Bifurcation analysis of Duffing oscillator with a fifth-order nonlinear factor

Abstract

ABSTRACTThis paper aims to investigate and analyze the dynamic behavior of a Duffing equation with a non-linear factor of the fifth-order, which is similar to the one developed by Zhao and Zhang but with a more general bifurcation analysis. While Zhao and Zhang use numerical methods to explore the behavior of the model, this paper employs both analytical and numerical approaches to analyze the bifurcation behavior. The paper determines normal form coefficients of codim-1 bifurcation points, including fold, pitchfork, and Hopf, as well as codim-2 bifurcation points such as Bogdanov-Takens and cusp. The sub-criticality or super-criticality of these bifurcation points is also calculated analytically. The analytical results are then confirmed through numerical simulation and continuations, which reveal more dynamic behaviors of the system. Furthermore, the paper investigates the external forcing effect on the model and extracts its dynamic behaviors.KEYWORDS: Duffing oscillatorbifurcationHopfBogdanov-takenscuspChenciner Disclosure statementNo potential conflict of interest was reported by the author(s).AcknowledgmentsCompliance with ethical standards, the authors declare that they have no conflict of interest concerning the publication of this manuscript.Additional informationNotes on contributorsJ. AlidoustiJ. Alidousti received his M.Sc. and Ph.D. degrees in applied mathematics from Amirkabir University of Technology and Shahrekord University in 2009 and 2016, respectively. Now, he is an assistant professor of mathematical department at Shahrekord University, Shahrekord, Iran. He has published about several scientific papers in international journals. His research interests include dynamical systems and bifurcation analysis.K. SalehiK. Salehi received his Ph.D. in computer science from University of Tabriz in 2019, M.Sc. and B.Sc. in computer science from Sharif University of Technology and Yazd University in 2010 and 2007, respectively. He is currently an assistant professor of computer science in Shahrekord University, Iran. His main research involves the challenges and solutions of the interaction of machine learning and formal methods as well as scientific computation.Z. EskandariZ. Eskandari received her M.Sc. and Ph.D. degrees in applied mathematics from Isfahan University of Technology, in 2006 and 2019. She worked for two years in the Shahrekord University as a postdoctoral researcher. During her research, she was a scholar visitor with applied mathematics Group, University of Twente, Enchede, the Netherlands for 6 months. Now, she is an assistant professor of mathematical department at the Fasa University, Iran. Her research interests include dynamical systems and bifurcation analysis.E. Rafiean BorujeniE. Rafiean Borujeni received her B.Sc. and M.Sc. degrees in applied mathematics from Shiraz University and Shahrekord University in 2009 and 2023, respectively. She is currently is pursuing Ph.D. degree in dynamical system at Shahrekord University under the supervision of Dr. Javad Alidousti. Her main research interest is bifurcation analysis.