Weak signal detection method based on nonlinear differential equations

Abstract

With the rapid development of computer network technology, it is often necessary to collect weak signals to collect favorable information. The development of signal detection technology is ongoing; however, various issues arise during the detection process. These issues include low efficiency and a high signal noise threshold. However, many problems will be encountered in the process of detection. In order to solve these problems, the nonlinear chaos theory is introduced to detect signals, and the simulation experiments of weak pulse signals and weak partial discharge signals are carried out respectively. The experimental results showed that the detection effect was remarkable in the quasi periodic state, and it had a good detection effect for weak pulse signals. At a signal-to-noise ratio of $-$ 25 dB, double coupling system, two-way ring coupling system, and single ring coupling system displayed detection success rates exceeding 98%. Meanwhile, the detection success rate of the strong coupling system was only 12%. Even at a noise signal ratio as low as $-$ 40 dB, the dual coupling system still maintained a detection success rate above 80%. The simulation results of partial discharge signal detection showed that there was a high fluctuation only at 2 ms, and the rest was basically stable at about 0 V, indicating that the system had a strong suppression effect on Gaussian white noise. When comparing the simulation results of the detection of the new chaotic system and the double coupling system, it was found that the new chaotic system has a superior impact in detecting weakly attenuated partial discharge signals. Through analysis of the system’s dynamic behavior, the research confirms its rich dynamic characteristics and sheds light on the reasons for phase state mutation and missed detection. The noise system is utilized for comparing the performance of various systems, with the goal of enhancing the system’s detection capability.

Keywords

Nonlinear system differential equation chaos electrical signal pulse

1. Introduction

A weak signal is characterized by a very small amplitude and is easily disturbed by noise. Such signals can be found in various fields and contain a wealth of information. As a result, the detection of weak signals is of utmost importance [1]. As a comprehensive category method, this technology includes mathematics, physics, computer and other disciplines. The main purpose of this method is to obtain the important information hidden behind it from the weak signal. In order to obtain this information better, it is necessary to improve the accuracy and reduce the influence of noise in the signal [2]. The main focus of weak signal detection technology lies in controlling the signal-to-noise ratio (SNR). Real-time detection and accuracy are essential while controlling the SNR. Generally, the sensitivity of the system towards weak signals strengthens with a lower SNR and detected signal frequency, enabling weaker signals to be identified [3].

Chaos is a fundamental occurrence in nature that possesses qualities of unpredictability and irregularity, occurring without human influence. It highlights the intricacy and multidimensionality of nature, and represents the relationship between order and disorder, as well as necessity and chance within society [4]. Chaos theory was first developed in the 1960s, and its emergence is considered a significant systems theory which revolutionises the world. It thrives with nonlinear science progress. This theory describes random motion that typically occurs in a deterministic nonlinear system, but displays deterministic behaviour. Furthermore, a characteristic feature of nonlinear science is the phenomenon of chaos. Nonlinear systems exhibit the most common phenomenon, where absolute sensitivity to initial conditions is the most notable feature of their evolution. As the initial value changes constantly, the system’s periodic and quasi-periodic orbits correspondingly change at any given time [5]. Qiao and Shu studied the advantages of noise in coupled neurons for weak feature recognition, and proposed a multi-objective optimization method based on adaptive coupled neurons [6]. This approach overcomes the limitations of ineffective noise technology and eliminates significant background noise. This has high feasibility in detecting machinery faults at an early stage. Additionally, the research investigates the suppression of white noise due to the strong sensitivity of chaotic systems to weak signals and their anti-noise capabilities. Consequently, signal detection employs chaos theory based on nonlinear differential equations.

In recent years, the field of detecting weak signals has encountered significant challenges, including identifying difficulties arising from noise interference and a low signal-to-noise ratio. Moreover, actual signals tend to exhibit nonlinear characteristics, which further complicates detecting weak signals. The novelty of this study lies in proposing the utilization of distinct peak variances in the state variables of a chaotic system for detecting low-frequency signals. Hence, this paper proposes a weak noise detection system which merges nonlinear and chaotic models. This combination enables the mining of the signal’s nonlinear features while minimising the impact of errors and interference via randomness derived from chaos theory. The paper provides a detailed description of the chaos theory and related mechanical characteristics. Subsequently, a detection model based on chaotic systems is established, followed by simulation experiments on detecting weak pulse and partial discharge signals through the chaotic system.

This study is divided into five sections. Section 2 presents the current research status of chaos theory in nonlinear differential equations and weak signal detection both domestically and internationally. Using this theory, a model for weak signal detection is then established. In Section 4, the detection performance of four different models for partial discharge signal detection is compared. The study evaluates the detection probability and detection effects for each model. The model is then employed in the detection of faint signals. The findings indicate that utilising chaotic systems is effective in detecting weak signals and can significantly enhance the SNR. Moreover, this approach presents novel concepts for addressing issues in diverse fields.

2. Related works

With the rapid development of Internet technology, chaos theory has become increasingly precise in numerical simulation and has been applied to a range of complex network system problems. In order to enhance the variety of multi-wing chaotic attractors in higher-order chaotic systems, Wang proposed a novel three-dimensional higher-order chaotic system. The study has identified that the system’s chaotic behavior had a minimum order of 0.84. This provided conclusive evidence for the stability and controllability of the errors included [7]. Sharma and Sinha proposed a controller design method for controlling chaotic systems in order to make general nonlinear systems achieve the desired periodic or quasi-periodic motion. Based on chaos theory, the method replaces the quasi-periodic system with an approximate periodic system with an appropriately large principal period. Experimental results showed that the method had achieved success [8]. In order to verify the effectiveness and robustness of the new ASHLN-ELM method and the proposed method, Anh and Kien proposed a new nonlinear parameterized model. The biggest advantage of this model is that it can ensure that the system state residual is fast convergence to 0, and not affected by time-varying disturbance, the experimental results also confirmed the feasibility and effectiveness of the method [9]. On the basis of one-dimensional chaos, Huang et al. designed a new two-dimensional chaotic system through mapping, which realized a wider chaotic range and more complex chaotic behavior. The experimental results proved the effectiveness and reliability of the method [10]. To solve the threat of chaos oscillation in power system to the overall stability of power grid, Ma et al. have studied the dynamic behavior of power grid based on chaos theory by means of Lyapunov index map and fork diagram, and discussed the chaotic domain in power grid. The experimental results showed that the proposed method had good accuracy and availability [11].

Sambas et al. proposed a new type of linearly balanced three-dimensional chaotic system, and conducted a series of analyses on its dynamic properties. The results show that the throughput of the new chaotic system based on FPGA is 462.731 Mbps, and the system can effectively resist external attacks and has strong stability [12]. Chen et al. carried out additional research on chaotic systems and discovered that chaotic attractors and Lyapunov coexist in such systems. The results of the study indicate that chaotic behaviors may occur prior to the trajectory reaching the equilibrium point, which is valuable for better comprehension and practical applications of chaotic systems [13]. When studying chaotic system theory and FPGA, Sambas et al. introduced a three-dimensional chaotic system equipped with a closed butterfly curve with equilibrium points. The research showed that a new multi-stable chaotic model was finally realized using FPGA [14]. Bulut et al. discussed the availability of chaos synchronization in secure communication, realized master-slave synchronization through the active controller method, and carried out real-time simulation of five different chaotic systems, such as Lorenz, Sprott, Rucklidge, Moor-Spiegel and Rossler. Experimental results showed that the simulation output and real-time results were achieved successfully [15]. Xiong et al. studied a fixed parameter chemical oscillation chaotic system with different attractors, and built a new chaotic oscillation electronic circuit for the system, and the experimental results showed that it achieved effective control and implementation of the chaotic circuit [16]. Aqeel et al. carried out a thorough investigation into controlling chaotic behaviour in geomagnetic Krause and Roberts systems. They integrated the Krause-Roberts system with state space linearization technology, and the results of the experiment demonstrated the efficacy of state space linearization [17]. Xian and colleagues proposed a two-parameter fractal sorting vector for controlling iterative node relationships in spatio-temporal chaotic systems. The experimental results demonstrated that this method exhibited superior dynamic characteristics, improved encryption effectiveness, and better abilities to cope with diverse security analysis and simulation attacks [18].

Previous studies have shown that the research on chaos theory based on nonlinear ordinary differential equations has been very mature. And this theory has been widely used in the field of Internet, but it is rarely used in the field of signal detection. The research shows that this theory can be well used in signal detection. This research will analyze its detection methods and results.

3. Signal detection based on chaos theory in nonlinear systems

The research combines nonlinear models with chaotic models, which are ideal for describing nonlinear behaviors and interactions within intricate systems. It comprehensively captures nonlinear characteristics in the signal, thus enhancing sensitivity of the detection system to weak noise. The inclusion of the chaotic model can increase the anti-interference capability and stability of the detection system to weak noise, thereby enabling more precise and trustworthy noise detection results. This approach enables a comprehensive examination of the non-linear characteristics in the signal, and mitigates the impact of errors and interference caused by the random behavior in chaos theory.

3.1 Chaos theory and state judgment methods in nonlinear systems

The definition of chaos refers to an uncertain motion set in nonlinear theory. At present, the definition of chaos in the academic circle is still controversial, and the most common definitions are as follows.

Defined $I$ as a closed interval in the field of real numbers with mappings $f:I\to I$ , $\exists x\in I$ , such that $f({x_{n}})\leqslant x<f(x)<f({x_{2}})$ or $f({x_{3}})\geqslant x>f(x)>f({x_{2}})$ , and $f({x_{n}})=f({f({x_{n-1}})})$ . Assuming that the function is periodic and the period does not have an upper bound, $f$ has a periodic point of any positive integer period, there is an irreducible subset $S\subset I$ in the closed interval $I$ , and $S$ does not contain a periodic point, $\forall x,y\in S,x\neq y$ must satisfy the Eq. (1).

$\displaystyle\left\{{\begin{array}[]{l}\mathop{\lim}\limits_{n\to\infty}\sup|{% f({x_{n-1}})-f({y_{n-1}})}|>0\\ \mathop{\lim}\limits_{n\to\infty}\inf|{f({x_{n-1}})-f({y_{n-1}})}|=0\\ \end{array}}\right.$ (1)

For $\forall x\in S$ and periodic points $p\in I$ , there is Eq. (2).

$\displaystyle\mathop{\lim}\limits_{n\to\infty}\sup|{f({x_{n-1}})-f({p_{n-1}})}% |>0$ (2)

In Eq. (2), $x_{n}=f({x_{n-1}})$ and $y_{n}=f({y_{n-1}})$ are iteration sequences of $f$ and $S$ starting points respectively. $x_{0},y_{0}\in S$ . It is disorderly inside. During the iterative process of any two initial positions in the limit expression in Eq. (1), the highest value of the distance between the iterative points is a positive number. $S$ And the limit of its infimum is 0. The limit in Eq. (2) means that for $\forall x_{n}=f({x_{n-1}})$ both will not approach the periodic point [19].

Based on the definition, it is evident that the iterative sequence is non-convergent. As the number of iterations increases, there are instances when the sequences converge, while at other times they diverge. This phenomenon highlights that the phase trajectory of the chaotic system is an intricate structure with significant randomness. The system’s inherent randomness pertains to its non-periodic occurrence. Crucially, distinguishing between chaotic and standard random behaviour hinges on this feature of inherent randomness.

In constructing a state space via time series analysis, the Lyapunov exponent is crucial for objective measurement. For kinetic equations, a combination of both approaches is needed. Additionally, to determine variability, calculating the Lyapunov exponent and divergence index rate between adjacent trajectories of data-constructed state variables is necessary. Taking one-dimensional mapping as an example, if $x_{n+1}=f({x_{n}})$ there is a point around the $x_{0}+\delta x_{0}$ initial point, $x_{0}$ will be shown in Eq. (3) after n iterations.

$\displaystyle\delta x_{n}=|{f^{(n)}({x_{0}+\delta x_{0}})-f({x_{0}})}|=\frac{% df^{(n)}({x_{0}})}{dx}\delta x_{0}=e^{\lambda n}\delta x_{0}$ (3)

In Eq. (3), the Lyapunov exponent is shown in Eq. (4).

$\displaystyle\lambda=\frac{1}{n}\ln\frac{\delta x_{n}}{\delta x_{0}}=\frac{1}{% n}\ln\frac{df^{(n)}({x_{0}})}{dx}$ (4)

After using the differential rule of compound function, it is shown in Eq. (5).

$\displaystyle\frac{df^{(n)}({x_{0}})}{dx}=\frac{df({x_{0}})}{dx}\cdot\frac{df(% {x_{1}})}{dx}\ldots\frac{df({x_{i}})}{dx}\ldots\frac{df({x_{n-1}})}{dx}$ (5)

In Eq. (5), $x_{i}=f({x_{i-1}})=f^{(i)}({x_{0}})$ , as shown in Eq. (6).

$\displaystyle\lambda=\frac{1}{n}\ln\left|{\mathop{\prod}\limits_{i=0}^{n-1}f^{% \prime}({x_{i}})}\right|=\frac{1}{n}\sum\limits_{i=0}^{n-1}{\ln|{f^{\prime}({x% _{i}})}|}$ (6)

Or as shown in Eq. (7).

$\displaystyle\lambda=\mathop{\lim}\limits_{n\to\infty}\frac{1}{n}\sum\limits_{% i=0}^{n-1}{\ln|{f^{\prime}({x_{i}})}|}$ (7)

Usually, the number of Lyapunov exponents is $\lambda$ determined by the dimensionality of the complex system. At the same time, the Lyapunov exponent also determines the formation of stable orbits. $\lambda<0$ , the nonlinear system is in a non-chaotic state. $\lambda\geqslant 0$ , the nonlinear system is in a chaotic state. This method also has a major disadvantage. It often takes a long time to discriminate, requires a lot of computation, and is prone to error. According to the knowledge of mathematics and physics, entropy can be a good measure of the randomness or chaos of the phase transition state of a dynamical system. Entropy is also related to $\lambda$ existence. Kolmogorov entropy is usually recorded as K-entropy. There are two main methods for calculating K-entropy in academic circles: maximum likelihood method and correlation integral method [20]. When using the associative integral method, the definition of n-order Renyi entropy is introduced first, as shown in Eq. (8).

$\displaystyle K_{n}=-\mathop{\lim}\limits_{\tau\to 0}\mathop{\lim}\limits_{r% \to 0}\mathop{\lim}\limits_{m\to\infty}\frac{1}{m\tau({n-1})}\ln\sum\limits_{i% _{1},\ldots,i_{m}}{P^{n}({i_{0},i_{1},\ldots,i_{m}})}$ (8)

In Eq. (8), $P^{n}({i_{0},i_{1},\ldots,i_{m}})$ is the n-level joint probability distribution. The correlation integral $C({r,m})$ is shown in Eq. (9).

$\displaystyle C({r,m})=\mathop{\lim}\limits_{N\to\infty}\frac{1}{N({N-1})}\sum% \limits_{i=1}^{N}{\sum\limits_{j=1}^{N}{\theta({r-\|{X_{i}-X_{j}}\|})}},i\neq j$ (9)

And the Heaviside function is shown in Eq. (10).

$\displaystyle\theta(x)=\left\{{\begin{array}[]{l}1,x>0\\ 0,x\leqslant 0\\ \end{array}}\right.$ (10)

In Eq. (9), N represents the reconstructed value of the number of phase space vectors; $\|{X_{i}-X_{j}}\|$ represents the distance of different state vectors in Euclid space, and $C({r,m})$ represents the ratio of the number of point pairs whose distance is not greater than r in the phase space to the number of all point pairs. Kolmogorov entropy calculated based on correlation integral $C({r,m})$ is shown in Eq. (11).

$\displaystyle K_{2}=-\mathop{\lim}\limits_{\tau\to 0}\mathop{\lim}\limits_{r% \to 0}\mathop{\lim}\limits_{m\to\infty}\frac{1}{m\tau}\ln C({r,m})$ (11)

Since the time delay and the embedding dimension are constant, it can be simplified as shown in Eq. (12).

$\displaystyle K_{2}=-\mathop{\lim}\limits_{r\to 0}\frac{1}{m\tau}\ln C({r,m})$ (12)

When $r\to 0$ , the Kolmogorov entropy can be obtained as Eq. (13).

$\displaystyle K_{2}=\frac{1}{m\tau}\ln\frac{r^{D_{2}}}{C({r,m})},r\to 0$ (13)

When the K-entropy is equal to 0, the system is in a periodic state; when the K-entropy approaches infinity, the system is in random motion; when the K-entropy is a constant greater than 0, the system is in a chaotic state.

The chaotic attractor is generated by multiple iterations of the phase trajectory of the dynamical system and has the properties of self-similarity. There is a correlation dimension within the attractor, which is usually used to reflect the information contained in the chaotic attractor and the complexity of the system. When solving the correlation dimension, GP algorithm is usually used to arbitrarily take the system state variable as the time series $\{{x_{1},x_{2},\ldots,x_{m}}\}$ . Spatial reconstruction is carried out on the obtained time series, as shown in Eq. (14) [21].

$\displaystyle x_{i}({\tau,m})=({x_{i},x_{i+\tau},\ldots,x_{i+({m-1})\tau}}),% \tau=1,2,\ldots,m$ (14)

Equation (14), $\tau$ represents the time delay, and m represents the embedding dimension. Then the correlation dimension of the phase space of the system is shown in Eq. (15).

$\displaystyle D_{2}=\mathop{\lim}\limits_{r\to 0}\frac{\ln C({r,m})}{\ln r}$ (15)

From the GP algorithm, both the time delay and the embedding dimension will seriously affect the accuracy of the associative dimension. The associative dimension is usually an integer, but there are a few moments when it is an integer. When the system is in a periodic state, it is an integer, and when the system is in a non-periodic state, it is a non-integer [22].

3.2 Chaos theory detection principle and establishment of detection model

The accuracy of the associative dimension is significantly affected by both the time delay and the embedding dimension, as in the GP algorithm. Although it is generally an integer, there are certain occasions when it can be non-integer as well. Specifically, the dimension becomes an integer in a periodic state and a non-integer in a non-periodic state [22].

Figure 1.

Simulation model of weak signal detection based on chaos theory.

As a second-order nonlinear system, the chaotic system has the characteristics of simple structure and diverse parameters, and its dynamic equation is shown in Eq. (16).

$\displaystyle\left\{{\begin{array}[]{l}\dot{x}=y+a+\gamma\sin(\omega t)\\ \dot{y}=(-xy/b)-x^{3}+c\\ \end{array}}\right.$ (16)

In Eq. (16), the system state variable $x, y$ represents a fixed parameter in the $\gamma\sin(\omega t)$ system, i.e., the internal drive signal of the system. The system usually has two forms: a chaotic state and a periodic state, and the internal driving signal of the chaotic system gradually increases from zero. Typically, chaotic systems are very sensitive to weak signals, so the system often switches between two states. Before and after the critical point, the threshold in the critical chaotic state is usually called the critical chaotic value, and vice versa the critical periodic value. If there is no mutation point, the chaotic system will not be able to detect weak signals. Before detection, it is necessary to set the initial phase of the built-in drive signal of the system to the initial value of zero, and the drive signal is as shown in Eq. (17).

$\displaystyle F(t)=\gamma_{d}\sin t+A\sin({({1+\Delta\omega})t+\Delta\varphi})% =\gamma^{\prime}\sin({t+\theta})$ (17)

In Eq. (17),

$\displaystyle\left\{{\begin{array}[]{l}\gamma^{\prime}=\sqrt{\gamma_{d}^{2}+2% \gamma_{d}A\cos({\Delta\omega t+\Delta\varphi})+A^{2}}\\ \theta=\arctan\left({\frac{A\sin({\Delta\omega t+\Delta\varphi})}{\gamma_{d}+A% \cos({\Delta\omega t+\Delta\varphi})}}\right)\\ \end{array}}\right.,$

it is also necessary to determine many parameter relationships between the signal to be detected, the driving signal inside the system, and each phase. It should also be noted that there is no frequency or phase difference between the driving signal inside the chaotic system and the signal to be detected, and their system periodic state is shown in Fig. 2.

Figure 2.

Periodic state of the system when there is no frequency difference and no phase difference.

As shown in Fig. 2, when the frequency of the signal to be detected is equal to the internal driving signal of the system, the signal to be detected and the driving signal are linearly superimposed, and the amplitude of the superimposed signal is much greater than the critical chaotic threshold. At this point, the system changes from a chaotic state to a periodic state. Indicates that an incoming signal has been detected. The phase transition state of the system when there is no frequency difference and phase difference is shown in Fig. 3.

Figure 3.

System phase transition state when there is no frequency difference and phase difference.

When the frequency of the signal being detected is the same as the internal driving signal of the system (as shown in Fig. 2), the signal being detected and the driving signal are overlaid in a linear fashion, resulting in a superimposed signal amplitude that exceeds the critical chaotic threshold. This state change indicates the system has switched from a chaotic to a periodic state and signifies the detection of an incoming signal. Although the difference between the built-in drive signal and the signal to be measured is important, it only determines whether the system behaves in a periodic state or a chaotic state, and has no effect on the overall phase transition state. However, the appearance of frequency difference will cause the system to be in an unstable state, that is, it will show intermittent chaotic phenomenon [23]. Therefore, the intermittent chaotic state when the system has a frequency difference is shown in Fig. 4.

Figure 4.

Intermittent chaotic state when the system has frequency difference.

As shown in Fig. 4, when the frequency difference exists, the state of the system is changeable in motion and in most cases shows periodic dynamic changes. When setting up the detection model, it is also necessary to determine the critical chaotic threshold of the chaotic system. Combined with the system dynamics Eq. (16), the phase transition state of the system when the frequency difference exists is shown in Fig. 5.

Figure 5.

System phase transition state with frequency difference.

Figure 6.

Phase transition state of critical threshold of new chaotic system.

As shown in Fig. 5, the critical chaos threshold of the system is obtained from the figure $\gamma_{d}=$ 20541234755. From the phase diagram, it can be oberved that the amplitudes of the internal drive signal and the detection signal are superimposed, and when the frequency difference between the two is small, the transition frequency between the chaotic state and the periodic state is also small. So each state exists for a long time, which makes it easy to detect. At the same time, as the frequency difference between the two gradually increases, the switching frequency between the periodic state and the chaotic state also increases, the switching speed becomes faster and faster, and the duration of each state becomes shorter and shorter. When the state is detected, it is very easy to miss the detection.

When constructing the simulation detection model, the critical chaotic threshold of the new chaotic system should be determined first. Then set the fixed parameters of the chaotic system. The settings $a=3,b=10,c=2$ and the initial state of the system are $({x(0),y(0)})=({0,0})$ , the angular frequency of the driving signal $\omega_{o}=$ 1 rad/s, the simulation time $T=$ 800 s, the noise variance $\sigma^{2}=$ 0.001 of the Gaussian white noise, and the critical chaotic threshold is obtained $\gamma_{d}=$ 2.541234755, such as shown in Fig. 6.

As shown in Fig. 6, the error of the obtained threshold is extremely small, and the minimum error is only $10^{-9}$ . Although the error is small, small changes can make a big difference. Usually, the obtained threshold is directly used in the new chaotic system as the critical chaotic threshold, which can effectively ensure the accuracy of weak signal detection. Secondly, to avoid the situation of missing detection in the signal to be detected with any phase difference, two forward and backward chaotic systems can be set together as the same monitoring model. Finally, the signals to be detected are grouped according to the frequency band position, and the simulation step sequence is set $a_{n}$ , and other parameters remain unchanged.

Figure 7.

Detection results of positive and negative alternating pulse signal.

4. Analysis of simulation results of weak signal detection based on nonlinear chaos theory

To confirm the proposed method’s effectiveness and demonstrate the superiority of the enhancement, the MATLAB/Simulink is employed to develop a simulation model for detecting weak signals in a chaotic system using a nonlinear model. Additionally, the MATLAB/Simulink is utilized to simulate the alignment period’s dual-coupled system, while a radio frequency signal transmitter generated the necessary weak signal for our investigation. When detecting the positive and negative alternating square wave pulse signal, a double coupling system needs to be used separately. There are two groups of positive and negative bidirectional pulse signals hidden in the square wave pulse signal $s(t)$ . The pulse signal width is 50 mm; the variance of the $\delta^{2}=$ 0.01 added noise is $n(t)$ , and the initial state of the double-coupling system is set as $(x_{1},y_{1})=(x_{2},y_{2})=(0,0)$ . The detection results of positive and negative alternating pulse signals are shown in Fig. 7.

As shown in Fig. 7, the pulse signals at 150 s, 250 s, 350 s and 450 s were detected. And when the synchronisation difference is caused, the reverse oscillation does not appear in the results, and the detection effect is good. To investigate the influence of SNR on the detection performance of the dual-coupling system, the change of SNR was achieved by changing the variance in this experiment. The simulation experiment was performed 100 times for each group, and the detection probability of each group was calculated. If the output SNR is greater than 7.95 dB, the detection is considered successful. By comparing several different detection models, the results are shown in Table 1.

Table 1
Detection probabilities of several different chaotic systems under different signal-to-noise conditions

System type	Pulse type	SNR/dB
		$-$ 8	$-$ 15	$-$ 20	$-$ 25	$-$ 40
Double coupling system	Square wave	100	100	100	300	81
Bidirectional ring coupling system	Square wave	100	99.8	99.6	/	/
Unidirectional ring coupling system	Square wave	99.5	99.0	98.4	/	/
Strongly coupled system	Square wave	100	92.3	12	/	/

As shown in Table 1, under the condition of $-$ 20 dB, both single-phase and two-way coupled systems have detection rates as high as 97 or more. Compared to the other systems, the single-coupling system has a weaker detection capability for low SNR signals. The detection probability of the double coupling system is still above 80% at $-$ 40 dB, indicating that this system has a good effect on noise suppression. From the simulation results, it can be seen that the double coupling system has a good anti-noise performance and can detect pulse signals in most states, especially in the quasi-periodic state. Through previous researches, it is found that there are mainly four mathematical models commonly used in the detection of partial discharge signals. The schematic diagram of the model is shown in Fig. 8.

Figure 8.

Four mathematical models for partial discharge signal detection.

As shown in Fig. 8, among the four models, the single exponential decaying oscillation model and the double exponential decaying oscillation model both have the property of a sinusoidal function image. By comparing the principles of the new chaotic system, they are highly compatible with it. Therefore, a new type of chaotic system is chosen to capture the mathematical models of the partial discharge signals in the two damped oscillation modes mentioned above. The initial state of the doubly coupled system is first set to a quasi-periodic state. Then two kinds of signal peaks are introduced respectively, which are $A_{1}=A_{2}$ partial discharge signals with $=$ 5 mv and attenuation coefficient of $\tau_{1}=\tau_{2}=10^{-3}$ .

This study only considers the background noise of white noise, which obeys a normal distribution with a mean of 0 and a variance of $\sigma^{2}\cdot\sigma{}^{2}=$ 0.01. Gaussian white noise is added together with the partial discharge signal with a SNR of $-$ 25.10 dB. Figure 9 shows the difference in pulse synchronization caused by the transient out-of-synchronization phenomenon obtained from the two partial discharge signals in the attenuation form with the addition of white noise.

Figure 9.

Detection results of PD signal with two attenuation forms.

Table 2

Detection success rate comparison

		Detection success rate (%)
Algorithms	Number of experiments	Average overall success rate	Conventional weak signal	Add Gaussian white noise	Add Gaussian white noise and local electrical signal interference
Linear detection algorithm	1	60.10	75.24	61.54	45.37
	2		74.86	59.67	42.75
	3		77.12	62.02	46.39
	4		75.21	60.34	41.02
	5		76.11	61.27	42.57
	Average		75.71	60.97	43.62
Algorithm based on wavelet	1	72.36	78.24	75.36	65.21
packet transformation	2		79.34	74.21	64.38
	3		79.48	73.54	64.27
	4		80.01	72.95	63.59
	5		78.92	73.58	62.35
	Average		79.20	73.93	63.96
Adaptive filtering algorithm	1	79.36	80.21	79.62	75.24
	2		80.69	78.68	76.38
	3		81.29	80.04	77.95
	4		81.67	79.53	77.24
	5		83.01	79.97	78.94
	Average		81.37	79.57	77.15
Algorithm of this study	1	95.82	98.27	95.68	94.37
	2		97.68	96.37	93.68
	3		97.69	94.21	93.51
	4		98.84	95.67	93.92
	5		98.02	95.78	93.58
	Average		98.10	95.54	93.81

Figure 10.

Detection results of partial discharge signals in two damped oscillation forms.

As shown in Fig. 9, it is a schematic diagram of the pulse synchronisation difference of the single-exponential decay type and the double-exponential decay type, respectively. It can be seen that in the quasi-periodic state, the double-coupling system has a strong suppression effect on Gaussian white noise and also detects two weak signals relatively quickly in terms of detection speed. The SNR of the signal obtained by this system is greatly optimized compared to the SNR of the original signal. In terms of sinusoidal signal detection, the difference between the new chaotic system and other systems is that this system has certain requirements for the frequency of the detection signal. The frequency of the signal inside the system must be the same as the frequency of the detection signal, and signals of other frequencies will not be detected by the system. Because of this feature of the system, we can precisely locate the frequency of the signal we need to detect, which greatly reduces the interference signal that occurs during the detection process. Three kinds of periodic interference signals with frequencies of 50 kHz, 250 kHz and 1 MHz and amplitudes of 0.2 V, 0.1 V and 0.1 V were selected respectively, and Gaussian white noise was added in the detection process $\sigma^{2}=$ 0.01. Then the critical threshold is adjusted to be 10,000 times higher than the signal peak value, so as to achieve the scale transformation of the new chaotic system. Typically, the amplitudes of these detection signals are extremely small, the smallest being in the order of microvolts. The corresponding frequency is also extremely high. When detecting weak signals with other detection methods, it is usually necessary to amplify the signal and reduce the frequency. The advantage of a chaotic system is that this step is not necessary, and the same effect can be achieved by scaling. Finally, the optimized phase state discrimination method is used to identify and judge the system state, and the detection results of the two electrical signals are obtained, as shown in Fig. 10.

As shown in Fig. 10, after the phase state discrimination method and the phase diagram are improved, both reflect the system state after adding white noise, and both can be used as a criterion for judging the system state. Obviously, the same demerit occurred when two different partial discharge signals were detected, and there were no two different judgment criteria. The simulation results show that when the local electrical signal processed by the noise is detected by the detection model, one of the two systems of the forward chaotic system and the reverse chaotic system must enter the periodic state, and it can be detected in real time because of the existence of this phenomenon. An optimized phase state determination method. The study has identified various weak signal detection techniques for assessing their detection accuracy rates vis-à-vis the method proposed in the research. These techniques include a linear detection algorithm, a wavelet packet transform based algorithm, and an adaptive filtering algorithm. The success rate of signal detection for all four methods was empirically evaluated under different conditions. The findings are illustrated in Table 2. It is apparent from the table that the proposed algorithm achieves a much higher detection accuracy for weak signals than other algorithms, with a rate of 95.82%. This surpasses the other three algorithms’ accuracies by 16.46% to 35.72%.

5. Conclusion

Most traditional methods lack strong anti-noise capabilities in signal detection and are easily susceptible to interference from noise. To effectively address this issue, chaos theory based on nonlinear differential equations is introduced in signal detection research. The simulation experiment for detecting pulse signals yielded positive and negative alternating square wave pulse signals. The detection quality was excellent and no reverse oscillation occurred. By analyzing the detection conditions at various SNRs, it becomes evident that the dual-coupling system has a detection probability of over 96% at $-$ 25 dB. Even at a low SNR of $-$ 40 dB, the detection probability remains above 75%, depicting a powerful anti-noise effect of the system. In summary, the integration of chaos theory has enhanced the identification of faint signals and reduced noise interference to a certain extent. Additionally, several domains can be converted into nonlinear chaotic systems to tackle complex problems, as observed in other disciplines. The issue discussed here presents a viable resolution.

The study focuses on detecting weak signals using a chaotic system built on nonlinear differential equations, and achieved favourable results. However, some limitations in this research were noted, such as the prolonged time taken by the system to shift from the chaotic state to the periodic state. Moreover, the identified weak signal requires a certain pulse width to be detected, which can be observed in the concluded outcome. The information provided is not yet comprehensive and requires further research to improve its quality. This study aims to continue enhancing our methodology and applying it to a broader range of fields. Additionally, it is essential to acknowledge that the weak signal detection system utilized in this study is intricate and susceptible to parameter selection, thus necessitating further improvements in future studies.

Footnotes

Conflict of interest

It is declared by the authors that this article is free of conflict of interest.

Data availability statement

The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request.

References

Liu

Zhang

Wang

Ren

. Adaptive fuzzy prescribed finite-time tracking control for nonlinear system with unknown control directions. IEEE T Fuzzy Syst. 2022; 30(6): 1993-2003.

Sabir

Raja

Kamal

Guirao

JLG

Saeed

Salama

. Neuro-swarm heuristic using interior-point algorithm to solve a third kind of multi-singular nonlinear system. Math Biosci Eng. 2021; 18(5): 5285-5308.

Siuly

Alcin

Bajaj

Sengur

Zhang

. Exploring hermite transformation in brain signal analysis for the detection of epileptic seizure. IET Sci Meas Technol. 2019; 13(1): 35-41.

Salehimehr

Taheri

Razavi

Parpaei

Faghihlou

. A new power swing detection method based on chaos theory. Electr Eng. 2020; 102(2): 663-681.

Jiang

Tang

Liu

Sivakumar

Panga

. A hybrid wavelet-Lyapunov exponent model for river water quality forecast. J Hydroinform. 2021; 23(4): 864-878.

Qiao

Shu

. Coupled neurons with multi-objective optimization benefit incipient fault identification of machinery. Chaos Soliton Fract. 2021; 145: 110813.

Wang

. Dynamic behavior analysis and robust synchronization of a novel fractional-order chaotic system with multiwing attractors. J Math-UK. 2021; 2021: 6684906.

Sharma

Sinha

. Control of nonlinear systems exhibiting chaos to desired periodic or quasi-periodic motions. Nonlinear Dynam. 2020; 99(1): 559-574.

Anh

HPH

Kien

. Robust extreme learning machine neural approach for uncertain nonlinear hyper-chaotic system identification. Int J Robust Nonlin. 2021; 31(18): 9127-9148.

10.

Huang

Yang

. An efficient symmetric image encryption by using a novel 2D chaotic system. IET Image Process. 2020; 14(6): 1157-1163.

11.

Min

Wang

. Nonlinear dynamic analysis and surface sliding mode controller based on low pass filter for chaotic oscillation in power system with power disturbance. Chinese J Phys. 2018; 56(5): 2488-2499.

12.

Sambas

Vaidyanathan

Zhang

Koyuncu

Bonny

Tuna

Alcin

Zhang

Sulaiman

Awwal

Kumam

. A novel 3D chaotic system with line equilibrium: Multistability, integral sliding mode control, electronic circuit, FPGA implementation and its image encryption. IEEE Access. 2022; 10: 68057-68074.

13.

Chen

Liu

Wei

Feng

. New insights into a chaotic system with only a Lyapunov stable equilibrium. Math Method Appl Sci. 2020; 43(15): 9262-9279.

14.

Sambas

Vaidyanathan

Bonny

Zhang

, Sukono Hidayat

Gundara

Mamat

. Mathematical model and FPGA realization of a multi-stable chaotic dynamical system with a closed butterfly-like curve of equilibrium points. Appl Sci-Basel. 2021; 11(2): 788.

15.

Bulut

Catalbas

Guler

. Chaotic systems based real-time implementation of visual cryptography using LabVIEW. Trait Signal. 2020; 37(4): 639-645.

16.

Xiong

Yin

Zhang

. Stabilization and circuit implementation of a novel chemical oscillating chaotic system. Circuit World. 2019; 45(2): 93-106.

17.

Aqeel

Azam

Ayub

. Control of chaos in krause and roberts geomagnetic chaotic system. Chinese J Phys. 2022; 77(1): 1331-1341.

18.

Xian

Wang

Teng

Yan

Wang

. Cryptographic system based on double parameters fractal sorting vector and new spatiotemporal chaotic system. Inform Sciences, 2022; 596: 304-320.

19.

Zhao

Cao

. Periodic bursting oscillations in a hybrid Rayleigh-Van der Pol-Duffing oscillator. Nonlinear Dynam. 2023; 111(3): 2263-2279.

20.

Mangiarotti

Peyre

Zhang

Huc

Roger

Kerr

. Chaos theory applied to the outbreak of COVID-19: An ancillary approach to decision making in pandemic context. Epidemiol Infect. 2020; 148: e95.

21.

Fodchuk

Balovsyak

Novikov

Yanchuk

Romankevych

. Reconstruction of spatial distribution of strains in crystals using the energy spectrum of X-ray Moire patterns. Ukr J Phys Opt. 2020; 21(3): 141-151.

22.

Sahasrabuddhe

Laiphrakpam

. Multiple images encryption based on 3D scrambling and hyper-chaotic system. Inform Sciences, 2020; 550: 252-267.

23.

Mahatab

. Sixth norm of a Steinhaus chaos. Int J Number Theory. 2020; 16(6): 1295-1306.

Weak signal detection method based on nonlinear differential equations

Abstract

Keywords

1. Introduction

2. Related works

3. Signal detection based on chaos theory in nonlinear systems

3.1 Chaos theory and state judgment methods in nonlinear systems

Table 1 Detection probabilities of several different chaotic systems under different signal-to-noise conditions

Footnotes

Conflict of interest

Data availability statement

References

Table 1
Detection probabilities of several different chaotic systems under different signal-to-noise conditions