Perspective Chapter: Behavioral Analysis of Nonlinear Systems and the Effect of Noise on These Systems

Abstract

One of the crucial concepts in determining the structure of dynamic systems is to recognize the behavior of nonlinear systems, which is one of the current issues in engineering sciences. In general, nonlinear systems exhibit behaviors such as stability, periodic, quasi-periodic and chaotic. Since in nonlinear systems, changing parameters can have a great effect on changing the behavior of nonlinear systems, for this reason, it has been studied how different parameters affect the behavior of a system. Due to the importance of determining the behavior of nonlinear systems, in this chapter, first, various criteria for estimating the behavior of nonlinear systems are discussed and then the effect of these parameters on these systems is examined.

1. Introduction

For researchers in engineering sciences, investigating and determining the qualitative results of dynamic systems is of particular importance, which is often studied and examined using theoretical theories, differential equations, and other tools. One of the key concepts in analyzing dynamic systems is identifying the behavior of nonlinear systems, which is among the current topics in engineering sciences. For this purpose, many researchers in engineering sciences are interested in studying how different parameters affect the behavior of a system.

Nonlinear dynamics and complex systems are a very broad subject that essentially falls into an interdisciplinary research field. This issue includes mathematics, physics, chemistry, medicine, engineering sciences, and so on. Complex systems are composed of numerous components, each of which may interact with one another and even external factors, resulting in diverse interactions. For example, water and air, human organs, living organisms, infrastructure such as electrical grids, complex software, electronic systems, ecological systems, cellular systems, and ultimately all kinds of networks can be considered complex systems, each with their own components and external interactions. Modeling the behavior of complex systems is challenging due to their interdependence and interactions among components or between a specific system and its environment. Complex systems possess distinctive features such as nonlinearity, self-organization, feedback, robustness, adaptability, Emergence, and network-like structures that result from diverse interactions among their components. In the past, researchers discovered that the relationship between nonlinear dynamics and complex systems in engineering sciences is such that simple systems (with a few variables) exhibit stable behavior, while complex systems (with many variables) exhibit unstable behavior. However, it has recently been found that even a simple nonlinear system can exhibit irregular behavior [1, 2, 3, 4]. This means that a system displaying irregular behavior is considered a complex system, but it is possible for a simple nonlinear system to exhibit irregular behavior. In general, nonlinear systems exhibit behaviors such as stability, periodic, semi-periodic, and chaotic, which are detailed in this section (refer to Figure 1) [1, 2, 3, 4].

Figure 1.

Behavioral nonlinear systems.

The distinction between the behaviors of linear and nonlinear systems is about their stability. The behavior of non-switching linear systems is not sensitive to initial conditions, while in some cases of nonlinear systems, the equilibrium point of the system may be stable. However, for some initial conditions, the system response may be converged (stable), while for others, it may be divergent (unstable), and identifying these types of systems is not an easy task [2, 3, 4, 5, 6]. One type of nonlinear systems behavior is chaos [3], which is unpredictable and represents order within disorder [1, 2]. Chaotic systems, although they appear random, belong to the category of deterministic systems, which are distinct from stochastic systems that are random by nature. In chaotic systems, a small change in initial conditions can lead to significant changes in the output [2, 3, 4, 5, 6, 7], which is a characteristic feature of chaos. Another behavior of nonlinear systems is periodic. In general, this behavior occurs in dynamic nonlinear systems around a fixed point with a specific amplitude and frequency [6, 7, 8, 9]. Another behavior of nonlinear systems is semi-periodic, which is formed from the combination of several periodic behaviors [9, 10, 11, 12, 13, 14].

2. Detecting chaos in the time domain

One of the common methods for detecting chaos is to use the patterns present in the time series [15]. One of the simplest methods in the time domain is to plot the time series in phase space and then observe the created pattern to determine the signal behavior. Detecting periodic behavior in phase space is very simple, but the behavior of chaotic and quasi-periodic systems in phase space is very complex and unpredictable. This method alone is not a precise way to detect the presence of chaos, as random systems and real dynamic systems mixed with noise also exhibit similar behavior in phase space.

Lyapunov exponents of a system are a set of non-changing geometric measures that directly express the system’s dynamics. One of its applications is in detecting the phenomenon of chaos in a system and also as a measure of the chaotic nature of behavior. The topic of chaotic behavior is qualitative as far as it relates to sensitivity to initial conditions and structural instability. However, in studying a system, we have information about its behavior in the form of a differential equation, a recursive mapping, or a time series, and therefore it is necessary to have analytical or quantitative methods for detecting chaos in any system so that we can distinguish chaotic behavior from random noise-like behavior. Additionally, this method should be able to provide both a measure and a quantity for the degree of chaos in the system. Describing the quantitative sensitivity of a system’s behavior to initial conditions in chaotic situations is possible by introducing Lyapunov exponents [5].

Lyapunov exponents are a set of non-changing geometric measures that directly express the dynamics of systems. The Lyapunov exponent is calculated as follows:

Consider two neighboring points in phase space at times zero and t, where the distance between the points in the direction of (i) is $‖\delta x_i\left(0\right)‖$ and $‖\delta x_i\left(t\right)‖$, respectively. The Lyapunov exponent is defined as follows:

$$\frac{‖\delta x_i\left(t\right)‖}{‖\delta x_i\left(0\right)‖}=e^{{\lambda }_{i}t}\to {\lambda }_i=\underset{t\to \infty }{lim}\left(\frac{1}{t}ln\frac{‖\delta x_i\left(t\right)‖}{‖\delta x_i\left(0\right)‖}\right)$$E1

In this equation, ${\lambda }_i$ represents the Lyapunov exponent. As can be seen, two points with infinitesimally small proximity in the initial state, diverge significantly from each other in the direction of the (i). This phenomenon is referred to as “sensitivity to initial conditions.”

1. If the ${\lambda }_i<0$, then we will have a stable fixed point or a stable periodic cycle. In other words, all selected initial points will converge toward a fixed point or a periodic cycle. These systems are called asymptotically stable. With a negative increase, ${\lambda }_i\to -\infty$, the stability of the system increases, such that for ${\lambda }_i=-\infty$, there exists a super-stable fixed point or periodic cycle.

2. If ${\lambda }_i=0$, the system only oscillates around a fixed point. In this case, every selected initial point oscillates around a stable limit cycle.

3. If ${\lambda }_i<0$, there are no stable fixed points or limit cycles; in fact, the points are unstable, but the system is bounded and chaotic. In other words, if the largest Lyapunov exponent in the system is positive, the system is chaotic; otherwise, it is not chaotic [7, 11, 16, 17].

Chaotic systems have unique properties that distinguish them from other dynamics. One of the distinctive features of chaotic systems is their strong dependence on initial conditions. One powerful tool for detecting chaos is the Lyapunov exponent.

A variety of methods have been presented for calculating the Lyapunov exponent, one of which is the calculation of the Lyapunov exponent using time series [7]. However, Lyapunov exponent is highly sensitive to noise, which is why using Lyapunov exponent as a criterion for detecting chaos in noisy environments is not a good measure [7, 8, 9]. As seen, chaotic dynamics create complex attractors that are limited to a part of space and cannot cover the entire space. Various methods have been proposed to calculate the dimensions of chaos and hidden dynamics, including correlation dimensions and fractal dimensions. One of the disadvantages of these methods is their computational complexity and their dependence on noise. Another method is to use R/S analysis and the Hurst exponent [18]. This method is used to distinguish between random time series and non-random and chaotic time series. The disadvantages of this method include computational complexity and the inability to detect chaos in noisy environments. Another method for analyzing chaotic signals is to use the Kolmogorov-Sinai entropy. This law talks about the disorder in a system. A random signal has the most disorder, but a deterministic system has the most order. This entropy is related to the Lyapunov expressions of the signal. This entropy is the average of the positive Lyapunov exponents of the system. One of the limitations of this method is its high sensitivity to noise [19].

3. Detection of nonlinear systems behavior

Many issues in various fields, including electrical engineering, are inherently nonlinear and are modeled using partial and ordinary differential equations. Only a limited number of these equations have exact solutions, and most of these problems do not provide precise answers, necessitating the use of novel methods for their analysis. Based on the observed time series of a process, detecting the presence of nonlinearity is quite challenging. Therefore, efforts have been made to provide tools for identifying the behavior of nonlinear systems. The behavior of nonlinear systems in phase space is highly intricate. A simple approach to understanding this behavior is to plot the time series in phase space and analyze the patterns created. In this section, various criteria for analyzing the behavior of nonlinear systems are introduced.

3.1 Calculation of the Lyapunov exponent based on the analytical method of differential transformation and behavioral analysis of nonlinear systems with respect to unknown parameters

In order to analyze the behavior of nonlinear dynamic systems, a method for calculating the Lyapunov exponent based on various analytical differential transformation methods has been proposed. In this process, using the differential transformation method, the time series of the desired system is calculated and replaced in relation to the Lyapunov exponent, and then the behavior of the system is determined using the calculated Lyapunov exponents. In this section, the Lyapunov exponent method based on three representative limit cycle algorithms has been proposed, with the aim of clarifying them further. Since the behavior of nonlinear dynamic systems depends on the changes in their parameters, for example, in a Colpitts oscillator, the behavior of the oscillator depends on the changes in parameters such as Inductor and capacitor value, therefore, to ensure that a nonlinear system has the desired behavior, the most effective approach is to accurately adjust the parameters, if possible. Consequently, the analysis of the behavior of nonlinear dynamic systems with respect to changes in various parameters of the systems has been studied using the Lyapunov exponent method based on the differential transformation method.

3.1.1 Calculation of the Lyapunov exponent based on the classical analytical differential transformation method

The aim of this section is to determine the different behaviors of nonlinear dynamic systems using the Lyapunov exponent method based on time series. Accordingly, changes in the Lyapunov exponent for unknown parameters have also been investigated. This process, using the classical differential transformation method and the Lyapunov exponent method based on time series, is described in the context of method-1.

Method-1: Let us assume that the time series $\left\{x_1\left(t\right),x_2\left(t\right),\dots ,x_n\left(t\right)\right\}$, is the solution of the main Eq. (1), and the following expression,

$${\lambda }_j=\frac{1}{T}\mathit{ln}\left|\frac{x_j\left(r+T\right)-x_j\left(s+T\right)}{x_j\left(r\right)-x_j\left(s\right)}\right|,j=1,2,\dots ,N.$$E2

${\lambda }_j$ represents the Lyapunov exponent of the system under consideration, in which T, the time of evolution, and r and s are two selected nearby sample points on the path. In this case, by applying the differential transformation method, the Lyapunov exponent of the system in the time series space will be as follows.

$${\lambda }_j=\frac{1}{T}\mathit{ln}\left|\frac{X_j\left(1\right)\left(r-s\right)+{\sum }_{k=2}^n{X}_j\left(k\right)\left[{\left(r+T\right)}^k-{\left(s+T\right)}^k\right]}{X_j\left(1\right)\left(r-s\right)+{\sum }_{k=2}^n{X}_j\left(k\right)\left[r^k-s^k\right]}\right|,j=1,2,\dots ,N$$E3

Proof: Clearly, by substituting the point “N” from the time series into the differential transformation method $\left(x\left(N\right)={\sum }_{k=0}^n{X}_j\left(k\right)N^k\right)$ for the definition of the corresponding Lyapunov coefficient, the desired relationship is obtained.

In continuation, with the help of presenting the algorithmic steps, we describe the use of method-1.

Algorithm 1: Calculation of the Lyapunov Exponent Based on the Classical Differential Transformation Method.

First step: Applying the differential transformation method to the system state equations (calculating the recursive relationships).

$$X_j\left(k\right)=\frac{1}{k!}{\left[\frac{d^k{x}_j\left(t\right)}{{\mathit{dt}}^k}\right]}_{t=t_0},j=1,2,\dots ,N.$$E4

Second step: Calculating the limited time series using the differential transformation method.

$$x_j\left(t\right)=\sum _{k=0}^N{X}_j\left(k\right){\left(t-t_0\right)}^k,j=1,2,\dots ,N.$$E5

Third step: Calculating the Lyapunov exponent based on the differential transformation method using method-1.

$${\lambda }_j=\frac{1}{T}\mathit{ln}\left|\frac{X_j\left(1\right)\left(r-s\right)+{\sum }_{k=2}^n{X}_j\left(k\right)\left[{\left(r+T\right)}^k-{\left(s+T\right)}^k\right]}{X_j\left(1\right)\left(r-s\right)+{\sum }_{k=2}^n{X}_j\left(k\right)\left[r^k-s^k\right]}\right|,j=1,2,\dots ,N$$E6

Fourth step: Analysis of the changes in Lyapunov exponents with respect to unknown parameters and identification of different system behaviors [20, 21, 22].

3.2 Identification of behavior based on analysis in the frequency domain

One method for detecting chaotic behavior from periodic behavior involves using frequency domain analysis. This process entails plotting the frequency spectrum of a time series, revealing features that may not be easily observed in the time domain. In periodic signals, energy concentrates at specific frequencies, while chaotic behaviors exhibit a frequency spectrum with non-zero values across various frequencies, creating a wide band. In deterministic systems, a wide band spectrum can signal the onset of chaos. However, it’s essential to note that relying solely on frequency analysis is not always accurate for determining the presence of chaos. For instance, power spectrum characteristics are also employed in frequency domain analysis. Additionally, it’s worth mentioning that the frequency spectrum of a random time series or a time series from real dynamic systems affected by noise will also exhibit a wide band, making it challenging to distinguish between these scenarios based solely on the frequency spectrum [23].

3.3 Nonlinear system behavior detection based on time-frequency analysis

As mentioned, the methods for detecting chaos in the time and frequency domains are not robust against noise. Another method used for chaos detection is time-frequency analysis. One of these methods is based on the short-time Fourier transform. The main idea of the short-time Fourier transform is to multiply the input signal x(t) by a window function $w\left(\tau \right)$, which changes its location with time. In other words, the signal is divided into short-time segments, and the Fourier transform is applied to each segment. In this way, each frequency spectrum shows the frequency content in a short-time period. Such a spectrum includes the change of frequency content with time. The short-time Fourier transform is defined as follows [24, 25]:

$$\mathit{\text{STFT}}\left(t,f\right)=\underset{-\infty }{\overset{\infty }{\int }}x\left(t+\tau \right)w\left(\tau \right)e^{-j2\mathit{\pi f\tau }}\mathit{d\tau }$$E7

where $x\left(t\right)$ is the input signal, $w\left(\tau \right)$ is the window function with the width of the T, and X(τ,ω) is the complex-valued spectrum.

$$w\left(\tau \right)=\left\{\begin{array}{lll}1& \text{for}& m\left(t\right)\ge c\left(t\right)\\ 0& \text{for}& m\left(t\right)\le c\left(t\right)\end{array}\right.$$E8

Short-time Fourier transform determines which frequency components and at what times are present in the signal. The algorithm for detecting chaos based on short-time Fourier transform is as follows:

1. Calculate the short-time Fourier transform

2. Estimate the dominant spectral components frequency

   The frequency at which the short-time Fourier transform frequency of the signal is maximized is calculated according to the following equation:

   $$f_m\left(t\right)=arg{max}_f\text{STFT}\left(t,\mathrm{f}\right)$$E9

   where, f represents the frequency and t the time.

3. We define the $u_{\mathrm{\Omega }}\left(t,f\right)$ as follows:

   $$u_{\mathrm{\Omega }}\left(t,f\right)=\left\{\begin{array}{l}1\text{STFT}\left(t,\mathrm{f}\right)\ge 0.01{\mathit{max}}_f\left|\text{STFT}\left(t,\mathrm{f}\right)\right|\\ 0\text{elsewhere},\end{array}\right.$$E10

4. In order to detect chaos, we define the following function:

   $$\mathrm{m}\left(\mathrm{t}\right)=\underset{0}{\overset{f_m\left(t\right)}{\int }}{u}_{\mathrm{\Omega }}\left(t,f\right)\mathrm{d}f$$E11

5. To detect chaos, we operate as follows:

$$d\left(t\right)=\left\{\begin{array}{l}1\text{for}\mathrm{m}\left(\mathrm{t}\right)\ge \mathrm{c}\left(\mathrm{t}\right)\\ 0\text{for}\mathrm{m}\left(\mathrm{t}\right)\le \mathrm{c}\left(\mathrm{t}\right)\end{array}\right.$$E12

If $d\left(t\right)=1$, it indicates a chaotic signal and if $d\left(t\right)=0$, it indicates a periodic signal. In this case, $\mathrm{c}\left(\mathrm{t}\right)$ is the threshold value that depends on the selected window width. This method is resistant to noise and is used to detect chaos in a noisy environment. One of the drawbacks of this method is its strong dependence on the selected window width. On the other hand, this method cannot detect random signals such as noise. Another new method that has been recently used is the use of energy distribution based on continuous wavelet transform in chaotic systems [26]. Continuous wavelet transforms of :

$$W_x\left(a,b\right)=⟨x,W⟩=\frac{1}{\sqrt{\left|a\right|}}\underset{-\infty }{\overset{\infty }{\int }}x\left(t\right)\overline{\mathrm{\Psi }}\left(\frac{t-b}{a}\right)\mathit{dt}$$E13

In which a and b are the scale and shift parameters, respectively and $\overline{\mathrm{\Psi }}\left(\frac{t-b}{a}\right)$ is the complex conjugate of the $\mathrm{\Psi }\left(\frac{t-b}{a}\right)$ function.

The concept of shift in the wavelet transform is similar to the concept of time shift in the short-time Fourier transform. The shift represents the amount of window displacement and contains the time information of the transform. However, unlike the short-time Fourier transform, the wavelet transform does not have a direct frequency parameter. Instead, it uses a scale parameter that is inversely related to the frequency. In other word: $a=\frac{1}{f}$

A function ψ is called a wavelet if:

1. It has a nonzero value only in a certain range (small wave).

2. It has a limited frequency range.

In this case, the energy distribution based on continuous wavelet transform to detect chaos is used. The algorithm for detecting chaos is as follows:

First, we sample the signal. In this case, we divide the signal into N parts which are equal. Then, for each part, we apply the continuous wavelet transform and calculate the wavelet coefficients and the energy of each part from the coefficients of the continuous wavelet transform according to Parseval’s theorem. In chaotic systems, the energy distribution changes irregularly. This method is not resistant to noise [26].

4. A recent criterion for distinguishing chaos from noise

In recent years, various nonlinear electronic systems that exhibit chaotic behavior have been studied. Some behaviors that are hidden in normal conditions due to the presence of noise are real examples of chaotic behavior of a completely deterministic nature. One of these systems is oscillators. The output signal of ideal oscillators is a periodic function in the time domain and especially at high frequencies is usually sinusoidal. Also, regardless of factors that are usually negligible such as heat and wear, it can be said that the frequency and amplitude of this signal are always constant. But anyway, in practical oscillators, these undesirable noise effects cause minor disturbances in the frequency, phase, and amplitude of the output signal. On the other hand, oscillators can show chaotic behavior for some parameters. For this reason, chaos analysis in oscillators and its separation from noise is of great importance.

Chaotic systems exhibit behavior that resembles random processes, yet they are non-random. Chaotic series are a subset of nonlinear processes known for their high complexity and irregular behavior. Although chaotic time series may appear random, they possess distinct properties that set them apart from truly random series. One key characteristic of chaotic processes, which differentiates them from random processes, is their sensitivity to initial conditions. Even a slight error in measuring the initial state can lead to exponential growth in the Lyapunov exponent in future values of the time series. In most cases, the frequency spectrum and autocovariance function of chaotic series resemble white noise. In fact, chaotic processes often share first and second-moment properties with white noise and colored noise. The frequency spectrum obtained from a random time series and a time series associated with real dynamic systems mixed with noise both exhibit a wide band. Therefore, distinguishing between these cases based solely on the frequency spectrum is not possible. A criterion for detecting and distinguishing chaos from noise is presented below [27].

Theorem 1: The variance of the autocorrelation coefficient of the energy signal of chaos is greater than the variance of the autocorrelation coefficient of the energy signal of noise.

Proof: Consider the following continuous-time system:

$$\dot{x}=f\left(x,\phi \right)$$E14

In this case, according to Lyapunov exponent, it can be written

$$‖\mathit{\delta x}\left(t\right)‖=‖\mathit{\delta x}\left(0\right)‖e^{\mathit{\lambda t}}\stackrel{‖\mathit{ax}‖=‖a‖‖x‖}{\to }‖\mathit{\delta x}\left(t\right)‖=‖\mathit{\delta x}\left(0\right)e^{\mathit{\lambda t}}‖\to \mathit{\delta x}\left(t\right)=\mathit{\delta x}\left(0\right)e^{\mathit{\lambda t}}$$E15

where,$\lambda$ is the largest Lyapunov exponent and $\mathit{\delta x}\left(t\right)$ is the variation of the signal $x$. In chaotic systems, as we know, $\lambda \succ 0$.

According to the definition of the norm of the signal and energy, we can write:

$${‖x‖}^2=E_x\left(t\right)\to E_x\left(t\right)={\left(‖\mathit{\delta x}\left(0\right)‖e^{\mathit{\lambda t}}\right)}^2$$E16

On the other hand, the spectrum of color noise power in general can be considered as follows:

$$S\left(\alpha \right)=\frac{A}{f^{\alpha }}\to X\left(f\right)=\frac{\sqrt{A}}{f^{\frac{\alpha }{2}}}\to {\mathrm{E}}_n\left(t\right)=\int X{\left(t\right)}^2\mathit{dt}\to {\mathrm{E}}_n\left(t\right)=0.31\mathit{Aln}\left(t\right)$$E17

To show that $E_n\left(t\right)\le {\mathrm{E}}_x\left(t\right)$, we must show that $E_n\left(t\right)-E_x\left(t\right)\le 0$. For this purpose, the Taylor series of the multivariable function $E_n\left(t\right)-E_x\left(t\right)\le 0$ around the variables $A,t,‖\mathit{\delta x}\left(0\right)‖,\lambda$ is written as follows:

$$E_n\left(t\right)-E_x\left(t\right)=0.31Aln\left(t\right)-{‖\mathit{\delta x}\left(0\right)‖}^2-2{‖\mathit{\delta x}\left(0\right)‖}^2\mathit{\lambda t}-4{‖\mathit{\delta x}\left(0\right)‖}^2{\lambda }^2{t}^2$$E18

As can be seen, for $t\prec 1$ this relation is negative and for t > 1 for $A<\frac{{‖\mathit{\delta x}\left(0\right)‖}^2}{0.31ln\left(t\right)}$ we can conclude: $E_n\left(t\right)\le {\mathrm{E}}_x\left(t\right)$

By multiplying both sides of the above relation by, $e^{-\mathit{st}}$, we have:

$$\begin{array}{l}{E}_n\left(t\right)\le {\mathrm{E}}_x\left(t\right)\to E_n\left(t\right)e^{-\mathit{st}}\le {\mathrm{E}}_x\left(t\right)e^{-\mathit{st}}\to \\ \underset{0}{\overset{\infty }{\int }}{E}_n\left(t\right)e^{-\mathit{st}}\le \underset{0}{\overset{\infty }{\int }}{\mathrm{E}}_x\left(t\right)e^{-\mathit{st}}\to E_n\left(s\right)\le {\mathrm{E}}_x\left(s\right)\end{array}$$E19

According to the definition of autocorrelation coefficient $R_{{\mathrm{E}}_x\left(t\right)}=L^{-1}\left({\mathrm{E}}_x{\left(s\right)}^2\right)$, the above relation can be written as subscript:

$$R_{{\mathrm{E}}_n\left(t\right)}\le R_{{\mathrm{E}}_x\left(t\right)}$$E20

Using the properties of the probability distribution function:

$$E_n\left(t\right)\le {\mathrm{E}}_x\left(t\right)\to f_n\le f_x$$E21

Where $f_n$ is the noise energy distribution function and $f_x$ is the chaotic signal energy distribution function. From the two relations (20) and (21):

$$\int R_{{\mathrm{E}}_n}{f}_n\le \int R_{{\mathrm{E}}_x}{f}_x\to {\mu }_{R_{{\mathrm{E}}_n}}\le {\mu }_{R_{{\mathrm{E}}_x}}$$E22

Where ${\mu }_{R_{{\mathrm{E}}_x}}$ is the mean of the autocorrelation coefficient of chaos and ${\mu }_{R_{{\mathrm{E}}_n}}$ is the mean of the autocorrelation coefficient of noise signal. From the two relations (20) and (22) we can conclude:

$$R_{{\mathrm{E}}_n}\left(t\right)-{\mu }_{R_{{\mathrm{E}}_n}}\le R_{{\mathrm{E}}_x}\left(t\right)-{\mu }_{R_{{\mathrm{E}}_x}}\to {\left(R_{{\mathrm{E}}_n}\left(t\right)-{\mu }_{R_{{\mathrm{E}}_n}}\right)}^2{f}_n\le {\left(R_{{\mathrm{E}}_x}\left(t\right)-{\mu }_{R_{{\mathrm{E}}_x}}\right)}^2{f}_x$$E23

Therefore, according to the definition of variance, it can be written:

$$\int {\left(R_{{\mathrm{E}}_n}\left(t\right)-{\mu }_{R_{{\mathrm{E}}_n}}\right)}^2{f}_n\mathit{dt}\le \int {\left(R_{{\mathrm{E}}_x}\left(t\right)-{\mu }_{R_{{\mathrm{E}}_x}}\right)}^2{f}_x\mathit{dt}\to {\delta }_{R_{{\mathrm{E}}_n}}^2\le {\delta }_{R_{{\mathrm{E}}_x}}^2$$E24

This means that in a noisy environment, the variance of the autocorrelation coefficient of the energy signal of chaos is greater than the variance of the autocorrelation coefficient of the energy signal of noise.

Method 1: Compare the variance of the autocorrelation coefficient of the energy signal of chaos in a noisy environment to the variance of the autocorrelation coefficient of the energy signal of noise.

Proof: For the chaotic signal $x\left(t\right)$ in a noisy environment as $y\left(t\right)=x\left(t\right)+n\left(t\right)$, according to the variance properties, we can write:

$$var\left(x+y\right)=var\left(x\right)+var\left(y\right)+2cov\left(x,y\right)$$E25

The energy of the chaotic signal in the noise medium is calculated as follows:

$$E\left(x\left(t\right)+n\left(t\right)\right)=‖x\left(t\right)+n\left(t\right)‖=E\left(x\left(t\right)\right)+E\left(n\left(t\right)\right)+2\int x\left(t\right)n\left(t\right)\mathit{dt}$$E26

Where, $n\left(t\right)$ is a noise signal.

So:

$$var\left(E\left(x\left(t\right)+n\left(t\right)\right)\right)>var\left(E\left(n\left(t\right)\right)\right)$$E27

In this relation, $E\left(\dots \right)$ represents energy.

As seen in Theorem 1 and Method 1, due to the dependence of the chaotic signal on the nonlinear dynamics of the system, it can be said that chaos has smoother changes than noise, but due to the random nature of noise, noise follows more severe changes. For this reason, it can be said that the variance of the energy of the chaotic signal is greater than the variance of the energy of the noise signal in different frequency sub-bands. As it is clear, by separating the high-frequency part in the chaotic signal mixed with noise, the effect of noise on the signal is reduced. Because noise often appears in high frequencies, by taking the signal information in the frequency domain and attenuating the high frequencies, which is equivalent to attenuating noise, chaos detection in a noisy environment can be better followed by examining the variance of energy in frequency sub-bands.

5. Chaos detection based on autocorrelation coefficient of energy distribution using static discrete wavelet transform

This method is based on a time-frequency analysis of the signal. By using static violet transformation, low-frequency components (signal general) and high-frequency components (signal details) are separated. The static wavelet transform is the same as the discrete wavelet transform. Figure 2 shows the general structure of the signal decomposition algorithm into low and high-frequency components using the discrete wavelet transform.

Figure 2.

General structure of signal decomposition using discrete wavelet transform.

Stationary wavelet transform is similar to discrete wavelet transform with the difference that sampling is not used in it (like Figure 3).

Figure 3.

General structure of signal decomposition using static wavelet transform.

The algorithm for detecting chaos is used to detect chaos.

1. The high-frequency components of the signal (signal details) are decomposed in several steps using the static wavelet transform (according to Figure 2).

2. Calculation of energy from detail coefficients (calculation of energy distribution in different frequency sub-bands) using Parswal’s relation.

   $$\mathrm{E}\left({\mathrm{d}}_{\mathrm{i}}\right)=\sum _{i=1}^{\mathrm{m}}{\left|{\mathrm{d}}_{\mathrm{i}}\right|}^2$$E28

3. Calculation of autocorrelation coefficient of energy distribution in different frequency sub-bands.

4. The variance of the autocorrelation coefficient of the energy distribution in different frequency sub-bands of the chaotic signal is more than the variance of the autocorrelation coefficient of the energy distribution in the frequency sub-bands of the noise signal.

For example, the fourth-order chaotic oscillator circuit based on memristor is shown in Figure 4. In Figures 5 and 6, the autocorrelation coefficient of energy distribution in different frequency sub-bands of chaos oscillator and the autocorrelation coefficient of energy distribution in frequency sub-bands of Gaussian white noise are considered.

Figure 4.

The circuit schematic of chaotic oscillator based on memristor.

Figure 5.

Autocorrelation coefficient of energy distribution in frequency sub-bands of chaos signal.

Figure 6.

Autocorrelation coefficient of energy distribution in frequency sub-bands of Gaussian white noise signal with mean 1 and variance 5.

6. Conclusions

In this chapter, we introduce a method for calculating the Lyapunov exponents using the differential transformation method to explore how various system parameters influence system behavior. We also propose a new criterion, based on the static violet transform, to differentiate between noise and chaos. Additionally, a method for detecting chaos based on energy distribution in various frequency sub-bands is described. Simulation results demonstrate the effectiveness of these methods in distinguishing chaos from noise.

Conflict of interest

There is no conflict of interest between the authors.

Declarations

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. This research did not receive any specific grant from any funding agency in the public, commercial, or not-for-profit sectors.