## Homotopy analysis method for approximations of Duffing oscillator with dual frequency excitations.(English)Zbl 1448.34077

Summary: In this paper, the classical Duffing oscillator under dual frequency excitations is studied by the homotopy analysis method(HAM). Analytical study of the low-order approximations is firstly conducted and the saddle node(SN) bifurcation boundary for the initial guess solution is obtained. The maximum value bifurcation plot of the high order approximations with the bifurcation parameters $$f_1$$ and $$\lambda_1$$ are obtained and compared with the numerical solutions based on the Runge-Kutta method. The results show that the initial guess solution can qualitatively reflect the trend of the numerical solution, and the high order approximations agree well with the numerical solutions. The maximum value bifurcation plots of high order approximations show periodic and quasi-periodic solutions, which agree well with the numerical ones.

### MSC:

 34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations 34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations 34C60 Qualitative investigation and simulation of ordinary differential equation models 34A45 Theoretical approximation of solutions to ordinary differential equations

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