Sultanov, Oskar A. Bifurcations of autoresonant modes in oscillating systems with combined excitation. (English) Zbl 1461.34058 Stud. Appl. Math. 144, No. 2, 213-241 (2020). The nonlinear oscillating systems perturbated by the slow varying periodic force are called to be in the autoresonant mode, provided that they admit solutions adjusting to the pumping and keeping the state for the sufficiently long time by certain initial conditions. Here, the planar nonlinear system, which is changed from a second order differential equation by the time rescale and averaging over the fast variable in the generalized polar coordinates \((\rho,\Psi)\), is well investigated. For the critical case that the parametric and external excitations act equally, the author first provides the asymptotic expansions of particular solutions with the unbounded amplitude and bounded phase mismatch, then does the stability analysis of such solutions, and gives the asymptotic analysis via Lyapunov functions at last. Reviewer: Hao Wu (Nanjing) Cited in 4 Documents MSC: 34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations 34C23 Bifurcation theory for ordinary differential equations 37C60 Nonautonomous smooth dynamical systems 34D20 Stability of solutions to ordinary differential equations Keywords:autoresonance; asymptotics; bifurcation; nonlinear equations; stability PDFBibTeX XMLCite \textit{O. A. Sultanov}, Stud. Appl. Math. 144, No. 2, 213--241 (2020; Zbl 1461.34058) Full Text: DOI arXiv