{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,30]],"date-time":"2026-03-30T13:37:10Z","timestamp":1774877830306,"version":"3.50.1"},"reference-count":82,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2021,9,14]],"date-time":"2021-09-14T00:00:00Z","timestamp":1631577600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100007129","name":"Natural Science Foundation of Shandong Province","doi-asserted-by":"publisher","award":["ZR2018MA025"],"award-info":[{"award-number":["ZR2018MA025"]}],"id":[{"id":"10.13039\/501100007129","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This paper is about the dynamical evolution of a family of chaotic jerk systems, which have different attractors for varying values of parameter a. By using Hopf bifurcation analysis, bifurcation diagrams, Lyapunov exponents, and cross sections, both self-excited and hidden attractors are explored. The self-exited chaotic attractors are found via a supercritical Hopf bifurcation and period-doubling cascades to chaos. The hidden chaotic attractors (related to a subcritical Hopf bifurcation, and with a unique stable equilibrium) are also found via period-doubling cascades to chaos. A circuit implementation is presented for the hidden chaotic attractor. The methods used in this paper will help understand and predict the chaotic dynamics of quadratic jerk systems.<\/jats:p>","DOI":"10.3390\/axioms10030227","type":"journal-article","created":{"date-parts":[[2021,9,14]],"date-time":"2021-09-14T23:34:11Z","timestamp":1631662451000},"page":"227","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":20,"title":["Chaotic Dynamics by Some Quadratic Jerk Systems"],"prefix":"10.3390","volume":"10","author":[{"given":"Mei","family":"Liu","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences, Liaocheng University, Liaocheng 252059, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3978-2225","authenticated-orcid":false,"given":"Bo","family":"Sang","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Liaocheng University, Liaocheng 252059, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4453-8588","authenticated-orcid":false,"given":"Ning","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Electrical and Information Engineering, Tianjin University, Tianjin 300072, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Irfan","family":"Ahmad","sequence":"additional","affiliation":[{"name":"Sirindhorn International Institute of Technology, Thammasat University, Pathum Thani 12000, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,9,14]]},"reference":[{"key":"ref_1","unstructured":"Thompson, J.M.T., and Stewart, H.B. 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