{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,1]],"date-time":"2026-04-01T02:43:53Z","timestamp":1775011433334,"version":"3.50.1"},"reference-count":29,"publisher":"MDPI AG","issue":"14","license":[{"start":{"date-parts":[[2022,7,12]],"date-time":"2022-07-12T00:00:00Z","timestamp":1657584000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Sensors based on chaotic oscillators have a simple design, combined with high sensitivity and energy efficiency. Among many developed schemes of such sensors, the promising one is based on the Duffing oscillator, which possesses a remarkable property of demonstrating chaotic oscillations only in the presence of a weak sine wave at the input. The main goal of this research was to evaluate the maximal sensitivity of a practically implemented metal detector based on the Duffing oscillator and compare its sensitivity with conventional sensors. To achieve high efficiency of the Duffing-based design, we proposed an algorithm which performs a bifurcation analysis of any chaotic system, classifies the oscillation modes and determines the system sensitivity to a change in different parameters. We apply the developed algorithm to improve the sensitivity of the electronic circuit implementing the Duffing oscillator, serving as a key part of a three-coil metal detector. We show that the developed design allows detecting the presence of metal objects near the coils more reliably than the conventional signal analysis techniques, and the developed detector is capable of sensing a large metal plate at distances up to 2.8 of the coil diameter, which can be considered a state-of-the-art result.<\/jats:p>","DOI":"10.3390\/s22145212","type":"journal-article","created":{"date-parts":[[2022,7,12]],"date-time":"2022-07-12T23:02:01Z","timestamp":1657666921000},"page":"5212","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Sensitivity Optimization and Experimental Study of the Long-Range Metal Detector Based on Chaotic Duffing Oscillator"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9860-8211","authenticated-orcid":false,"given":"Timur","family":"Karimov","sequence":"first","affiliation":[{"name":"Youth Research Institute, St. Petersburg Electrotechnical University \u201cLETI\u201d, 5 Professora Popova St., 197376 Saint Petersburg, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0672-3475","authenticated-orcid":false,"given":"Olga","family":"Druzhina","sequence":"additional","affiliation":[{"name":"Youth Research Institute, St. Petersburg Electrotechnical University \u201cLETI\u201d, 5 Professora Popova St., 197376 Saint Petersburg, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7679-3014","authenticated-orcid":false,"given":"Valerii","family":"Vatnik","sequence":"additional","affiliation":[{"name":"Department of Computer-Aided Design, St. Petersburg Electrotechnical University \u201cLETI\u201d, 5 Professora Popova St., 197376 Saint Petersburg, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ekaterina","family":"Ivanova","sequence":"additional","affiliation":[{"name":"Department of Computer-Aided Design, St. Petersburg Electrotechnical University \u201cLETI\u201d, 5 Professora Popova St., 197376 Saint Petersburg, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5075-2292","authenticated-orcid":false,"given":"Maksim","family":"Kulagin","sequence":"additional","affiliation":[{"name":"Youth Research Institute, St. Petersburg Electrotechnical University \u201cLETI\u201d, 5 Professora Popova St., 197376 Saint Petersburg, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Veronika","family":"Ponomareva","sequence":"additional","affiliation":[{"name":"Department of Computer-Aided Design, St. Petersburg Electrotechnical University \u201cLETI\u201d, 5 Professora Popova St., 197376 Saint Petersburg, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4022-9742","authenticated-orcid":false,"given":"Anzhelika","family":"Voroshilova","sequence":"additional","affiliation":[{"name":"School of Public Administration and Entrepreneurship, Institute of Economics and Management, Ural Federal University Named after the First President of Russia B.N.Yeltsin, 51 Lenina Ave., 620075 Yekaterinburg, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6515-0224","authenticated-orcid":false,"given":"Vyacheslav","family":"Rybin","sequence":"additional","affiliation":[{"name":"Youth Research Institute, St. Petersburg Electrotechnical University \u201cLETI\u201d, 5 Professora Popova St., 197376 Saint Petersburg, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,7,12]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"130","DOI":"10.1016\/j.ymssp.2016.05.012","article-title":"Defect detection of pipes using Lyapunov dimension of Duffing oscillator based on ultrasonic guided waves","volume":"82","author":"Wu","year":"2017","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1016\/j.cnsns.2014.12.010","article-title":"Noise activated dc signal sensor based on chaotic Chua circuit","volume":"24","author":"Korneta","year":"2015","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Karimov, T., Druzhina, O., Karimov, A., and Butusov, D. (2020, January 20\u201324). Axial movement sensor based on chaotic oscillator and planar coil. Proceedings of the 2020 26th Conference of Open Innovations Association (FRUCT), Yaroslavl, Russia.","DOI":"10.23919\/FRUCT48808.2020.9087471"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Hu, W. (2011). The Design of Improved Duffing Chaotic Circuit Used for High-Frequency Weak Signal Detection. Electronics and Signal Processing, Springer.","DOI":"10.1007\/978-3-642-21697-8_107"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"440","DOI":"10.1109\/41.753783","article-title":"The application of chaotic oscillators to weak signal detection","volume":"46","author":"Wang","year":"1999","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"104168","DOI":"10.1016\/j.ijpvp.2020.104168","article-title":"Defect detection in pipe structures using stochastic resonance of the Duffing oscillator and ultrasonic guided waves","volume":"187","author":"Wu","year":"2020","journal-title":"Int. J. Press. Vessel. Pip."},{"key":"ref_7","first-page":"425","article-title":"Weak infrared radiation intensity measurement based on extended Duffing oscillator. AOPC 2021: Infrared Device and Infrared Technology","volume":"12061","author":"Zhao","year":"2021","journal-title":"Proc. SPIE"},{"key":"ref_8","unstructured":"Hu, W., and Liu, Z. (2010, January 6\u20139). Study of metal detection based on chaotic theory. Proceedings of the 2010 8th World Congress on Intelligent Control and Automation, Jinan, China."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"012186","DOI":"10.1088\/1755-1315\/100\/1\/012186","article-title":"A new kind of metal detector based on chaotic oscillator","volume":"100","author":"Hu","year":"2017","journal-title":"IOP Conf. Ser. Earth Environ. Sci."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Hu, W.J., and Liu, Z. (2017). A New Metal Detection Method Based on Duffing Chaotic System. Destech Trans. Eng. Technol. Res.","DOI":"10.12783\/dtetr\/iceta2016\/6997"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Karimov, T.I., Druzhina, O.S., Ostrovskii, V.Y., Karimov, A.I., and Butusov, D.N. (2020, January 27\u201330). The Study on Multiparametric Sensitivity of Chaotic Oscillators. Proceedings of the 2020 IEEE Conference of Russian Young Researchers in Electrical and Electronic Engineering (EIConRus), St. Petersburg and Moscow, Russia.","DOI":"10.1109\/EIConRus49466.2020.9039312"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1631\/jzus.2002.0047","article-title":"Using chaos to improve measurement precision","volume":"3","author":"He","year":"2002","journal-title":"J. Zhejiang Univ.-Sci. A"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Teodorescu, H.N.L. (2017, January 5\u20136). Sensors based on nonlinear dynamic systems\u2014A survey. Proceedings of the 2017 International Conference on Applied Electronics (AE),  Pilsen, Czech Republic.","DOI":"10.23919\/AE.2017.8053572"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Korneta, W., Gomes, I., Picos, R., and Z\u00e1bovsk\u00fd, M. (2021). Biomimetic Chaotic Sensor for Moderate Static Magnetic Field. Sensors, 21.","DOI":"10.3390\/s21216964"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1295","DOI":"10.1007\/s11071-021-07062-2","article-title":"Single-coil metal detector based on spiking chaotic oscillator","volume":"107","author":"Karimov","year":"2021","journal-title":"Nonlinear Dyn."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1650226","DOI":"10.1142\/S0218127416502266","article-title":"A very simple method to calculate the (positive) largest Lyapunov exponent using interval extensions","volume":"26","author":"Mendes","year":"2016","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"241","DOI":"10.15388\/NA.2008.13.2.14582","article-title":"Analogue electrical circuit for simulation of the Duffing\u2013Holmes equation","volume":"13","author":"Mykolaitis","year":"2008","journal-title":"Nonlinear Anal. Model. Control"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Karimov, T., Nepomuceno, E.G., Druzhina, O., Karimov, A., and Butusov, D. (2019). Chaotic oscillators as inductive sensors: Theory and practice. Sensors, 19.","DOI":"10.3390\/s19194314"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Karimov, T., Butusov, D., Andreev, V., Karimov, A., and Tutueva, A. (2018). Accurate synchronization of digital and analog chaotic systems by parameters re-identification. Electronics, 7.","DOI":"10.3390\/electronics7070123"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Karimov, T.I., Druzhina, O.S., Karimov, A.I., Kolev, G.Y., and Butusov, D.N. (2021, January 26\u201329). Comparison of Bifurcation Diagrams for Numerical and Analog Chaotic Systems. Proceedings of the 2021 IEEE Conference of Russian Young Researchers in Electrical and Electronic Engineering (ElConRus), Moscow and St. Petersburg, Russia.","DOI":"10.1109\/ElConRus51938.2021.9396558"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1970","DOI":"10.1103\/PhysRevLett.74.1970","article-title":"Extracting messages masked by chaos","volume":"74","author":"Cerdeira","year":"1995","journal-title":"Phys. Rev. Lett."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"32","DOI":"10.1016\/j.physd.2007.06.004","article-title":"Return map analysis of chaotic phase synchronization","volume":"233","author":"Tsukamoto","year":"2007","journal-title":"Phys. D Nonlinear Phenom."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Ostrovskii, V.Y., Tutueva, A.V., Rybin, V.G., Karimov, A.I., and Butusov, D.N. (2020, January 3\u20136). Continuation Analysis of Memristor-Based Modified Chua\u2019s Circuit. Proceedings of the 2020 International Conference Nonlinearity, Information and Robotics (NIR), Innopolis, Russia.","DOI":"10.1109\/NIR50484.2020.9290157"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Ostrovskii, V.Y., Tutueva, A.V., Andreev, V.S., and Rybin, V.G. (2020, January 27\u201329). Phase Bifurcation Analysis of Nonlinear Dynamical Systems. Proceedings of the 2020 XXIII International Conference on Soft Computing and Measurements (SCM), St. Petersburg, Russia.","DOI":"10.1109\/SCM50615.2020.9198810"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Guo, Y.X., Shao, Z.B., and Li, T. (2015). An analog-digital mixed measurement method of inductive proximity sensor. Sensors, 16.","DOI":"10.3390\/s16010030"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Ma, M., Wang, Y., Liu, F., Zhang, F., Liu, Z., and Li, Y. (2019). Passive wireless LC proximity sensor based on LTCC technology. Sensors, 19.","DOI":"10.3390\/s19051110"},{"key":"ref_27","unstructured":"Sosnicki, O., Michaud, G., and Claeyssen, F. (2010, January 18\u201319). Eddy current sensors on Printed Circuit Board for compact mechatronic application. Proceedings of the Sensoren und Messsysteme, Meylan, France."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"4463","DOI":"10.1109\/TMAG.2009.2021855","article-title":"Extending the operating distance of inductive proximity sensor using magnetoplated wire","volume":"45","author":"Mizuno","year":"2009","journal-title":"IEEE Trans. Magn."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"200","DOI":"10.3311\/PPee.9676","article-title":"LTCC based planar inductive proximity sensor design","volume":"60","author":"Zuk","year":"2016","journal-title":"Period. Polytech. Electr. Eng. Comput. Sci."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/22\/14\/5212\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:49:01Z","timestamp":1760140141000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/22\/14\/5212"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,7,12]]},"references-count":29,"journal-issue":{"issue":"14","published-online":{"date-parts":[[2022,7]]}},"alternative-id":["s22145212"],"URL":"https:\/\/doi.org\/10.3390\/s22145212","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,7,12]]}}}