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The amplitude-frequency and phase-frequency response equations of the main resonance of the system are determined by the harmonic balance method. The amplitude-frequency and phase-frequency response equations of the steady solution to the system under the combined action of parametric excitation and forced excitation are obtained by using the average method, and the stability conditions of the steady solution are obtained based on Lyapunov's first method. The necessary conditions for heteroclinic orbit cross section intersection and chaos of the system are given by the Melnikov method. Based on the separation of fast and slow variables, the bifurcation phenomena of the system under different conditions are obtained. The amplitude-frequency characteristics of the total response of the system under different excitation frequencies are investigated by analytical and numerical methods, respectively, which shows that the two methods achieve consistency in the trend. The influence of fractional order and fractional derivative term coefficient on the amplitude-frequency response of the main resonance of the system is analyzed. The effects of nonlinear stiffness coefficient, parametric excitation term coefficient, and fractional order on the amplitude-frequency response of subharmonic resonance are discussed. Through analysis, it is found that the existence of parametric excitation will cause the subharmonic resonance of the Mathieu\u2013Duffing oscillator to jump. Finally, the subcritical and supercritical fork bifurcations of the system caused by different parameter changes are studied. Through analysis, it is known that the parametric excitation coefficient causes subcritical fork bifurcations and fractional order causes supercritical fork bifurcations.<\/jats:p>","DOI":"10.1063\/5.0138864","type":"journal-article","created":{"date-parts":[[2023,2,17]],"date-time":"2023-02-17T14:44:20Z","timestamp":1676645060000},"update-policy":"https:\/\/doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":52,"title":["Resonance and bifurcation of fractional quintic Mathieu\u2013Duffing system"],"prefix":"10.1063","volume":"33","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0413-6821","authenticated-orcid":false,"given":"Jiale","family":"Zhang","sequence":"first","affiliation":[{"name":"College of Mathematics and Statistics, Taiyuan Normal University 1 , Jinzhong 030619, Shanxi, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1323-5402","authenticated-orcid":false,"given":"Jiaquan","family":"Xie","sequence":"additional","affiliation":[{"name":"College of Mathematics and Statistics, Taiyuan Normal University 1 , Jinzhong 030619, Shanxi, China"},{"name":"Institute of Advanced Forming and Intelligent Equipment, Taiyuan University of Technology 2 , Taiyuan 030024, Shanxi, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wei","family":"Shi","sequence":"additional","affiliation":[{"name":"Institute of Advanced Forming and Intelligent Equipment, Taiyuan University of Technology 2 , Taiyuan 030024, Shanxi, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0269-9586","authenticated-orcid":false,"given":"Yiting","family":"Huo","sequence":"additional","affiliation":[{"name":"College of Mathematics and Statistics, Taiyuan Normal University 1 , Jinzhong 030619, Shanxi, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhongkai","family":"Ren","sequence":"additional","affiliation":[{"name":"Institute of Advanced Forming and Intelligent Equipment, Taiyuan University of Technology 2 , Taiyuan 030024, Shanxi, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Dongping","family":"He","sequence":"additional","affiliation":[{"name":"Institute of Advanced Forming and Intelligent Equipment, Taiyuan University of Technology 2 , Taiyuan 030024, Shanxi, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"317","published-online":{"date-parts":[[2023,2,17]]},"reference":[{"issue":"12","key":"2023081022414891000_c1","doi-asserted-by":"publisher","first-page":"2150181","DOI":"10.1142\/S0218127421501819","article-title":"Parametric frequency analysis of Mathieu\u2013Duffing equation","volume":"31","year":"2021","journal-title":"Int. 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