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Averaging oscillations with small fractional damping and delayed terms. (English) Zbl 1142.34385
Summary: We demonstrate the method of averaging for conservative oscillators which may be strongly nonlinear, under small perturbations including delayed and/or fractional derivative terms. The unperturbed systems studied here include a harmonic oscillator, a strongly nonlinear oscillator with a cubic nonlinearity, as well as one with a nonanalytic nonlinearity. For the latter two cases, we use an approximate realization of the asymptotic method of averaging, based on harmonic balance. The averaged dynamics closely match the full numerical solutions in all cases, verifying the validity of the averaging procedure as well as the harmonic balance approximations therein. Moreover, interesting dynamics is uncovered in the strongly nonlinear case with small delayed terms, where arbitrarily many stable and unstable limit cycles can coexist, and infinitely many simultaneous saddle-node bifurcations can occur.

MSC:
34K23 Complex (chaotic) behavior of solutions to functional-differential equations
70K65 Averaging of perturbations for nonlinear problems in mechanics
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
34C29 Averaging method for ordinary differential equations
34K18 Bifurcation theory of functional-differential equations
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