{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:57:42Z","timestamp":1760245062808,"version":"build-2065373602"},"reference-count":19,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2021,12,6]],"date-time":"2021-12-06T00:00:00Z","timestamp":1638748800000},"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":["61603121"],"award-info":[{"award-number":["61603121"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper considers the Modified Autonomous Van der Pol\u2013Duffing equation subjected to dynamic state feedback, which can well characterize the dynamic behaviors of the nonlinear dynamical systems. Both the issues of local stability switches and the Hopf bifurcation versus time delay are investigated. Associating with the \u03c4 decomposition strategy and the center manifold theory, the delay stable intervals and the direction and stability of the Hopf bifurcation are all determined. Specifically, the computation of purely imaginary roots (symmetry to the real axis), the positive real root formula for cubic equation and the sophisticated bilinear form of adjoint operators are proposed, which make the calculations mentioned in our discussion unified and simple. Finally, the typical numerical examples are shown to illustrate the correctness and effectiveness of the practical technique.<\/jats:p>","DOI":"10.3390\/sym13122336","type":"journal-article","created":{"date-parts":[[2021,12,7]],"date-time":"2021-12-07T02:48:13Z","timestamp":1638845293000},"page":"2336","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["On Stability Switches and Bifurcation of the Modified Autonomous Van der Pol\u2013Duffing Equations via Delayed State Feedback Control"],"prefix":"10.3390","volume":"13","author":[{"given":"Tiao-Yang","family":"Cai","sequence":"first","affiliation":[{"name":"Collage of Engineering, Hebei Normal University, Shijiazhuang 050024, China"}]},{"given":"Hui-Long","family":"Jin","sequence":"additional","affiliation":[{"name":"Collage of Engineering, Hebei Normal University, Shijiazhuang 050024, China"}]},{"given":"Hong","family":"Yu","sequence":"additional","affiliation":[{"name":"Collage of Engineering, Hebei Normal University, Shijiazhuang 050024, China"}]},{"given":"Xiang-Peng","family":"Xie","sequence":"additional","affiliation":[{"name":"Institute of Advanced Technology, Nanjing University of Posts and Telecommunications, Nanjing 210003, China"}]}],"member":"1968","published-online":{"date-parts":[[2021,12,6]]},"reference":[{"key":"ref_1","unstructured":"Lebovitz, N.R. (1877). Bifurcation and stability problems in astrophysics. Applications of Bifurcation Theory, Acadamic Press."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"394","DOI":"10.1109\/PROC.1981.11973","article-title":"Multimode oscillations in a modified van der Pol oscillator containing a positive nonlinear conductance","volume":"69","author":"Shinriki","year":"1981","journal-title":"Proc. IEEE"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"3092","DOI":"10.1103\/PhysRevA.46.3092","article-title":"Bistable chaos. I. Unfolding the cusp","volume":"46","author":"King","year":"1992","journal-title":"Phys. Rev. A"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1016\/j.jmaa.2007.09.067","article-title":"Bifurcations, chaos and synchronization in ADVP circuit with parallel resistor","volume":"341","author":"Matouk","year":"2008","journal-title":"J. Math. Anal. Appl."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"149563","DOI":"10.1155\/2009\/149563","article-title":"Bifurcation analysis of a Van der Pol-Duffing circuit with parallel resistor","volume":"2009","author":"Braga","year":"2009","journal-title":"Math. Probl. Eng."},{"key":"ref_6","first-page":"436","article-title":"Horseshoe in a modified Van der Pol-Duffing circuit","volume":"210","author":"Fan","year":"2009","journal-title":"Appl. Math. Comput."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"363","DOI":"10.1007\/s40435-014-0118-1","article-title":"Hyperchaos and bifurcations in a driven Van der Pol-Duffing oscillator circuit","volume":"3","author":"Vincent","year":"2015","journal-title":"Int. J. Dyn. Control"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"517","DOI":"10.1016\/j.cnsns.2016.01.001","article-title":"Hopf-bifurcation-delay-induced bursting patterns in a modified circuit system","volume":"36","author":"Han","year":"2016","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1007\/s11071-016-3232-0","article-title":"Nonlinear dynamics of a \u03d56 modified Duffing oscillator: Resonant oscillations and transition to chaos","volume":"88","author":"Miwadinou","year":"2017","journal-title":"Nonlinear Dyn."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physd.2017.07.008","article-title":"A modified hybrid Van der Pol-Duffing-Rayleigh oscillator for modelling the lateral walking force on a rigid floor","volume":"358","author":"Kumar","year":"2017","journal-title":"Phys. D Nonlinear Phenom."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2287","DOI":"10.1140\/epjst\/e2019-900043-4","article-title":"Analysis and electronic implementation of an absolute memristor autonomous Van der Pol-Duffing circuit","volume":"228","author":"Rajagopal","year":"2019","journal-title":"Eur. Phys. J. Spec. Top."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1950067","DOI":"10.1142\/S0218127419500676","article-title":"Hyperchaos and coexisting attractors in a modified van der Pol-Duffing oscillator","volume":"29","author":"Rajagopal","year":"2019","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_13","first-page":"126522","article-title":"Complex mixed-mode vibration types triggered by the pitchfork bifurcation delay in a driven van der Pol-Duffing oscillator","volume":"411","author":"Ma","year":"2021","journal-title":"Appl. Math. Comput."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Panigoro, H., Suryanto, A., Kusumawinahyu, W., and Darti, I. (2021). Dynamics of an eco-epidemic predator-prey model involving fractional derivatives with power-law and mittag-leffler kernel. Symmetry, 13.","DOI":"10.3390\/sym13050785"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Isidori, A. (1986). Control of nonlinear systems via dynamic state feedback. Algebraic and Geometric Methods in Nonlinear Control Theory, Riedel.","DOI":"10.1007\/978-94-009-4706-1_8"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1016\/0375-9601(95)00208-K","article-title":"An electronic analog of the Mackey-Glass system","volume":"201","author":"Namajunas","year":"1995","journal-title":"Phys. Lett. A"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1137\/0307017","article-title":"On the \u03c4 decomposition method of stability analysis for retarded dynamical systems","volume":"7","author":"Lee","year":"1969","journal-title":"SIAM J. Control"},{"key":"ref_18","first-page":"77","article-title":"On zeros of some transcendental equations","volume":"29","author":"Cooke","year":"1986","journal-title":"Funkcialaj Ekvacioj"},{"key":"ref_19","unstructured":"Hassard, B.D., Kazarinoff, N.D., and Wan, Y.H. (1981). Theory and Applications of Hopf Bifurcation, Cambridge University Press."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/12\/2336\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:40:05Z","timestamp":1760168405000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/12\/2336"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,12,6]]},"references-count":19,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2021,12]]}},"alternative-id":["sym13122336"],"URL":"https:\/\/doi.org\/10.3390\/sym13122336","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2021,12,6]]}}}