A New Chaotic Weak Signal Detection Method Based on a Simplified Fractional-Order Genesio–Tesi Chaotic System
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsThis study explores the application of a simplified fractional-order Genesio-Tesi chaotic system for detecting weak signals masked by Gaussian white noise. The key innovation lies in using the chaos-divergence state as a criterion for signal identification, which provides a clearer distinction between signal and noise compared to traditional chaos-periodic states. Numerical simulations demonstrated the method's effectiveness, successfully detecting a signal with a frequency of 100 rad/s at a signal-to-noise ratio as low as −40.83 dB. This approach enhances the practical applicability of weak signal detection by simplifying the evaluation of the divergence state and improving the accuracy of frequency and amplitude estimation. The proposed method establishes a foundation for further research into chaotic systems with divergent states and their application in weak signal detection tasks.
To expand the research presented in the paper, an interesting question is the consideration of another noise model – non-Gaussian noise. Such noise is more typical in real technical systems. Replacing Gaussian noise with other models can significantly affect the effectiveness of the proposed detection methods, as the statistical properties of these noises can alter the dynamics of the system and its sensitivity to weak signals. This will allow for an assessment of the robustness of the proposed method in more complex and realistic noise environments.
Author Response
Comments: This study explores the application of a simplified fractional-order Genesio-Tesi chaotic system for detecting weak signals masked by Gaussian white noise. The key innovation lies in using the chaos-divergence state as a criterion for signal identification, which provides a clearer distinction between signal and noise compared to traditional chaos-periodic states. Numerical simulations demonstrated the method's effectiveness, successfully detecting a signal with a frequency of 100 rad/s at a signal-to-noise ratio as low as −40.83 dB. This approach enhances the practical applicability of weak signal detection by simplifying the evaluation of the divergence state and improving the accuracy of frequency and amplitude estimation. The proposed method establishes a foundation for further research into chaotic systems with divergent states and their application in weak signal detection tasks.
To expand the research presented in the paper, an interesting question is the consideration of another noise model – non-Gaussian noise. Such noise is more typical in real technical systems. Replacing Gaussian noise with other models can significantly affect the effectiveness of the proposed detection methods, as the statistical properties of these noises can alter the dynamics of the system and its sensitivity to weak signals. This will allow for an assessment of the robustness of the proposed method in more complex and realistic noise environments.
Response: We fully agree with your suggestion to expand our research by considering non-Gaussian noise models. Although the primary focus of the study is on a novel approach to detecting weak chaotic signals through chaotic-to-divergent state transitions, exploring different types of noise can further demonstrate the effectiveness of this method. Poisson noise, for instance, represents random counting errors that arise from discrete probability distributions in real-world technological systems. Additionally, when signals travel through multiple paths to reach the receiver in a multipath propagation environment, they can interfere with each other, resulting in Rayleigh noise in the received signal strength. Consequently, we have included both Rayleigh and Poisson noise and have presented the results of the noise resistance performance in a new table.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
Comments and Suggestions for AuthorsIn the manuscript under review, the authors propose a new technique for detecting weak signals based on fractional order Genesio-Tesi chaotic system. It is interesting. However, authors need to answer the following questions:
1. One of the main weaknesses of the study is the lack of comparative analysis with alternative approaches to detecting weak signals.The fractional order mathematical framework is quite computationally expensive. Can the required dynamical modes be obtained, for example, by varying the symmetry coefficient (as in "Inducing multistability in discrete chaotic systems using numerical integration with variable symmetry") or the dissipation coefficient ("Scaling properties of the Lorenz system and dissipative Nambu mechanics")? Can a frequency-amplitude adaptive oscillator ("Adaptive frequency oscillators and applications") be built based on the Genesio-Tesi chaotic system to directly display the characteristics of the input signal?
2. The second weakness of the study is that it only presents single-parameter diagrams. Could the authors construct two-parameter diagrams (as in "Revealing hidden features of chaotic systems using high-performance bifurcation analysis tools based on CUDA technology") to clearly visualize the areas of dynamical modes? In the demonstration of frequency dependence (Figure 5), it may be useful to display inter-peak intervals instead of state variable peaks.
3. How can the authors explain the fact that in Figure 2 the LLE > 0 after a > 0.93, while in the bifurcation diagram the chaotic regime occurs only around a = 0.97.
Minor comments for publication purposes include:
- In Figures 2 and 4, it is recommended to plot the Lyapunov exponents in different colors.
- In Figure 4, there should be a different parameter on the x-axis. For a bifurcation diagram, a higher resolution for the variable parameter is reccomended.
- Abbreviations for the complexity metrics should be introduced.
- There is one reference missing on page 9.
- Figure 5 shows several windows with divergence. Please specify the parameters more precisely in the text on page 9.
Comments on the Quality of English LanguageThe text contains syntax and stylistic errors. Proofreading by a native speaker is required to improve the quality of English.
Author Response
In the manuscript under review, the authors propose a new technique for detecting weak signals based on fractional order Genesio-Tesi chaotic system. It is interesting. However, authors need to answer the following questions:
Comments 1: One of the main weaknesses of the study is the lack of comparative analysis with alternative approaches to detecting weak signals. The fractional order mathematical framework is quite computationally expensive. Can the required dynamical modes be obtained, for example, by varying the symmetry coefficient (as in "Inducing multistability in discrete chaotic systems using numerical integration with variable symmetry") or the dissipation coefficient ("Scaling properties of the Lorenz system and dissipative Nambu mechanics")? Can a frequency-amplitude adaptive oscillator ("Adaptive frequency oscillators and applications") be built based on the Genesio-Tesi chaotic system to directly display the characteristics of the input signal?
Response 1: Thank you for your feedback. Fractional-order systems do present the challenge of high computational complexity. However, in this study, we focused more on the rich dynamic behavior of fractional-order systems and leveraged this advantage to achieve better weak signal detection results. Of course, to apply these results in practical settings in the future, we will explore ways to optimize computational efficiency through algorithms. Additionally, we would like to emphasize that integer-order chaotic systems are a special case of fractional-order chaotic systems, therefore fractional-order systems still able to retain the dynamic modes found in integer-order systems. When compared to dynamic behaviors derived from varying the symmetry and dissipation coefficients, chaotic-to-divergent patterns are a more common feature in chaotic systems. However, after reviewing the literature you provided, we believe that the dynamic patterns achieved by adjusting the symmetry and dissipation coefficients could also be effective for weak signal detection. In response to the reviewer’s suggestion regarding the method for constructing a frequency-amplitude adaptive oscillator, we note that this method is suitable for integer-order Genesio-Tesi chaotic systems. In fractional-order systems, however, the time-dependent memory effects inherent in fractional calculus mean that adjusting the frequency often requires recalculating the numerical solution. In this article, we review several existing methods for frequency detection and ultimately adopt the variance method based on state variable features. Further research will focus on methods for detecting weak signal frequencies in fractional-order chaotic systems. We have added this explanation to the appropriate section of the manuscript.
Comments 2: The second weakness of the study is that it only presents single-parameter diagrams. Could the authors construct two-parameter diagrams (as in "Revealing hidden features of chaotic systems using high-performance bifurcation analysis tools based on CUDA technology") to clearly visualize the areas of dynamical modes? In the demonstration of frequency dependence (Figure 5), it may be useful to display inter-peak intervals instead of state variable peaks.
Response 2: Thank you for the reviewer's suggestions. The two-parameter plot can certainly provide a clearer visualization of dynamic patterns. However, the primary focus of this study is weak signal detection in fractional-order chaotic systems. In this work, the two-parameter plots—fractional order and coefficients at the point of weak signal addition—hold particular research value. When calculating fractional order, using a large interval may overlook some important dynamic patterns, while a very small interval can lead to excessive computation time. This is because changes in fractional order affect the approximate solution algorithm, which has not been addressed in the existing references. Additionally, if we consider the frequency and amplitude of weak signals as parameters influencing fractional-order chaotic systems, we could develop new methods for detecting weak signal frequencies, which is a direction we intend to explore in future research. Regarding Figure 5(now Figure 6), we understand that the reviewer is interested in the frequency information represented by the peak-to-peak interval. However, the information presented in Figure 5 reflects a scenario where the weak signal frequency is fixed while the driving frequency of the chaotic detection system is continuously varied. The dynamic behavior of the fractional-order chaos detection system changes only within a narrow range around the weak signal frequency. Specifically, Figure 5 demonstrates that when we use a weak signal with a frequency of 1 rad/s, the dynamic model's frequency varies between 0.9777 and 1.0049. The occasional divergence state at 1.0223 does not affect the determination of the detection range.
Comments 3: How can the authors explain the fact that in Figure 2 the LLE > 0 after a > 0.93, while in the bifurcation diagram the chaotic regime occurs only around a = 0.97.
Response 3: Thank you for your feedback. We have rechecked the approximate solution algorithm for fractional order chaotic systems and redrawn Figure 2 to ensure that the Lyapunov exponent spectrum results are consistent with the bifurcation diagram.
Minor comments for publication purposes include:
- In Figures 2 and 4, it is recommended to plot the Lyapunov exponents in different colors.
- In Figure 4, there should be a different parameter on the x-axis. For a bifurcation diagram, a higher resolution for the variable parameter is reccomended.
- Abbreviations for the complexity metrics should be introduced.
- There is one reference missing on page 9.
- Figure 5 shows several windows with divergence. Please specify the parameters more precisely in the text on page 9.
Response: Thank you for your review comments. We have rechecked the manuscript and made revisions to address any issues.
Author Response File: Author Response.pdf
Round 2
Reviewer 2 Report
Comments and Suggestions for AuthorsIn the response letter, the authors duly addressed the main comments from the previous round of review, also making appropriate changes to the manuscript. However, the minor comments on Figure 4 were not fully taken into account: the bifurcation diagram still looks too sparse, and the x-axis label suggests the gamma parameter (not F).
Author Response
Comments: In the response letter, the authors duly addressed the main comments from the previous round of review, also making appropriate changes to the manuscript. However, the minor comments on Figure 4 were not fully taken into account: the bifurcation diagram still looks too sparse, and the x-axis label suggests the gamma parameter (not F).
Response: Thank you for your valuable feedback, which pointed out our oversight in the manuscript. For this purpose, we have redrawn the bifurcation diagram in Figure 4 and corrected the labels on the x-axis to ensure that it accurately reflects the driving amplitude γ. In addition, we have re selected the interval of the bifurcation diagram and used a finer interval of 0.00000175 between 0 and 0.0007.
Author Response File: Author Response.pdf
