Liènard chaotic system based on Duffing and the Sinc function for weak signals detection
Keywords:Sinc-Duffing, Liènard system, Melnikov detector, weak signal detection
This article presents a modified Duffing system based on Liènard´s Theorem and the integral of Melnikov, the first is used to propose the interpolation ´Sinc´ as a non-linear damping function and the second is used to assure an asymptotically stable limit cycle. The Sin-Duffing system is driven into chaos by using its corresponding bifurcation diagram, Lyapunov exponents, and the Theory of Melnikov. Furthermore, the system is placed in a critical state which produced chaotic and periodic sequences, driving it into a regimen of intermittence between chaos and the self-sustained oscillations near the stable limit cycle. Intermittence is achieved by searching and tuning all involved parameters when a very systematic procedure is used. Also, such a regimen is presented here as a useful mechanism to estimate the frequency of a very low weak signal for detection applications. The latest is made possible because the system capabilities to distinguish the intermittent periods were strengthened by a new method based on Melnikov´s function that only depends on the most influential parameter in the type-Liènard system. The complete system formed by the new Sinc-Duffing oscillator showed higher sensitivity compere to other chaotic systems such as the traditional Duffing or the Van der Pol-Duffing for weak signal detection with a signal-to-noise ratio down to -70 dB.
M. K. J S Armand Eyebe Fouda, J Yves Effa, “The three-state test for chaos detection in discrete maps. applied soft computing journal,” Applied Soft Computing, vol. 13, no. 1, pp. 4731–4737, 2013.
A. A.-H. E. Zambrano-Serrano, “A novel antimonotic hyperjerk system: Analysis, synchronization and circuit design,” Physica D: Nonlinear Phenomena, vol. 424, 2021.
X. G. D. Chen, S. Shi, “Weak signal frequency detection using chaos theory: A comprehensive analysis,” IEEE Transactions on Vehicular Technology, vol. 70, p. 8950–8963, 2021.
S. P. D.L. Birx, “Chaotic oscillators and complex mapping feed forward networks (cmffns) for signal detection in noisy environments,” 1992.
J. L. Guanyu Wang, Dajun Chen, “The application of chaotic oscillators to weak signal detection,” IEEE Transactions on industrial electronics, vol. 61, pp. 440–444, 1999.
R. Z. Yanhua Wu, “Analysis of internal frequency’s influence on blind detection weak psk signal by using duffing oscillator,” 2017.
H. Y. B.L. Jian, X.Y. Su, “Bearing fault diagnosis based on chaotic dynamic errors in key components,” IEEE Access, vol. 9, p. 53509–53517, 2021.
V. M. V. Gupta, M. Mittal, “Chaos theory and artfa: Emerging tools for interpreting ecg signals to diagnose cardiac arrhythmias,” Wireless Personal Communications, vol. 118, p. 3615–3646, 2021.
H. Y. G. Li, Y. Hou, “A new duffing detection method for underwater weak target signal,” Alexandria Engineering Journal, vol. 61,p. 2859–2876, 2022.
P. M. J.S. Muthu, “Review of chaos detection techniques performed on chaotic maps and systems in image encryption,” SN Computer Science, vol. 2, 2021.
E. C. R.J. Escalante-González, “Emergence of hidden attractors through the rupture of heteroclinic-like orbits of switched systems with self-excited attractors,” Complexity, pp. 1–24, 2021.
M. B. I. Kovacic, The Duffing Equation: Nonlinear Oscillators and their Behaviour. John Wiley and Sons, 2011.
T.L.Lou, “Frequency estimation for weak signals based on chaos theory,”pp. 361–364, IEEE, 2008.
L. T. M. Richardson, “A sinc function analogue of chebfun,” SIAM Journal on Scientific Computing, vol. 33, no. 5, pp. 2519—-2535, 2011.
J. Giné, “Center conditions for polynomial liènard systems,” Qualitative Theory of Dynamical Systems, vol. 16, no. 11, pp. 119–126, 2017.
B. J. M. Sinha, F. Dorfler, “Synchronization of liènard-type oscillators in uniform electrical networks,” 2016 American Control Conference (ACC), pp. 4311—-4316, 2016.
S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer, 2003.
S. Lynch, Dynamical Systems with Applications using Python. In Dynamical Systems with Applications using Python. No. 1, Springer International Publishing AG, 2018.
M. M. J Awrejcewicz and Holicke, “Smooth and non-smooth high dimensional chaos and melnikov type methods,” Journal of Vibration and Control, no. 1, p. 318, 2007.
S. Strogatz, ANFIS: adaptive-network-based fuzzy inference system, vol. 2. CRC Press, 2015.
P. W. M. Siewe, C. Tchawoua, “Melnikov chaos in a periodically driven rayleigh–duffing oscillator,” Mechanics Research Communications, vol. 37, no. 4, pp. 363—-368, 2010.
Z. Y. F. Cheng, “A new method to determine the bifurcation threshold value of the duffing chaos detection system,” pp. 1143–1146, IEEE, 2012.
Q. D. C.Wang, C. Fan, “Constructing discrete chaotic systems with positive lyapunov exponents,” International Journal of Bifurcation and Chaos, vol. 28, no. 7, p. 1850084, 2018.
M. L. A. Venkatesan, “Bifurcation and chaos in the double-well duffing–van der pol oscillator: Numerical and analytical studies,” Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, vol. 56, no. 6, pp. 6321––6330, 1997.
B. Y. H.H. Peng, X.M. Xu, “Implication of two-coupled differential van der pol duffing oscillator in weak signal detection,” Journal of the Physical Society of Japan, vol. 85, no. 4, pp. 1–8, 2016.
Y. S. Z. Zhihong, “Application of van der pol–duffing oscillator in weak signal detection,” Computers & Electrical Engineering, vol. 41, pp. 1–8, 2015.
Y. L. Y. Xie, T. Lin, “Weak signal frequency detection based on intermittent chaos,” 2014.
J. M.-C. C. Bermúdez-Gómez, R. Enriquez-Caldera, “Chirp signal detection using the duffing oscillator,” pp. 344–349, IEEE, 2012.
R. E.-C. P. Pancóatl-Bortolotti, A. Costa, “Design and analysis of a new chaotic system inspired on duffing,” pp. 160–165, 2021.
R. E.-C. P. Pancóatl-Bortolotti, A. Costa, “A novel chaotic system based on binomial functions for detection of ultra weak signals,” pp. 102–107, 2021.