Symmetry-breaking and pull-down motion for the Helmholtz–Duffing oscillator

Abstract

An accurate frequency of the Helmholtz–Duffing oscillator is obtained by a sophisticated modification of He’s frequency formulation. The pull-down instability existing in the symmetric breaking phenomenon is a newly discovered dynamic motion for oscillators with even nonlinearities. A criterion for predicting the asymmetrical amplitude motion and the pull-down instability is built by measuring the amplitude change. The good matching performance between the analytic results and numerical ones indicates that the criterion offers a starting point for future research into the pull-down phenomenon, and opens a new path for studying more complex nonlinear oscillators with even nonlinearities.

Application of homotopy perturbation method to solve a nonlinear mathematical model of depletion of forest resources

“…In 2008, He employed the HPM to solve boundary value problems [20]. In 2007, Javidi and Golbabai used a revised version of the HPM to solve non-linear Fredholm integral equations [21].Recently, HPM with small variations has been applied to study fractal duffing oscillator problems under arbitrary conditions [22], modified HPM for nonlinear oscillators Anjum and He [23], attachment oscillator arising in nanotechnology [24], conservative nonlinear oscillators [25], non-linear oscillator problems in a fractal space [26] and HPM including Aboodh transformation to solve fractional calculus Tao et al [27], vibrating magnetic inverted pendulum Moatimid et al [28], Symmetry-breaking and pull-down motion for the helmholtz-duffing oscillator Niu et al [29], nonlinear fractional Drinfeld-Sokolov-Wilson Equation Nadeem and Alsayaad [30], trajectory analysis of a zero-pitch-angle e-Sail Niccolai et al [31], natural convection between two concentric horizontal circular cylinders Abdulameer and Ali Al-Saif [32], nonlocal initialboundary value problems for parabolic and hyperbolic Al-Hayani and Younis [33], multi-step iterative methods for solving nonlinear equations Saeed et al [34], telegraph equation Moazzzam et al [35], triangular linear diophantine fuzzy system of equations Shams et al [36], condensing coagulation model and Lifshitz-Slyzov equation Arora et al [37], singular nonlinear system of boundary value problems Pathak et al [38], rikitake-yype system Ene and Pop [39], heat and mass transfer with 2D unsteady squeezing viscous flow problem Abdul-Ameer and Ali Al-Saif [40], variable Speed Wind Turbine Control Shalbafian and Ganjefar [41], radial thrust problem Niccolai et al [42], special third grade fluid flow with viscous dissipation effect over a stretching sheet Swain et al [43], and the frequency-amplitude relationship of a nonlinear oscillator with cubic and quintic nonlinearities He et al [44]. The HPM has become a widely-used technique to solve a large variety of problems in different fields and many research papers have been published each year using this method as evidenced by a simple search on Google Scholar.…”