Original Article Journal of Low Frequency Noise, Vibration and Active Control 2021, Vol. 0(0) 1–25 © The Author(s) 2021 Article reuse guidelines: sagepub.com/journals-permissions DOI: 10.1177/14613484211026407 journals.sagepub.com/home/lfn Hybrid rayleigh–van der pol–dufﬁng oscillator: Stability analysis and controller Chun-Hui He 1,2 , Dan Tian 3 , Galal M Moatimid 4 , Hala F Salman 5 and Marwa H Zekry 6 Abstract The current study examines the hybrid Rayleigh–Van der Pol–Dufﬁng oscillator (HRVD) with a cubic–quintic nonlinear term and an external excited force. The Poincar´ e–Lindstedt technique is adapted to attain an approximate bounded solution. A comparison between the approximate solution with the fourth-order Runge–Kutta method (RK4) shows a good matching. In case of the autonomous system, the linearized stability approach is employed to realize the stability performance near ﬁxed points. The phase portraits are plotted to visualize the behavior of HRVD around their ﬁxed points. The multiple scales method, along with a nonlinear integrated positive position feedback (NIPPF) controller, is employed to minimize the vibrations of the excited force. Optimal conditions of the operation system and frequency response curves (FRCs) are discussed at different values of the controller and the system parameters. The system is scrutinized numerically and graphically before and after providing the controller at the primary resonance case. The MATLAB program is employed to simulate the effectiveness of different parameters and the controller on the system. The calculations showed that NIPPF is the best controller. The validations of time history and FRC of the analysis as well as the numerical results are satisﬁed by making a comparison among them. Keywords hybrid Rayleigh–Van der Pol–Dufﬁng oscillator, Poincar´ e–Lindstedt technique, linearized stability, multiple scales method, nonlinear integral positive position feedback Introduction Nonlinear vibration provides an interesting potential example of the mathematical description of the nonlinear behavior of many phenomena in science, physics, and practical engineering; for example, the N/MEMS system vibrates nonlinearly. 1-7 The nonlinear wave equation of Kundu–Mukherjee–Naskar equation can be ﬁnally converted into a Dufﬁng-like equation. 8 Fangzhou oscillator is a generalized Dufﬁng equation (DE) with a singular term. 9-11 The nonlinear vibration systems in a porous medium can be converted to a fractal modiﬁcation of the DE. 12-14 The gecko-like vibration plays an important role in the accurate 3-D printing process. 15 The inherent pull-in instability of MEMS systems can be completely overcome by the fractal vibration theory. 16-18 The homotopy perturbation method (HPM) 19 and the Hamitonian approach 20 are two main analytical tools for nonlinear vibration systems. The combination of the Laplace transforms, Lagrange multiplier, fractional 1 Xi’an University of Architecture & Technology, Xi’an, PR China 2 National Engineering Laboratory for Modern Silk, College of Textile and Clothing Engineering, Soochow University, Suzhou, China 3 School of Science, Xi’an University of Architecture and Technology, Xi’an, China 4 Department of Mathematics, Faculty of Education, Ain Shams University, Cairo, Egypt 5 Department of Basic Sciences, Faculty of Computers and Artiﬁcial Intelligence, Cairo University, Giza, Egypt 6 Department of Mathematics and Computer Science, Faculty of Sciences, Beni-Suef University, Beni-Suef, Egypt Corresponding author: Dan Tian, School of Science, Xi’an University of Architecture and Technology, Xi’an, China. Email: tiandan@xauat.edu.cn Creative Commons CC BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License (https:// creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as speciﬁed on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/open-access-at-sage).