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Purpose The planar dynamical motion of a double-rigid-body pendulum with two degrees-of-freedom close to resonance, in which its pivot point moves in a Lissajous curve has been addressed. In light of the generalized coordinates, equations of Lagrange have been used to construct the controlling equations of motion. Methods New innovative analytic approximate solutions of the governing equations have been accomplished up to higher order of approximation utilizing the multiple scales method. Resonance cases have been classified and the solvability conditions of the steady-state solutions have been obtained. The fourth-order Runge–Kutta method has been utilized to gain the numerical solutions for the equations of the governing system. Results The history timeline of the acquired solutions as well as the resonance curves have been graphically displayed to demonstrate the positive impact of the various parameters on the motion. The comparison between the analytical and numerical solutions revealed great consistency, which confirms and reinforces the accuracy of the achieved analytic solutions. Conclusions The non-linear stability analysis of these solutions have been examined and discussed, in which the stability and instability areas have been portrayed. All resonance cases and a combination of them have been examined. The archived results are considered as generalization of some previous works that are related to one rigid body and for fixed pendulum’s pivot point.

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In the current paper, the problem of sextic anharmonic oscillators is investigated. There are three integrable cases of this problem. Emphasis is placed on two integrable cases, and a full description of each one is provided. The separated functions of the first and second integrability cases are transformed from a higher degree to the third and fourth degrees. Respectively, the periodic solution is obtained using Jacobi’s elliptic functions. The topology of phase space and Liouville tori’s bifurcations are discussed. The phase portrait is studied to determine singular points and classify their types in addition to the graphic representation for each of them. Finally, the numerical illustrations are introduced using the Poincaré surface section to emphasize the problem’s integrability.