{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,17]],"date-time":"2026-02-17T03:43:33Z","timestamp":1771299813748,"version":"3.50.1"},"reference-count":29,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2023,12,13]],"date-time":"2023-12-13T00:00:00Z","timestamp":1702425600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["Grant Nos. 11874302"],"award-info":[{"award-number":["Grant Nos. 11874302"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["JCKYS2021604SSJS013"],"award-info":[{"award-number":["JCKYS2021604SSJS013"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Stable Supporting Fund of Acoustics Science and Technology Laboratory","award":["Grant Nos. 11874302"],"award-info":[{"award-number":["Grant Nos. 11874302"]}]},{"name":"Stable Supporting Fund of Acoustics Science and Technology Laboratory","award":["JCKYS2021604SSJS013"],"award-info":[{"award-number":["JCKYS2021604SSJS013"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The transition from a chaotic to a periodic state in the Duffing chaotic oscillator detection system is crucial in detecting weak signals. However, accurately determining the critical threshold for this transition remains a challenging problem. Traditional methods such as Melnikov theory, the Poincar\u00e9 section quantitative discrimination method, and experimental analyses based on phase diagram segmentation have limitations in accuracy and efficiency. In addition, they require large computational data and complex algorithms while having slow convergence. Improved permutation entropy incorporates signal amplitude information on the basis of permutation entropy and has better noise resistance. According to the characteristics of improved permutation entropy, a threshold determination method for the Duffing chaotic oscillator detection system based on improved permutation entropy (IPE) and Poincar\u00e9 mapping (PM) is proposed. This new metric is called Poincar\u00e9 mapping improved permutation entropy (PMIPE). The simulation results and the verification results of real underwater acoustic signals indicate that our proposed method outperforms traditional methods in terms of accuracy, simplicity, and stability.<\/jats:p>","DOI":"10.3390\/e25121654","type":"journal-article","created":{"date-parts":[[2023,12,13]],"date-time":"2023-12-13T08:55:16Z","timestamp":1702457716000},"page":"1654","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Research on the Threshold Determination Method of the Duffing Chaotic System Based on Improved Permutation Entropy and Poincar\u00e9 Mapping"],"prefix":"10.3390","volume":"25","author":[{"given":"Jing","family":"Zhou","sequence":"first","affiliation":[{"name":"School of Marine Science and Technology, Northwestern Polytechnical University, Xi\u2019an 710072, China"},{"name":"Xi\u2019an Precision Machinery Research Institute, Science and Technology on Underwater Information and Control Laboratory, Xi\u2019an 710077, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5973-9780","authenticated-orcid":false,"given":"Yaan","family":"Li","sequence":"additional","affiliation":[{"name":"School of Marine Science and Technology, Northwestern Polytechnical University, Xi\u2019an 710072, China"}]},{"given":"Mingzhou","family":"Wang","sequence":"additional","affiliation":[{"name":"Xi\u2019an Precision Machinery Research Institute, Science and Technology on Underwater Information and Control Laboratory, Xi\u2019an 710077, China"}]}],"member":"1968","published-online":{"date-parts":[[2023,12,13]]},"reference":[{"key":"ref_1","first-page":"1405","article-title":"Coexistence, bifurcation and chaos of a periodically forced duffing system with absolute nonlinearity","volume":"228","author":"Chen","year":"2019","journal-title":"Springer J."},{"key":"ref_2","first-page":"42","article-title":"Study on weak signal detection method with Duffing oscillators","volume":"62","author":"Liu","year":"2013","journal-title":"Acta Phys. 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