{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:55:57Z","timestamp":1760151357456,"version":"build-2065373602"},"reference-count":21,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2022,2,25]],"date-time":"2022-02-25T00:00:00Z","timestamp":1645747200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper investigates the nonlinear dynamics of Transmission Control Protocol and Active Queue Management (TCP\/AQM) networks, including the local stability and periodic bifurcation. The parameter of transportation delay affects the stability of the dynamical systems. One of the purposes of our work is to determine the delay stable interval of the transportation network. It is found that there is only one critical value of network delay by the \u03c4 decomposition technique. When the delay passes the critical point, the system performs Hopf bifurcation with a pair of symmetry with purely imaginary roots (PIR). In addition, the other purpose is to consider the stability of bifurcating periodic solutions. Combining with \u03c4-decomposition strategy and central manifold theory, the issues of delay stable interval and stability of Hopf bifurcation are all tackled. Finally, numerical examples are illustrated to show the accuracy and effectiveness of the proposed method.<\/jats:p>","DOI":"10.3390\/sym14030463","type":"journal-article","created":{"date-parts":[[2022,2,27]],"date-time":"2022-02-27T20:49:03Z","timestamp":1645994943000},"page":"463","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["On the \u03c4 Decomposition Method for the Stability and Bifurcation of the TCP\/AQM Networks versus Time Delay"],"prefix":"10.3390","volume":"14","author":[{"given":"Hui-Long","family":"Jin","sequence":"first","affiliation":[{"name":"Collage of Engineering, Hebei Normal University, Shijiazhuang 050024, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tian-Le","family":"Di","sequence":"additional","affiliation":[{"name":"Collage of Engineering, Hebei Normal University, Shijiazhuang 050024, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hong","family":"Yu","sequence":"additional","affiliation":[{"name":"Collage of Engineering, Hebei Normal University, Shijiazhuang 050024, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ran-Ran","family":"Zhang","sequence":"additional","affiliation":[{"name":"Electric Power Supply Company of Handan Xinqu District, State Grid Corporation of China, Handan 056004, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,2,25]]},"reference":[{"key":"ref_1","unstructured":"Misra, V., Gong, W.-B., and Towsley, D. 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