frac{t^2}{2}}$$E3

where ω_o is the central frequency of the wavelet.

The Mexican hat wavelet, is commonly used for detecting and analyzing transient events. It has a peaked shape resembling a Mexican hat. The expression for the Mexican hat wavelet function is:

$$\mathrm{\Psi }\left(t\right)=\left(1-t^2\right)e^{-\frac{t^2}{2}}$$E4

Daubechies wavelets are a family of wavelets that are orthogonal. They are widely used in various applications. The expression for a specific Daubechies wavelet depends on the order chosen, such as Daubechies-4, Daubechies-6, etc. Daubechies wavelets have great application in signal analysis of vibrations, signals of turbulent phenomena, signals that have great variation over time such as transients and even chaotic signals, since it appears that for the analysis of these types of signals the choice should be the one that less unbalances the energy of the signal, that is, the one that needs the smallest number of coefficients to represent the signal [3].

CWT is a suitable tool for analyzing non-stationary and transient signals, as well as signals with time-varying frequency components.

Another method to characterize non-stationary signals is the Wigner-Ville distribution, which is part of a group of so-called bilinear integral transforms. This technique represented the first attempt to perform a joint analysis in time and frequency [2, 19]. According to [2], the bilinear Wigner-Ville distribution provides better resolution in the joint time-frequency domain compared to any linear transform. However, it suffers from a problem of cross-term interference, which does not represent any signal information. In other words, the WVD of two signals is not the sum of their individual WVDs [27].

For a continuous signal x(t) the Wigner-Ville distribution is defined as:

$$\mathit{WV}{D}_x\left(t,f\right)={\int }_{-\infty }^{\infty }x\left(t+\frac{\tau }{2}\right)x^{\ast }\left(t-\frac{\tau }{2}\right)e^{-j2\pi f_{\tau }}\mathit{d\tau }$$E5

A comprehensive study on the method is present in [2, 19].

In this chapter, we focus on utilizing the Synchrosqueezing Transform (SST) based on the CWT. The SST enhances the time-frequency representation obtained from the CWT by concentrating the energy of the signal in the time-frequency plane. However, it’s important to note that the SST can also be applied using the STFT as an alternative basis. In our formulation CWT was written in Eq. (2).

The SST involves three steps: First, the CWT is calculated. In the second step, we calculate an initial frequency f (a, b) by analyzing the oscillatory behavior of W_x(a, b) with respect to a [18, 21]. So that:

$$f\left(a,b\right)=-i{\left[W_x\left(a,b\right)\right]}^{-1}\frac{\partial W_x}{\partial a}\left(a,b\right)$$E6

In the third step, the transformation from the scale-time plane to the time-frequency plane is performed. Next, we reassign each value of W_x(a, b) to (a, f₁), where f₁ corresponds to the closest frequency to the initial frequency f (a, b) [18, 21]. This operation is described by Eq. (6).

$$T\left(a,f_1\right)={\left(\mathrm{\Delta }f\right)}^{-1}\sum _{b_k:\mathrm{f}\left(\mathrm{a},\mathrm{b}\_\mathrm{k}\right)‐\mathrm{f}\_\mathrm{l}\le \mathrm{\Delta }f/2}^{\infty }{W}_x\left(a,b_k\right)b_k^{-3/2}\mathrm{\Delta }{b}_k$$E7

In Eq. (7), ∆f denotes the width of each frequency band ∆f = f_lf_l − 1, and similarly for ∆b. SST based on CWT can provide a high-resolution time-frequency spectrum by compressing (reassigning) the result of the CWT. However, when the amplitude of high-frequency components is low, it becomes challenging to identify these components in the CWT or the SST spectrum derived from the CWT result. Unlike the CWT, the SST efficiently reveals the low amplitude and high-frequency components of a signal and allows for an inverse transformation without any loss of information. Nevertheless, the resolution of the SST is still not entirely satisfactory [18, 20, 21]. Initially proposed for wavelets, the SST can also be applied using the SSTF, known as STFT-based SST. The SST based on CWT is a TFA that reallocates the energy of the signal in the frequency domain, compensating for the propagation effects caused by the wavelet mother and avoiding distortions in the TFR [20]. Unlike other methods that perform reassignment in both frequency and time directions, synchrosqueezing reallocates power only in the frequency direction, preserving the signal’s time resolution. By preserving time, the inverse synchrosqueezing can reconstruct the original signal more accurately [18].

The SST is a post-processing operation that improves the TFR of a signal. While the CWT provides a global representation of the signal’s time-varying frequency content, the SST further refines this process by concentrating the energy in the time-frequency representation. By applying the SST, the energy of the signal is redistributed in such a way that it becomes more localized around its instantaneous frequency. This concentration of energy allows for a clearer and more precise identification of the signal’s dominant frequency components and their temporal variations. It helps to distinguish fine frequency modulations, identify transient events, and reveal nonlinear interactions that may be concealed in the original CWT TFR.

The MultiSynchrosqueezing Transform (MSST) is an extension of the Synchrosqueezing Transform (SST) that enhances the time-frequency representation of signals using a multiwavelet frame-work. The MSST overcomes certain limitations of the SST by providing an improved analysis of signals with more complex spectral characteristics [22]. In this chapter we used the MSST based on CWT. The MSST extends the SST to a multiwavelet framework by considering multiple analyzing wavelets. It improves the TFR obtained from the wavelet transform by redistributing the energy of the signal in a more localized manner. The MSST builds upon the STFT formulation, which can be expressed as follows:

$$G_x\left(u,w\right)={\int }_{-\infty }^{\infty }x\left(t\right)g\left(t-u\right)e^{-\mathit{i\omega }\left(t-u\right)}\mathit{dt}$$E8

By applying Taylor’s expansion, Eq. (8) can be rewritten as:

$$G_x\left(u,w\right)={\int }_{-\infty }^{\infty }A\left(u\right)e^{i\left(\varphi \left(u\right)+\varphi ’\left(u\right)\left(t-u\right)\right)}g\left(t-u\right)e^{-\mathit{i\omega }\left(t-u\right)}\mathit{dt}=A\left(u\right)e^{-\mathit{i\varphi }\left(u\right)}\widehat{g}\left(\omega -{\varphi }^{’}\left(u\right)\right)$$E9

The partial derivative of Eq. (9) yields:

$${\partial }_u{G}_x\left(u,\omega \right)=G_x\left(u,\omega \right){\mathit{i\varphi }}^{’}\left(u\right)$$E10

If G_x(u, ω) 0, the instantaneous frequency of the signal x(t) can be estimated as:

$$\widehat{\omega }\left(u,\omega \right)=\frac{{\partial }_u{G}_x\left(u,\omega \right)}{i{\partial }_u{G}_x\left(u,\omega \right)}$$E11

To enhance the energy concentration, the MSST employs frequency reassignment, as defined in Eq. (12) [28]:

$$F_{x,G}\left(u,\eta \right)={\int }_{-\infty }^{\infty }{G}_x\left(u,\omega \right)\delta \left(\eta -\widehat{\omega }\left(u,\omega \right)\right)\mathit{d\omega }$$E12

However, studies have shown that as the non-stationarity of the signal increases, the SST representation becomes increasingly blurry [28]. To address this issue, multiple SST operations are applied iteratively to smooth the TFR result, as presented in Eq. (13):

$$\left\{\begin{array}{l}{F}_{x,G}^{\left[2\right]}\left(u,\eta \right)={\int }_{-\infty }^{\infty }{F}_{x,G}^{\left[1\right]}\left(u,\omega \right)\delta \left(\eta -\widehat{\omega }\left(u,\omega \right)\right)\mathit{d\omega }\\ F_{x,G}^{\left[3\right]}\left(u,\eta \right)={\int }_{-\infty }^{\infty }{F}_{x,G}^{\left[2\right]}\left(u,\omega \right)\delta \left(\eta -\widehat{\omega }\left(u,\omega \right)\right)\mathit{d\omega }\\ F_{x,G}^{\left[N\right]}\left(u,\eta \right)={\int }_{-\infty }^{\infty }{F}_{x,G}^{\left[N-1\right]}\left(u,\omega \right)\delta \left(\eta -\widehat{\omega }\left(u,\omega \right)\right)\mathit{d\omega }\end{array}\right.$$E13

To reduce computation time, a new IF estimate is obtained, where N represents the number of iterations. So that he MSST method can be represented as:

$$F_{x,G}^{\left[N\right]}\left(u,\omega \right)={\int }_{-\infty }^{\infty }{G}_x\left(u,\omega \right)\delta \left(\eta -{\widehat{\omega }}^{\left[N\right]}\left(u,\omega \right)\right)\mathit{d\omega }$$E14

Additionally, the MSST retains the ability to reconstruct the original signal, which can be done using Eq. (15):

$$X\left(u\right)=\frac{{\int }_{-\infty }^{\infty }{F}_{x,G}^{\left[N\right]}\left(u,\omega \right)\mathit{d\omega }}{2\mathit{g\pi }}$$E15

The MSST approach is part of the post-processing operation of the STFT. By utilizing iterative reassignment procedures with multiple SST operations, the MSST improves the energy concentration in the time-frequency representation (TFR). For a comprehensive study on the MSST and the wavelet multisynchrosqueezed transform, refer to [22, 23].

The Synchroextracting Transform (SET) presents a promising approach for addressing the complexities associated with nonlinear dynamics. By extending the FT, the SET utilizes adaptive time-frequency analysis to capture the temporal and spectral characteristics of signals [24]. This enables the identification and isolation of distinct frequency components within a time series, facilitating the detection of nonlinear interactions and synchronization phenomena that may remain concealed when using classical methods [29].

At the core of the SET lies the concept of mode decomposition, which aims to decompose a given signal into a set of intrinsic mode functions (IMFs). These IMFs represent different frequency modes that collectively capture the signal’s temporal dynamics. The extraction of IMFs is achieved through an iterative sifting process, where local extrema and envelopes are identified and removed from the original signal until it exhibits well-behaved, zero-mean behavior. The fundamental idea of the SET involves extracting the maximum Short-Time Fourier Transform (STFT) coefficients within the Instantaneous Frequency (IF). This approach presents a hurdle to the Time-Frequency (TF) distribution. Mathematically, the SET can be expressed as:

$$S_e=G_e\left(t,\omega \right).\delta \left(\omega -{\omega }_0\left(t,\omega \right)\right)$$E16

where G_e(t, ω) represents the STFT of the signal x(t).

For the SET based on the Continuous Wavelet Transform (CWT), the formulation is as follows:

$$\mathcal{T}f\left(t,a\right)=Wf\left(t,a\right).a^{-\frac{3}{2}}.\delta \left(a-a_{\phi }\right)$$E17

For detailed formulation, please refer to [24].

It is noteworthy that the structure of the SET based on Wavelet transform is similar to the ideal TFA formulation. Moreover, in Eq. (17), the operator δ(a − a_φ) can be rewritten as [24]:

$$\delta \left(a-a_{\phi }\right)=\left\{\begin{array}{l}1,a=a_{\phi }\\ 0,a\ne a_{\phi }\end{array}\right.$$E18

Thus, T_f (t, a) can be rewritten as follows:

$${\mathcal{T}}_f\left(t,a\right)=\left\{\begin{array}{l}{W}_f\left(t,a\right).a^{-\frac{3}{2}},a=a_{\phi }\\ 0,a\ne a_{\phi }\end{array}\right.$$E19

From Eq. (19), we obtain the SET, which exhibits high time-frequency resolution and enables mode decomposition. A comprehensive formulation, including the reconstruction process of the signal x(t), can be found in [23]. In Appendix A, the algorithms for the implementation of pre-processing based methods are presented.

3. Applications

The signals used in this work are obtained through the Duffing-type oscillator with fractional damping model (Figure 1). The system consists of a structure of mass m₁, connected to a damper with fractional damping and a non-linear spring with non-linear cubic stiffness. The proposed system is excited by a non-ideal DC motor characterized by the moment of inertia J_M and the unbalanced mass m₀ with eccentricity r. It is a strongly non-linear system that was studied in comprehensive by the authors in [26].

Figure 1.

Duffing-type oscillator with fractional damping.

The equations of motion of the Duffing-type oscillator with fractional damping are given by:

$$\begin{array}{l}{m}_1{x}^{″}+F_d-k_{1}x+k_2{x}^3=m_{0}r\left({\left({\phi }^{\prime }\right)}^{2}sin\phi -{\phi }^{″}cos\phi \right)\\ \left(J+m_0{r}^2\right){\phi }^{″}=C_M\mathit{\varphi I}\left(t\right)-m_{0}r{x}^{″}cos\phi \\ I^{\prime }=-\frac{R_t}{L_t}I\left(t\right)-\frac{C_E\varphi }{L_t}{\phi }^{\prime }+\frac{U\left(t\right)}{L_t}\end{array}$$E20

The damping device exerts a force F_d given by:

$$F_d=c\frac{d^{p}x}{{\mathit{dt}}^p}$$E21

where c denotes a constant coefficient. As can be seen, F_d is proportional to the pth time derivative of the relative displacement, and it is observed that if p = 1 the force F_d corresponds to a linear viscous damping force [26].

It is convenient to work with dimensionless position and time, namely $u=\frac{x}{x_0}$ and τ = ω₀t, respectively, where x₀ is the static displacement, and introducing the following variable $\tilde{I}=\frac{I}{I_0}$, where I₀ is a rated current in the armature. The dimensionless fractional derivative is defined as: $\frac{d^p}{{\mathit{dt}}^p}\Rightarrow {\omega }_0^p\frac{d^p}{d{\tau }^p}$. In this way, it is possible to rewrite Eq. (20) in the dimensionless form as:

$$\begin{array}{l}\ddot{u}+\beta \left(p\right)u^{\left(p\right)}-u+\delta u^3=w_1\left({\dot{\phi }}^{2}sin\phi -\ddot{\phi }cos\varphi \right)\\ \ddot{\phi }=p_3\tilde{I}\left(\tau \right)-w_2\ddot{u}cos\phi \\ \dot{\tilde{I}}=-p_1\tilde{I}-p_2\dot{\phi }+U_1\left(\tau \right)\end{array}$$E22

where the dimensionless parameters are denoted by:

$$\begin{array}{l}{\omega }_0^2=\frac{k_1}{m_1+m_0}\beta \left(p\right)=\frac{c}{m_1{\omega }_0^{2-p}}\delta =\frac{k_2}{k_1}{x}_0^2{w}_1=\frac{m_{0}r}{m_1{x}_0}\\ w_2=\frac{m_0{x}_{0}r}{\left(J+m_0{r}^2\right)}{p}_1=\frac{R_t}{L_t{I}_0{\omega }_0}{U}_1=\frac{U}{L_t{I}_0{\omega }_0}{p}_2=\frac{C_E\varphi }{L_t{I}_0}\\ p_3=\frac{C_M\varphi I_0}{\left(J+m_0{r}^2\right){\omega }_0^2}\end{array}$$E23

The numerical integration of the Duffing-type oscillator with fractional damping was carried out considering the parameters of the DC motor and mechanical parameters available in [26]. The integration step is considered by h = π/200. Figure 2a shows the bifurcation diagram of the Duffing-type oscillator with fractional damping when the varying parameter is the parameter U₁ (dimensionless voltage in the armature). The broad distributions of points characterize the chaotic behavior while the countable few points imply the periodic behavior.

Figure 2.

Bifurcation diagram of the amplitude of the fractional non-ideal oscillator versus p.

Figure 3a shows the time domain response of the system, Figure 3b shows the phase portrait for U₁ = 3.6 and p = 1. The time domain response will be characterized in the frequency domain through the application of TFA. In this work we use the following TFA schemes: STFT, WSCL CWT, WVD, SST, MSST and SET. This procedure will be applied to all signals. In all STFT-based analyses, the Kaiser window with 256 data points was utilized. For the analyses based on the CWT, the Morlet wavelet was employed.

Figure 3.

Time domain analysis—(a) time response (b) phase portrait for U₁ = 4 and p = 1.

The TFA results (U₁ = 4 and p = 1) are presented in Figure 4. In all analyses, the periodic behavior of the system is well characterized. The TFRs based on STFT (Figure 4a) and WSCL (Figure 4b) characterize the fundamental frequency of the system but are not able to characterize the frequency from the engine. The other analyses present the characterization of the natural frequency and the external force in their TFRs. Note that the frequency spectra obtained by SST (Figure 4g), MSST (Figure 4g), and SET (Figure 4h) demonstrate better energy concentration in relation to the other methods. It should also be noted that all the schemes are capable of characterizing the transient regime of the signal.

Figure 4.

Time-frequency representation—(a) STFT, (b) WSCL, (c) CWT, (d) SST (e) IFR, (f) WVD, (g) MSST and (h) SET for U₁ = 3.6 and p = 1.

Figure 5a shows the system response in the time domain, while Figure 5b presents the phase portrait for U₁ = 4 and p = 1.2. The Phase portrait indicates the periodicity of the signal; however, the parameters U₁ = 4 and p = 1.2 clearly cause a period-doubling effect. The response in the time domain will be characterized in the frequency domain through the application of TFA schemes, as shown in Figure 6.

Figure 5.

Time domain analysis—(a) time response (b) phase portraits for U₁ = 4 and p = 1.2.

Figure 6.

Time-frequency representation—(a) STFT, (b) WSCL, (c) CWT, (d) SST (e) IFR, (f) WVD, (g) MSST and (h) SET for U₁ = 4 and p = 1.2.

The TFA results (U₁ = 4 and p = 1.2) are presented in Figure 6. The analyzes effectively the periodic behavior of the system. However, TFRs based on STFT (Figure 6a), WSCL (Figure 6b), and WVD (Figure 6f) are unable to distinguish the frequency components. The analysis using the CWT scheme identifies the frequency components present in the signal, but the energy dispersion in the TFR obscures some components that have very close values. On the other hand, the analyzes using SST (Figure 6g), MSST (Figure 6g), and SET (Figure 6h) are capable of characterizing and separating the multiple frequency components contained in the signal.

In the next analysis, Figure 7a shows the system response in the time domain, while Figure 7b presents the phase portrait for U₁ = 4 and p = 1. The Phase portrait indicates the chaotic dynamics of the signal. Again, the response in the time domain will be characterized in the frequency domain through the application of TFA schemes, as shown in Figure 8.

Figure 7.

Time domain analysis—(a) time response (b) phase portraits for U₁ = 4 and p = 1.

Figure 8.

Time-frequency representation—(a) STFT, (b) WSCL, (c) CWT, (d) SST (e) IFR, (f) WVD, (g) MSST and (h) SET for U₁ = 4 and p = 1.6.

Furthermore, Figure 8 presents TFA results from various schemes. It is noteworthy that the analyses with the STFT (Figure 8a) and WSCL (Figure 8b) schemes fail to characterize the signal. In the CWT analysis, the non-linearity characterization appears as a function of the presented frequency response, and distortions in the natural frequency and abrupt variations in the frequency spectrum are perceptible, despite the significant energy dispersion in the TFR. It’s worth emphasizing again that the natural frequency of the system is well characterized, despite the system’s non-periodic response. The WVD scheme (Figure 8f) shows a response similar to the CWT analysis. Lastly, the schemes based on post-processing, such as SST, MSST, and SET, present a TFR response that allows for a better characterization of the natural frequency distortion and the emergence of frequency components with abrupt variations.

Finally, in Figure 9a shows the system response in the time domain, and Figure 9b presents the phase portrait for U₁ = 3.85 and p = 1. The Phase portrait, again indicates the chaotic dynamics of the signal.

Figure 9.

Time domain analysis—(a) time response (b) phase portraits for U₁ = 3.85 and p = 1.

Figure 10 presents TFA results from various schemes. As with the previous analysis (U₁ = 4 and p = 1), the STFT (Figure 10a) and WSCL (Figure 10b) schemes fail in the characterization of the signal, as they are unable to demonstrate the variation of frequencies and the new components that are associated with the signal. In the CWT scheme, non-linearity is characterized by distortions in the natural frequency, and abrupt variations in the frequency spectrum are noticeable. The WVD scheme (Figure 10f) exhibits a response similar to the CWT analysis; however, the issue of cross terms is very evident in the frequency spectrum. Once again, the post-processing schemes present a TFR response that enables better characterization of the natural frequency distortion and the emergence of frequency components with abrupt variations (Figure 10d, g, and h), serving as a qualitative indicator for characterizing the chaotic dynamics associated with the signal.

Figure 10.

Time-frequency representation—(a) STFT, (b) WSCL, (c) CWT, (d) SST (e) IFR, (f) WVD, (g) MSST and (h) SET for U₁ = 3.85 and p = 1.

The results demonstrate that SST increases the energy concentration in the TFR using a reassignment procedure. The instantaneous frequency information derived from the CWT is employed to reassign the energy of each CWT coefficient to the corresponding time frequency bucket, resulting in a more accurate representation of the time interval. Variable spectral content of the signal. Its ability to provide a more time-focused representation of frequency makes it a suitable tool for analyzing signals with non-linear and non-stationary characteristics. Overall, SST enhances CWT by extracting and emphasizing crucial features from the time-varying spectral content of the signal. Its ability to focus energy on the time-frequency plane improves the analysis of complex signals, contributing to a deeper understanding of their underlying dynamics. However, the limitations of SST can be noticed, such as its reduced time-frequency resolution, especially when dealing with highly non-stationary signals. This reduced resolution can lead to blurred and smeared representations, complicating accurate identification and analysis of localized spectral components.

These limitations are overcome through the MSST schema, it is an improved version of the SST. MSST addresses the drawbacks of SST by incorporating multiple iterations of the SST process, leading to better time frequency resolution and better localization of spectral components. The signal undergoes iterative SST operations on MSST; each iteration refines the time-frequency representation by focusing even more on the energy of the components. This iterative process effectively enhances frequency peaks over time, thus improving resolution and allowing for more accurate and detailed analysis of non-stationary signals, as seen in the presented results.

By conducting multiple SST operations, the MSST scheme achieves superior energy concentration and spectral component localization, even with highly non-stationary signals. This allows for more accurate identification of instantaneous frequencies and improved tracking of signal dynamics over time. Additionally, MSST maintains the ability to reconstruct the original signal, ensuring that no information is lost during analysis. The reconstructed signal can be obtained by integrating the MSST coefficients. Regarding the SET scheme, the results show a higher concentration of energy in the TFR compared to the other tested schemes, and the separation of the frequency components is also superior, therefore, for the application presented, the SET scheme was superior.

4. Conclusions

In this chapter, time-frequency methods based on post-processing operations for the analysis of nonlinear systems are introduced and qualitatively compared with classical methods. The primary advantage of schemes based on post-processing is their superior characterization of instantaneous frequencies. In addition, these schemes focus energy in the time-frequency representation (TFR). Signals with nonlinear characteristics and chaotic dynamics were used for the comparison. The efficient capture capacity of instantaneous frequencies and the high concentration of energy in the TFR, associated with the results of the post-processing schemes, enable the characterization of nonlinearities and rich dynamics associated with the signals. The results demonstrate that the SET scheme holds potential for application in the analysis of signals from nonlinear systems, surpassing the results of the SST scheme. They also indicate that this technique can be used to extract relevant information and rich dynamics associated with the system’s dynamics.”

Acknowledgments

The authors would like to thank FAPESP, CAPES and CNPq for the financial supports.

A. Algorithms for post-processing based methods

A.1 Wavelet-based synchrosqueezing transform (SST)

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Input: Signal x(t), Wavelet parameters (mother wavelet ψ(t), scales a, translations b) Output: Synchrosqueezed Wavelet Transform (SST)

Step 1: Calculate the Continuous Wavelet Transform (CWT)

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1. 1.1 Choose a mother wavelet function ψ(t) and its associated scales a and translations b.

2. 1.2 Compute the CWT of the signal x(t) using the chosen wavelet and its parameters.

Apply the wavelet transform to x(t) over a range of scales and translations. - Calculate the CWT coefficients: $\mathit{CWT}\left(a,b\right)=\int x\left(t\right){\psi }^{\ast }\left(\frac{t-b}{a}\right)\mathit{dt}.$

Step 2: Calculate the Preliminary Frequency

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1. 2.1 Compute the partial derivative of the CWT coefficients with respect to scale (a):

   $‐\frac{\partial \mathit{CWT}}{\partial a}\left(a,b\right)=\frac{\partial }{\partial a}\mathit{CWT}\left(a,b\right).$

2. 2.2 Compute the inverse of the CWT coefficients: $‐{\mathit{CWT}}_{\mathrm{inv}}\left(a,b\right)=\frac{1}{\mathit{CWT}\left(a,b\right)}.$

3. 2.3 Calculate the preliminary frequency $f\left(a,b\right):‐f\left(a,b\right)=-i\cdot {\mathit{CWT}}_{\mathrm{inv}}\left(a,b\right)\cdot \frac{\partial \mathit{CWT}}{\partial a}\left(a,b\right).$

Step 3: Perform Transform Reassignment

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1. 3.1 Initialize an empty matrix to store the reassigned transform (SST).

2. For each CWT coefficient CWT (a, b):

   1. 3.1.1 Find the nearest frequency f₁ in the preliminary frequency range f (a, b).

   2. 3.1.2 Reassign the CWT coefficient to the corresponding location in the SST matrix: $‐\mathit{SST}\left(a,f_1\right)+=\frac{\mathit{CWT}\left(a,b\right)}{\frac{\partial \mathit{CWT}}{\partial a}\left(a,b\right)}.$

Step 4: Analysis and Visualization

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1. 4.1 Analyze and interpret the resulting SST matrix to extract relevant information about the signal’s time-frequency characteristics.

2. 4.2 Visualize the SST matrix using suitable plotting techniques to observe the time-frequency representation of the signal.

A.2 MultiSynchrosqueezing Transform (MSST)

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Input: Signal x(t), Wavelet parameters (mother wavelet ψ(t), scales a, translations b) Output: Modified Synchrosqueezing Transform (MSST)

Step 1: Calculate the Continuous Wavelet Transform (CWT)

————————————————————————————————————–

1. 1.1 Choose a mother wavelet function ψ(t) and its associated scales a and translations b.

2. 1.2 Compute the CWT of the signal x(t) using the chosen wavelet and its parameters.

Apply the wavelet transform to x(t) over a range of scales and translations. - Calculate the CWT coefficients: $\mathit{CWT}\left(a,b\right)=\int x\left(t\right)\cdot {\psi }^{\ast }\left(\frac{t-b}{a}\right)\mathit{dt}.$

Step 2: Calculate the Instantaneous Frequency

————————————————————————————————————–

1. 2.1 Compute the partial derivative of the CWT coefficients with respect to scale (a):

   $‐\frac{\partial \mathit{CWT}}{\partial a}\left(a,b\right)=\frac{\partial }{\partial a}\mathit{CWT}\left(a,b\right).$

2. 2.2 Compute the inverse of the CWT coefficients: $‐{\mathit{CWT}}_{\mathrm{inv}}\left(a,b\right)=\frac{1}{\mathit{CWT}\left(a,b\right)}.$

3. 2.3 Calculate the instantaneous frequency f (a, b) using the inverse CWT and the derivative: $‐f\left(a,b\right)=-i\cdot {\mathit{CWT}}_{\mathrm{inv}}\left(a,b\right)\cdot \frac{\partial \mathit{CWT}}{\partial a}\left(a,b\right).$

Step 3: Perform Transform Reassignment

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1. 3.1 Initialize an empty matrix to store the reassigned transform (MSST).

2. For each CWT coefficient CWT (a, b):

   1. 3.1.1 Find the nearest frequency f₁ in the instantaneous frequency range f (a, b).

   2. 3.1.2 Reassign the CWT coefficient to the corresponding location in the MSST matrix:

      $\mathit{\text{MSST}}\left(a,f_1\right)+=\frac{\mathit{CWT}\left(a,b\right)}{\frac{\partial \mathit{CWT}}{\partial a}\left(a,b\right)}.$

Step 4: Apply Post-processing (Optional)

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1. 4.1. Perform any desired post-processing on the MSST matrix, such as denoising or thresholding.

Step 5: Analysis and Visualization ————————————————————————————————————-

1. 5.1 Analyze and interpret the resulting MSST matrix to extract relevant information about the signal’s time-frequency characteristics.

2. 5.2 Visualize the MSST matrix using suitable plotting techniques to observe the time-frequency representation of the signal.

A.3 Synchroextracting Transform (SET)

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Input: Signal x(t), Wavelet parameters (mother wavelet ψ(t), scales a, translations b) Output: Synchroextracting Transform (SET)

Step 1: Calculate the Continuous Wavelet Transform (CWT)

————————————————————————————————————–

1. 1.1 Choose a mother wavelet function ψ(t) and its associated scales a and translations b.

2. 1.2 Compute the CWT of the signal x(t) using the chosen wavelet and its parameters. -

Apply the wavelet transform to x(t) over a range of scales and translations. - Calculate the CWT coefficients: $\mathit{CWT}\left(a,b\right)=\int x\left(t\right)\cdot {\psi }^{\ast }\left(\frac{t-b}{a}\right)\mathit{dt}.$

Step 2: Calculate the Instantaneous Frequency

————————————————————————————————————–

1. 2.1 Compute the partial derivative of the CWT coefficients with respect to scale (a):

   $‐\frac{\partial \mathit{CWT}}{\partial a}\left(a,b\right)=\frac{\partial }{\partial a}\mathit{CWT}\left(a,b\right).$

2. 2.2 Compute the inverse of the CWT coefficients: $‐{\mathit{CWT}}_{\mathrm{inv}}\left(a,b\right)=\frac{1}{\mathit{CWT}\left(a,b\right)}.$

3. 2.3 Calculate the instantaneous frequency f (a, b) using the inverse CWT and the derivative: $‐f\left(a,b\right)=-i\cdot {\mathit{CWT}}_{\mathrm{inv}}\left(a,b\right)\cdot \frac{\partial \mathit{CWT}}{\partial a}\left(a,b\right).$

Step 3: Perform Synchroextracting Transform

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1. 3.1 Initialize an empty matrix to store the synchroextracting transform (SET).

2. For each CWT coefficient CWT (a, b):

   1. 3.1.1 Find the maximum value of the CWT coefficients in the instantaneous frequency range f (a, b).

3. 3.1.2 Assign the maximum value to the corresponding location in the SET matrix: $‐\mathit{SET}\left(a,f_{\max }\right)=max\left\{\mathit{CWT}\left(a,b\right)|f\left(a,b\right)=f_{\max }\right\}.$

Step 4: Apply Post-processing (Optional)

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4.1. Perform any desired post-processing on the SET matrix, such as denoising or thresholding

Step 5: Analysis and Visualization

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1. 5.1 Analyze and interpret the resulting SET matrix to extract relevant information about the signal’s time-frequency characteristics.

2. 5.2 Visualize the SET matrix using suitable plotting techniques to observe the time-frequency representation of the signal.